Then C is a margin maximizing loss function in the sense that any convergence point of the normalized solutions β(λ) β(λ) p to the regularized problems (1) as λ → 0 is an l p marginmaximizing separating hyper-plane.

if this margin-maximizing hyper-plane is unique, then the solutions converge to it: Proof We prove the result separately for T = ∞ and T < ∞.


However, this assumption should perform the calculation in practice (as described in the last parts of this and the next sections).

we proceed with this assumption in the following parts.


We stress, however, that higher-order correlations do occur in the vine tree and depend on both the vine-tree selection and the copula families.

selecting the right vine-tree and copula families can -to a limited extent -account for higher-order correlations.


Our first key observation (already briefly mentioned in [26]) is that the critical step of rewriting the loss (1) in terms of the Laplacian (2) does not depend on any special properties of the rotation group other than the fact (a) rotation matrices are unitary (in fact, orthogonal) (b) if we follow one rotation by another, their matrices simply multiply.

in general, for any group G, a complex valued function ρ : G → C dρ×dρ which satisfies ρ(g 2 g 1 ) = ρ(g 2 )ρ(g 1 ) is called a representation of G. The representation is unitary if ρ(g −1 ) = (ρ(g)) −1 = ρ † , where M † denotes the Hermitian conjugate (conjugate transpose) of M . we have the following proposition.


The ability of neurons to adapt their responses to greatly varying sensory signal statistics is central to efficient neural coding [1,2,3,4,5,6,7].

accurate models for the underlying mechanisms can provide insight into the nature of neural coding itself.


After observing a transition (s, a) → s , the posterior distribution is also Dirichlet and can simply be updated as ρ s,a,s ← ρ s,a,s + 1.

to model the distribution of X s,π and X s,a,π we only need to maintain a set of hyperparameters µ s,0 , λ s , α s , β s and ρ s,a for each state s and action a encountered in the MCTS search tree and update them by using Bayes' rules.


It results in 340 concepts in the final concept graph.

this concept graph makes the symbolic graph reasoning layer can be identical across all three datasets and its weights can be easily shared to each other dataset.


However, they aim to execute a single goal task, while the goal of our problem is to maximize the cumulative reward in RL context.

the agent in our problem not only needs to consider dependencies among subtasks but also needs to infer the cost from the observation and deal with stochasticity of the environment.


Also, note that the joint distribution p(h, s, y) depends on the transformation defined by the encoder.

we can equivalently rewrite objective (1) as To analyze the equilibrium condition of the new objective (2), we first deduce the optimal discriminator D and the optimal predictor M for a given encoder E and then prove the global optimality of the minimax game.


For example, there are numerous online resources for images and video sequences of different types.

there has been an increased interest in exploiting the readily available unlabeled data to improve the performance of ConvNets.


In order to do so, it updates V such that where r t is the reward at time t and γ the discount factor.

a simple way to achieve that is to transform the problem into an optimization problem where the goal is to minimize: It is also useful at this point, introduce the TD effective reinforcement signal, equivalent to the dopaminergic signal [5]: ( ) ( ) A learning rule for the weights v of V can then be devised by finding the gradient of E with respect to the weights v. Here, V is the weighted sum of the activity of I and H. the gradient is given by [ ] Adding a learning rate and negating the gradient for minimization gives the update: Developing a learning rule for the actor units and their weights W using a cost function is a bit more complex.


Specifically, the Predictive Coding Network [12] always exploits the whole ground truth sequence before the current time step to predict the next frame.


This is because at the step of re-estimating a block A k from a set of patches {y i } i , some of these patches may be spatially overlapping in the full image.

the subspaces in A k are driven to explain the residual at some pixels multiple times.


Instead, consider absorbing this term into the diagonal component to preserve the bound.

formally, we look for a diagonal matrix F such that Σ ′′ i = Σ ′ i + F which also maintains x ⊤ Σ ′′ i x ≥ x ⊤ Σ i x for all x ∈ R d . we want to satisfy: where, for ease of notation, we take F(i) = F(i, i).


First, we assume that the distribution P (X) of the input feature x ∈ X is a product measure of P k (X) on each X k . That is, P X (dX This is typically assumed in the analysis of linear tensor estimation methods [13,6,3,21,1,36,27,37].

the Hereafter, with a slight abuse of notations, we denote by f L2 = f L2(P k ) for a function f : X k → R. The inner product in the space L 2 is denoted by f, g L2 := f (X)g(X)dP X (X).


The prediction for each galaxy does not change by permuting the members of the galaxy cluster.

we can treat each galaxy cluster as a "set" and use DeepSets to estimate the individual galaxy red-shifts.


Unless f is a polynomial itself, there exists an x 0 (usually uncountably many) for which f (x 0 ) = p n (x 0 ), so if A has an eigenvalue at x 0 we have Σ f (A) = Σ pn (A).

one cannot hope that the estimator (3), let alone the gradient estimator (5), to be unbiased for all matrices A.


if we let L(Y, Xw) denote the unregularized loss over the training set where X is the n × d matrix of instances and Y is the n-vector of class labels, we can express equation ( 1) in matrix form as Defining X to be XU (αΛ + β I) −1/2 where U and Λ are from the eigen-decomposition above, it is not hard to show that equation ( 3) is equivalent to the alternative ridge regularized learning problem That is, the two minimizers, w and w, yield the same predictions: Ŷ = Xw = X w. we can view feature network regularization as: 1) finding an embedding for the features using LLE in which all of the eigenvectors are used and scaled by the inverse square-roots of their eigenvalues (plus a smoothing term, βI, that makes the inverse well-defined); 2) projecting the data instances onto these coordinates; and 3) learning a ridge-penalized model for the new representation.


Unlike previous methods that use a restricted fixed kernel function, mapping functions in DHA are flexible across subjects because they employ multi-layer neural networks, which can implement any nonlinear function [12].

dHA does not suffer from disadvantages of the previous kernel approach.


Suppose there exists a central controller that collects the rewards of, and assigns the action to each individual agent, the problem reduces to the classical MDP with action space A and global reward function R c (s, a) = N −1 N i=1 R i (s, a).

without such a central controller, it is essential for the agents to collaborate with each other so as to solve the multi-agent problem based solely on local information.


These methods suffer from a critical limitation: an actually redundant channel would be mistakenly kept to minimize the reconstruction error of feature maps.

these methods may result in apparent drop in accuracy on more compact and deeper models such as ResNet [13] for large-scale datasets.


This is exactly the loss of the parity function PAR S on the agnostic parity learning problem.

if the algorithm U has regret O(poly(n), T 1−δ ), then the meta-algorithm for the online agnostic parity learning problem also has regret O(poly(n), T 1−δ ).


This is a very significant failing because, as mentioned previously, intra-dipole correlations are always present in each active dipole.

localization and orientation bias can occur because of convergence to a local minimum.


The response of each neuron r k (k = 1, • • • , N) is quantified as the number of spikes fired in a salient post-stimulus time window of a length τ .

the overall neuronal population response is represented as a spike count vector r = (r 1 , • • • , r N ).


On the other hand, KMTL-UCB can be viewed as a special case of Kernel-UCB on the augmented context space X .

the regret analysis of Kernel-UCB applies to KMTL-UCB, but it does not reveal the potential gains of multi-task learning.


Here, the random variables Y i (t) are conditionally independent when X(t) is given, and the conditional probabilities W [y i (t)|x(t)] are identical for all i and t. In this paper, we impose the binary assumptions to the problem, i.e., the data sequence {X(t)} and its noisy observations {Y i (t)} are all assumed to be binary sequences.

the stochastic matrix can be parameterized as where p ∈ [0, 1] represents the observation noise.


Stated differently, it is plausible that 'interior' points in a subset are going to be 'interior' points in the whole set.

a non-support vector of a subset has a good chance of being a non-support vector of the whole set and we can eliminate it from further analysis.


We set the hyper-parameter σ for the Gaussian kernel heuristically according to [39,8].

uAB ans SAB learning problems scale quadratically in the number of samples n. to reduce memory requirements we optimize by sampling mini-batches from the kernel matrix.


We make the natural assumptions that functional units are organized based on their responses to the categories of stimuli and the categories of stimuli can be characterized by the responses they induce in the units.

categories and units are interrelated and informative about each other.


p c (y|x) and p t (y|x) are equal to an adversarial loss w.r.t.

p α (y|x): let L MD = βL c DS (p c (y|x), p t (y|x)) + γL t DS (p t (y|x), p c (y|x)) and L JS be the Jensen-Shannon divergence, the value function U (c, t, d) of the minimax game can be rewritten as Here, L JS reaches the minimum if and only if p α (y|x) = p u (y|x) and L c DS (or L t DS ) reaches the minimum if and only if p c (y|x) = p t (y|x).


The most fundamental functional loss after a hemispheric stroke is the loss of fine motor control of the hand.

whether ipsilateral cortex encodes finger movements is critical to the potential feasibility of BCI approaches in the future.


Therefore, √ m ηk m,q , which is a plug-in estimator of the expected value of , converges in probability to 0 for any fixed q as m → ∞, and the theorem follows for this special case.

we will assume from now on that instab(A k , D) > 0.


The Hit-and-run, ball walk, and RWM algorithms are all thought to have a step size of roughly Θ(1) on sufficiently regular target distributions [21].

since most of the probability of a standard spherical Gaussian lies in a ball of radius √ d, one would expect all three of these algorithms to take roughly ( √ d) 2 = d steps to explore a sufficiently regular target distribution.


Since our algorithm is just Q-learning, it is online and does not store additional data besides the table of Q values (and a few integers per entry of this table).

it also enjoys a significant advantage over model-based algorithms in terms of time and space complexities.


Here, we compute the average similarity in S rather than the similarity of the best element in S only.

from the graph perspective (G) we sum over the weights of edges with at least one vertex in S. f sum cov.


The output of sequencing is a small number of noisy reads of each reference.

the reads that correspond to the same reference form a cluster.


The cosine SphereConv operator is a nonlinear function of θ (w,x) , with its g being the form of which can be reformulated as w T x w 2 x 2 .

it can be viewed as a doubly normalized convolutional operator, which bridges the SphereConv operator and convolutional operator.


The fact that an action leads to close next states can model the noise in a real system for instance.

problems such as the highway simulator [12], the mountain car or the inverted pendulum (possibly discretized) are particular cases of this type of Garnets.


However, when k > l, T l may not admit a symmetric orthogonal decomposition 4 .

it is not clear what TPM will return in this case and there are no guarantees it will even converge.


The proof, however, requires much more involved truncation arguments, which is beyond the scope of this paper.

we assume the uniform boundedness for convenience.


In both sets of experiments we use the following two performance measures for a learnt model A: Normalized error: To give an idea of how a good estimate A would perform under these two measures, we point out that a constant prediction xt+1 = x t leads to a normalized error of 1, and a random-walk prediction xt+1 = x t + ǫ t+1 , ǫ t+1 being a process, results in a nearly-zero cosine score.

when the true model is more than a simple random walk, a good estimate A should achieve a normalized error much smaller than 1 and a cosine score way above 0.


we can directly optimize over β j without updating β \j .


Given that DPP-MAX is too greedy, it may be desirable to allow for uncertainty in the observations.

we define DPP-SAMPLE which selects the batches via sampling subsets from DPPs, and show that the expected regret is smaller than that of DPP-MAX.


It is well-known that the geometric series of matrices, often called the Neumann series, I + λA × + (λA × ) 2 + • • • converges only if the maximum eigenvalue of A × , denoted by µ ×,max , is strictly smaller than 1/λ.

the geometric random walk kernel K GR is well-defined only if λ < 1/µ ×,max .


Since we evaluate the trust score on its precision at a given recall percentile level, we are interested in the relative ranking of the scores rather than their absolute values.

we do not compare against calibration techniques.


We set aside those hypotheses for now since the goal of the present work is to explore the limits of the totally unconstrained notion of identity-preserving transformation.

for the purposes of this paper, we consider background-variation as just another transformation.


. . , I n i , because each individual's gradient update is coupled with all the other agents' actions, which is a fundamental phenomenon in multi-agent learning.

special structures must be imposed on the games/reward functions.


This corresponds to the cost of forming ) −1 and α I .

if d max is not big, learning on large datasets is feasible via sparse GP models.


A match ( X, Ŷ ) in (X , Y) is a simple match if X ∪ Ŷ is non-empty and there exist no matches With the concept of the simple matches, we will show the Decomposition Theorem: every match can be decomposed as the union of a set of simple matches.

simple matches fully characterize the structure of matches.


(4) It is well-known that a Dirichlet sample can be generated by sampling from C independent Gammadistributed random variables with shape parameters α i and rate λ = 1; realizations of the class probabilities can be generated as follows: , where Therefore, the noisy Dirichlet likelihood assumed for each observation translates to C independent Gamma likelihoods with shape parameters either α i = 1 + α , if y i = 1, or α i = α otherwise.

in order to construct a Gaussian likelihood in the log-space, we approximate each Gamma-distributed x i with xi ∼ Lognormal(ỹ i , σ2 i ) through moment matching (mean and variance): for the parameters of the normally distributed logarithm we have: Note that this is the first approximation to the likelihood that we have employed so far.


That means, in the latter case, the reward distributions can be chosen adversarially.

it should be clear that the distribution-free lower bound is always higher than the distribution-dependent lower bound.


The second insight is a connection between the (1, 1)-norm and a notion of subspace sparsity introduced by [18].

any X ≽ 0 can be factorized (non-uniquely) as Consequently, any X ∈ F d that has at most s non-zero rows necessarily has ∥X∥ 1,1 ≤ s 2 d. ∥X∥ 1,1 is a convex relaxation of what [18] call row sparsity for subspaces.


Interestingly, such energy correlations are also prominent in other kinds of data, including brain signals [33] and presumably even financial time series which have strong heteroscedasticity.

developing a general model for energy-correlations of linear latent variables is an important problem in the theory of machine learning, and such models are likely to have a wide domain of applicability.


Each of these can be covered by ¾ balls of radius ¬ , and so on.

´Þ ¬µ can be covered by ¾ ÐÓ ¾ ¬ « ´ ¬ «µ balls of radius « ¾.


To give an intuition concerning the speed and the efficiency attained by sparsification, we should mention that the dimension of the feature space for the kernel used in Model 1 is 6 (Proposition 9.2 in [14]).

we deal with a kernel matrix of size 6 with sparsification versus a kernel matrix of size M = 5j , j = 1, . . . , 20 without sparsification.


This bound acts as a surrogate objective that is optimized using stochastic gradient descent (SGD) to find a locally optimal configuration of the split functions.

one complication introduced by this particular formulation is that the number of latent decision variables grows exponentially with the tree depth d. each gradient update will have a complexity of O(2 d p) for p-dimensional inputs.


2 The complexity of EMGD is on the same order as SAG and SDCA when κ ≤ n 1/2 , but higher in other cases.

in terms of computational cost, EMGD may not be the best one, but it has advantages in other aspects.


Similarly for SmoothGrad we have E sg (x) = w (the gradient is independent of the input, so averaging gradients over noisy inputs yields the same model weight).

integrated Gradients reduces to "gradient input" for this case: we see that the application of the basic gradient method to a linear model will pass our sanity check.


Hence, compression improvement becomes D KL [p(θ|ξ t−1 ) p(θ|ξ t )].

optimizing for compression improvement comes down to optimizing the KL divergence from the posterior given the past history ξ t−1 to the posterior given the total history ξ t . As such, we arrive at an alternative way to encode curiosity than information gain, namely D KL [p(θ|ξ t ) p(θ|ξ t , a t , s t+1 )], its reversed KL divergence.


This is because the threshold values arrive in non-increasing order.

we can reduce the optimistic setting to the Threshold Bandit problem with uncensored feedback, and use the DKWUCB proposed in Section 4 to solve it.


This happens whenever either L 1 < 1 (t) or L 2 < 2 (t), where L 1 and L 2 are the respective lengths of the first run and the first m runs of this arm.

unlike the previous version of the algorithm which consists in an exploration phase followed by an exploitation phase, the anytime version of the algorithm continuously switches between exploration and exploitation.


This issue becomes severe when one performs many parallel jobs (threadblocks) and leads to wrong inference results and operation failure, as reported by Masada et al.

their method is not easily scalable due to memory access conflicts.


We split instances in a node if knowing where the agent came from helps predict future discounted rewards.

the tree must keep what McCallum calls fringes, i.e., subtrees below the "official" leaves.


For other problems, such as the US Presidents, only 16% of individuals were assigned to the guessing strategy, indicating that knowledge about this domain was more widely distributed in our group of individuals.

the extension of the Thurstonia n model can eliminate individuals who are purely guessing the answers.


In this section we assume that the parameters of the target mixture distribution are known.

the learning algorithm is given A good starting point would be to study the performance of a linear combining rule.


Given a set of images, the location stack is defined as the set of values, with domain X , at a specific location across a set of images.

the empirical distribution at a given location of a DF is determined by the corresponding location stack.


Now, P true (t) and y are functions of w and a j for j ∈ I true .

they are independent of a j for any j ∈ I true .


Though being sound as a test set, it is inadequate for training, since the majority of identities in LFW have only one face image.

we rely on a larger outside dataset for training, as did by all recent highperformance face verification algorithms [4,2,21,20,13].


However, the scale of each code word is lost after each image patch is normalized to its invariant region.

it is possible for two features to match because they happen to look similar, even though in the physical world they actually have two different sizes.


. . , R s,M )} S s=1 be an auxiliary training set where (X s , A s ) ∼ µ and R s,m = R m (X s , A s ).

in each state-action pair, we generate T next states for each task, that is Y t s,m ∼ P m (•|X s , A s ) with t = 1, . . . , T . for any function Q we define the estimated transfer error as At each iteration, the algorithm Best Average Transfer (BAT) (Fig.


These connections allow for later stages to incorporate information that has been recovered by earlier stages into their estimate of the original image data.

these connections (between subsequent stages) allow for better optimization of the model.


This greedy algorithm starts with the empty set A 0 , and at each iteration i, it chooses an element e ∈ V that maximizes Let us denote this (centralized) greedy solution by A g . When f is integral (i.e., f : 2 V → N) it is known that the size of the solution returned by the greedy algorithm , where H(z) is the z-th harmonic number and is bounded by .

we have |A g | ≤ (1 + ln(max e f ({e})))|A c |, and obtaining a better solution is hard under natural complexity theoretic assumptions [7].


Quantities about x s such as expected values under the pseudomarginal can be computed using the samples x (i) s . However, for any given variable node x s , the incoming messages to x s given in (4) are defined in terms of the importance weights and sampled values of the neighboring variables.

we can compute an estimate of the messages and beliefs defined in (4)-( 5) at arbitrary values of x s , simply by evaluating (4) at that point.


We conclude that zero variance is achieved iff ¢ equals a constant.

the unbiasedness constraints behave orthogonally to the zero variance constraints: unbiasedness imposes a constraint on columns of whereas zero variance imposes a constraint on rows of .


However, these methods iteratively remove the observations or tunes the observations weights based on statistics obtained from the whole data set.

when a new data point is revealed, these methods need to re-explore all of the data and become quite expensive in computation and in storage.


Variable-length quantum coding requires the definition of a code length operator L, which is a positive matrix such that for any density matrix X, tr(XL) gives the expected dimension ("length") of the message assigned to X.

the quantum version of Kraft's inequality states that, ignoring rounding issues, for every variable-length quantum code with codelength operator L, there exists a density matrix W such that L = − log W . the matrix entropic loss can be interpreted as the (expected) code length of the observed outcome.


At each iteration we obtain D t by querying the label of a random variable not being labeled in D t−1 .

at each iteration, we learn using a weakly supervised structured prediction task that solves  and we query the variable that has the highest entropy, i.e., the highest uncertainty.


Next, we study the distributed implementation of PDA in (5a)-(5b) to solve (8).

let Φ(x) given the initial iterates x 0 , θ 0 , λ 0 and parameters γ > 0, τ i , κ i > 0 for i ∈ N , choosing D x and D y as defined in Definition 1, and setting ν x = ν y = 1, PDA iterations in (5a)-(5b) take the following form: Since K i is a cone, prox κiσ K i (.)


That is, at any particular time, the user does not want to listen to most of the songs in the world, which leads to a mean preference close enough to zero to approximate as zero.

the covariance kernel ¢¡ ¤£ ¦¥ £ simply turns into a correlation over a distribution of functions ¦ : ¡ ¤£ ¦¥ £ ¨ § © ¦ ¡ £ ¦ ¡ ¤£ .


We call such a particle system massless.

for NAG, it can also be verified that only h = Θ( √ η) results in a valid physical system and it is massive we provide a unified framework of choosing the correct time scaling factor h. In classic mechanics, the harmonic oscillator is one of the first mechanic systems, which admit an exact solution.


To update each column/row takes O(n 2 ) and there are n such columns/rows in total.

our algorithm has a computational complexity of O(Kn 3 ), where K is the number of sweeps through columns.


For simplicity we shall assume that each time the PDA synchronises all the outstanding updates are transferred.

after synchronisation the replica on the mobile device is consistent with the master copy on the server.


In order to understand the putative neural mechanism performing ICA derived from the formalism developed above, we need to find an online learning rule describing changes of the synapses as a function of pre-and postsynaptic activity.

taking the inverse of ( 5), we have for weights that solve the ICA problem we expect because of (3) that which defines the weight matrix W * at the solution.


In other words, node i survives from the influence of all parents (see Figure 2(c) for illustration).

the likelihood is a product of survival functions, i.e., Combining the above two scenarios together, we can obtain the overall likelihood of a cascade t c by multiplying together all individual likelihoods, i.e., the likelihood of all cascades is a product of the these individual cascade likelihoods, i.e.


This is specifically of great interest for the neurodegenerative disease diagnosis, where the features from different regions of the brain are extracted, but not all the regions are associated with a certain disease.

the most discriminative regions in the brain that utmost affect the disease would be identified, leading to a more reliable diagnosis model.


( 11) it becomes apparent that the continuous nearest neighbors effectively take k weighted averages over the database items.

prior work such as non-local networks [42], differentiable relaxations of the KNN classifier [41], or soft attention-based architectures [14] can be realized as a special case of our architecture with k = 1.


However, the randomized SVD method still needs to compute a full SVD for nuclear norm regularization and in our experiments, we found that it is slower than the full SVD method implemented in [22].

we only report the results of the orignial convex relaxations in [9,32,22,24].


( 2013 In general, these heuristic methods introduce conditional independencies into the inference network that are not present in the original distribution.

they cannot represent the true posterior even in the limit of infinite neural network capacities.


For our experiments, we use a modified Gated Recurrent Unit [5] in which the state update operation is a modular layer.

k modules are selected and applied at each time step.


As illustrated in Figure 1 this configuration does not attain the accuracy of its normalized counterpart.

seemingly, batch normalization yields faster training, higher accuracy and enable higher learning rates.


Again, we have m j=1 β j equal to the value of Gr p . Combining these two, we conclude that the expected value of Gr p is equal to As before, we show that f j (Opt j ) is upper-bounded by β j + i∈Opt j α i .

we can conclude that the competitive ratio of Gr p is 1 1+1/p = p 1+p .


This algorithm, also known as gradient descent or Landweber iteration, finds a weight β by performing a gradient descent iteratively.

we can interpret early stopping as shrinkage and the reciprocal of iteration number as shrinkage parameter, i.e., λ ≈ 1/t.


By using the re-parameterization w i = ab i , the learning machine eq.

( 4) results in This learner is a regular statistical model, hence both by comparing λ(a 0 , b 0 ) and −µ(a 0 , b 0 ) with N , let us clarify the effect of singularities.


Hence, maximizing the Tsallis entropy of a sparse MDN encourages exploration by providing bonus rewards to wide-spread mixture means and penalizing collapsed mixture means, while the causal entropy [1] of an MDN is less effective in terms of preventing the collapse of mixture means since there is no analytical form and its approximation is used in practice instead.

maximizing the Tsallis entropy of a sparse MDN has a clear benefit over the causal entropy in terms of exploration and mixture utilization.


. . , R 2 (i− 1, n) and flipping its j-th bit, which happens with probability at least Thus, if we have found L desired solutions in the i-th phase, the probability of finding a new desired solution in the next iteration is at least enPmax , where n−i+1 is the total number of desired solutions to find in the i-th phase.

then, the expected number of iterations E[T ] is O(kn log nP max ) until the k-th phase finishes, which implies that an optimal solution corresponding to max 1≤j≤n R 2 (k, j) has been found.


The coefficients on the edge variables µ i (x i , x i+1 ) are their reduced costs, For duals that obey the constraints of (6), it is clear that the maximal reduced cost is zero, when x i is set to the argmax used when constructing α i+1 (x i+1 ).

to a obtain a primal-optimal solution, we start at the end of the chain and follow the argmax indices to the beginning, which is the same backward sweep of the Viterbi algorithm.


This involves replacing the opacity value for the parent with that of the child, and then removing that child's entry from the associated grid cell's observation vector.

this process is identical to the first case, except that each removal of an entry from the global map is preceded by a single update to the same grid square.


Because C 1 and C 2 isolate the gates of transistors M 1 and M 2 respectively, these transistors are floating-gate devices.

the stored voltages V w+ and V w− are nonvolatile.


3), where the first term on the right-hand side can be efficiently estimated with the reparameterization trick (see Appendix C for the details) where v ⇠ Uniform(0, 1) and z ⌘ g(v, b, ✓) is the differentiable reparameterization for z|b (Appendix C).

using this to form the control variate and correcting with the reparameterization trick gradient, we arrive at where u, v ⇠ Uniform(0, 1), z ⌘ g(u, ✓), z ⌘ g(v, H(z), ✓), and ⌘ is a scaling on the control variate.


Finally, spatial audio may not always be experienced easily, e.g., when the video does not contain clean sound sources.

even videos with GT ambisonics were misclassified in both studies at a significant rate.


Running time and results depend on the size of S * .

we vary λ from 50 (S * ≈ V) to 9600 (S * = ∅) on a speech recognition data set [11].


Second, the storage of a individual context units also had an excitatory self-connection for maintenance).

the first stripe must learn to encode the first input, etc.


However, the task is non-trivial mainly because the two-layer sampling does not correspond to a typical empirical process: the documents for different queries are not identically distributed while the documents for the same query are not independent.

the empirical process techniques, widely used in previous work on generalization analysis, are not sufficient.


Its proof is in the supplementary material.

we only need to find x k such that (3.3) holds.


In itemset mining, a very large proportion of all feature subsets will have small supports.

restricting the application of the pruning criterion to potentially prunable patterns does not cause a loss of performance in practice.


The above-mentioned convex properties of objective function f imply that we only need to look for the value of δ * satisfying df If Otherwise, we accept the change described by Equation 1 with δ = δ * .

for objective function f we simply have where For the quadratic loss, we find where for the quadratic loss case, the optimal value δ * satisfies 2Cmδ For the exponential loss, we find where for the exponential loss case, the optimal value δ * satisfies After changing w 1 , we need to recompute 1 D w (i) ∀i ∈ {1, . . . , m}.


Note that while we can explicitly control the amount of sparsity in SMaLL, ProtoNN, and Bonsai, the methods that use `1 or elastic net regularization do not have this flexibility.

in order to get the different baselines on the same footing, we devised the following empirical methodology.


At each iteration k, this system is fully characterized by a state z k equal to the training set S k . The system is actioned by a control u k equal to the design x k+1 selected to be evaluated.

for a given state and control, the value of the objective function is unknown and modeled as a random variable W k , characterized by: where µ k (x k+1 ) and σ 2 k (x k+1 ) are the posterior mean and variance of the GP at x k+1 , conditioned on S k . We define a disturbance w k to be equal to a realization f k+1 of W k . w k = f k+1 represents a possible (simulated) value of the objective function at x k+1 .


This means that the system is locally asymptotically stable, or more precisely, locally exponentially stable (see Definition A.1 in Appendix A).

an important contribution of this paper is a proof of this seemingly simple fact: under some conditions, the Jacobian of the dynamical system given by the GAN update is a Hurwitz matrix at an equilibrium (or, if there are zero-eigenvalues, if they correspond to a subspace of equilibria, the system is still asymptotically stable).


Each window of LFP data is generated as a draw from this mixture of Gaussian processes.

all animals share an underlying brain state space, of which, all brain regions share the underlying components of the mixture model.


Since our algorithm significantly reduces the model size, it allows us to use larger dimensions of embedding vectors while still keeping our model size very small.

we also tried 600 and 1000 in LightRNN, and the results are showed in Table 2.


In the RANDOM condition, next trial is started from a random marker position so that the subject has to consider only immediate reward.

the optimal behavior is to collect a larger reward at each state (Fig.


These conditions are exactly satisfied in the case of reward uncertainty: the forgotten information (i) does not influence future rewards, (ii) does not influence future transition probabilities, (iii) is never known by the opponent, (iv) is not remembered later by the player.

we can construct the extensive-form game with the player restricted to Markovian policies and still solve it with CFR-BR.


Psychophysical investigations of human behaviour on object recognition tasks, measuring accuracies depending on image colour (greyscale vs. colour), image contrast and the amount of additive visual noise have been powerful means of exploring the human visual system, revealing much about the internal computations and mechanisms at work [42][43][44][45][46][47][48].

similar experiments might yield equally interesting insights into the functioning of DNNs, especially as a comparison to high-quality measurements of human behaviour.


Once we have computed Â, similarly to [22,7], we define the function which we optimize as ĝ(z) = f ( ÂT z) = g(A ÂT z).

we effectively work with an approximation f to f given by f (x) = ĝ( Âx) = g(A ÂT Âx).


However, a subtlety of pLSIand the crucial difference between it and LDA-is that d is a dummy index into the list of documents in the training set.

d is a multinomial random variable with as many possible values as there are training documents, and the model learns the topic mixtures p(zld) only for those documents on which it is trained.


[1] only provided tentative and inconclusive results on the utility of APM.

motivated by simple intuitions and previous evaluations of topic models, we propose a novel evaluation metric based on human evocation scores between word pairs (i.e.


The four tasks in the lower row allow for only one possible response (not counting no-go as a response), whereas the four tasks in the upper row demand that a choice be made between two possible responses.

the two rows differ in terms of the demands placed on response selection.


More generally, however, it is possible that there is some other approach for optimizing the function from the training samples, without learning a model.

at a high level, the question is whether a reasonable number of training samples suffices to minimize a submodular function.


Second, it learns to apply those deltas to the few provided examples of novel categories, unseen during training, in order to efficiently synthesize new samples from these categories.

in the few-shot scenario, we are able to synthesize enough samples of each new category to train a classifier in the standard supervised fashion.


To this end, let us fix a k and an i and notice that by the subgaussianity assumption on the rewards, the empirical mean holds for any α.

in particular, using this above formula for α the sum appearing in Lemma 1 can be bounded as where the last step follows from the fact 3 that ∞ k=0 e c √ k ≤ 2 c 2 holds for all c > 0.


f i (x) does not change if i is not involved in the exchange, i.e.

f i (x) is symmetric in {x \i } Second, exchanging the feature values of documents 1 and i exchanges their scores, i.e., the scoring function for the ith document can be expressed in terms of that of the first document.


While the Lancaster interaction has an interpretation in terms of joint central moments, Streitberg's correction corresponds to joint cumulants [22,Section 4].

a central moment expression like E X1...Xn [ k (1) Xn − µ Xn ] does not capture the correct notion of the interaction measure.


All of these metrics are based on the common assumption that a person should be able to understand a topic by understanding the abstract semantic connections between the word pairs.

evocation is a reasonable notion for evaluating topic modeling because it directly evaluates the level of semantic connection between word pairs.


One expects the probabilities of rare or unseen events in one context to be related to their probabilities in statistically similar contexts.

it should be possible to estimate more accurate probabilities by exploiting these regularities.


Whenever the largest w 2 j is not much larger than the average w 2 j , then ), where is the average document length.

the bound ( 9) is most useful for long documents.


In the presence of an ✏-strategic buyer, the rewards are no longer stochastic.

we need to analyze the regret of a seller in the presence of lies.


Then we derive the auxiliary "switch" vector for where ||v i || = 1.

the response of the or-node U i is defined as, Collaborative Edge: For any pair of leaf-nodes (L j , L j ′ ) respectively associated with two different or-nodes, we define the collaborative edge between them according to their contextual cooccurrence.


Comparing Equation 3with the entropy of the distribution rooted at a neighboring node 3, we see that they share a common term is positive, node 2 is a better root than 3, 2,3 is negative, we have the reverse situation.

when comparing neighboring nodes as root candidates, the difference in entropy of the resulting distribution is simply the difference in entropy their local distributions assign to their separator.


Also, we have N ≥ n = Θ( √ k/ε 2 ), so when k is larger than some absolute constant, the O(1/ √ N ) term in the final line above is Θ(ε/ 4 √ k) ≤ ε.

by Claims 2.6 and 2.7, either M fails to be accurate, so that ∃j ∈ [k] |a j − q j (P)| > ε, or we find a query q * such that q * (X) − q * (P) > ε.


Our work reveals an interesting connection between regularization and privacy: the larger the regularization constant, the less sensitive the logistic regression function is to any one individual example, and as a result, the less noise one needs to add to make it privacy-preserving.

regularization not only prevents overfitting, but also helps with privacy, by making the classifier less sensitive.


In such cases, derivatives of the objective function with respect to the parameters are unavailable.

we have seen a resurgence of interest in Derivative Free Optimization (DFO) [1,2,3,4,5,6,7,8].


These algorithms lead to networks in which groups of multiple neurons are coactivated to represent an object.

it is of great interest to find ways of using lateral inhibition to mediate winner-take-all competition between groups of neurons, as this could be useful for learning sparsely distributed representations.


Comparison between ( 14) and (12) shows that distinct from the HDP-LDA that combines a topic's global and local popularities in an additive manner as (n −ji jk + αr k ), the BNBP topic model combines them in a multiplicative manner as This term can also be rewritten as the product of n −ji •k and n −ji jk +rj c+n −ji •k +r• , the latter of which represents how much the jth document contributes to the overall popularity of the kth topic.

the BNBP and HDP-LDA have distinct mechanisms to automatically shrink the number of topics.


The problem is that when a decision pair is finally encountered for which that cue would lead to a correct decision, it is unlikely to be checked because other, more discriminating although less valid, cues are looked up before and already bring about a decision.

because one-reason decision making is intertwined with the learning mechanism and so influences which cues can be learned about, what mainly makes a cue come early in the order is producing a high number of correct decisions and not so much a high ratio of correct discriminations to total discriminations regardless of base rates.


For example, if we sample edges i 1 ∼ j 1 and i 2 ∼ j 2 , then we replace these edges with i 1 ∼ j 2 and i 2 ∼ j 1 .

when s = 0, the graph is the original discretization of a low-dimensional space; and as s increases to infinity, the graph becomes more and more like a uniformly chosen 4-regular graph (which is an expander [15] and which bears similarities with an Erdős-Rényi random graph [16]).


Even if a DBN representation shows little dependence among variables from one stage to another, the influence of variables tends to "bleed" through a DBN over time, and many variables become relevant to predicting value.

even using structured methods, we must often relax the optimality constraint and generate only approximate value functions, from which near-optimal policies will hopefully arise.


This is due to inaccuracies in the initial patch discovery stage: for a frequent configuration to be discovered, both of its patches must be found accurately.

we also mine additional positives from the set of remaining positive images P that did not produce any of the discovered configurations.


All N detections are labelled by λ as background and their conditional density is U N (A).

we have f y|σ (y|0) = 1 A N 1 N N where the summation is over the λ, δ compatible with σ = 0.


Similar to the classical gradient descent scheme of Nesterov [17], Algorithm 1 involves an extrapolation step (6).

the solution of the auxiliary problem (5) at iteration k + 1 is driven towards the extrapolated variable y k . As shown in [9], this step is in fact sufficient to reduce the number of iterations of Algorithm 1 to solve (1) when ε k = 0-that is, for running the exact accelerated proximal point algorithm.


It is important to note that TNs {(h κ i , m λ i )} i∈Nn are dependent on the weights w because the weighted Euclidean distance ( 1) is used in their definitions.

we need to properly update TNs in the optimization process.


Integrating property 3 against a function V § 42 q .

v "a b since S 4W is a positive operator; thus g § 4 32 is positive definite.


A primary insight is that, since Algorithm 3 regularizes with the L 2 norm and penalizes with hinge-loss, omitting the positive semidefinite requirement for M and vectorizing M makes the algorithm equivalent to a oneclass, linear support vector machine with O(n 3 ) input vectors.

the stochastic optimization is an instance of the PEGAGOS algorithm [19], albeit a cleverly constructed one.


In another words, the marginal distribution P (X) is a nuisance parameter; we maximize the joint log-likelihood, which is decomposed as log P (y, X|θ) = log P (y|X, θ) + log P (X), only with respect to θ [14].

no assumption about the generative model is necessary.


Although the gradient-based sampling is proposed to approximate LS solution in this paper, we believe that this sampling method can apply into other optimization problems for large-scale data analysis, since gradient is considered to be the steepest way to attain the (local) optima.

applying this idea to other optimization problems is an interesting study.


Our algorithm will begin by verifying that p 2 = Θ(n −1/2 ) and p 3 = Θ(n −2/3 ) using Lemma 2.2.

in the second case, we can assume that p 2 = Θ(n −1/2 ) and p 3 = Θ(n −2/3 ).


Estimating the parameters of an LDS-sometimes referred to as system identification-is a difficult problem, particularly if the goal is to obtain the maximum likelihood estimate of parameters.

spectral methods from the subspace identification literature, based on moment-matching rather than maximum likelihood, have become popular.


Note, however, that in this data set the true generating distribution is not known and is unlikely to be purely a CI model.

the average log KS measure over insample data has much lower values (see Figure 3 right, and plots in Figure 2).


Similar to the observational case, I essential graph of a DAG D is the graph union of all DAGs in the same I equivalence class; it is denoted by E I (D).

we have the following sequence: after a set of interventions I has been performed, the essential graph E I (D) is a graph with some oriented edges that captures all the causal relations we have discovered so far, using I.


Moreover, SSPtable takes advantage of bounded staleness to maximize ML algorithm performance, by reading the parameters ✓ from caches on the worker machines whenever possible, and only reading ✓ from the parameter server when the SSP model requires it.

workers (1) spend less time waiting for each other, and (2) spend less time communicating with the parameter server.


For instance, for the 0 − 1 loss, any η providing the same decisions than η is a minimum of L(η, η).

those values of η with η and U T V −1 Mη in the same decision region are not influenced by a bad choice of the ambiguity set.


But, of course, since the divergence and ELBO vary by a constant, maximizing the ELBO is equivalent to minimizing the divergence.

variational inference can be thought of as simultaneously solving two problems: • "probabilistic inference" or finding a distribution q(z) that is close to p(z|x) in KL-divergence.


The probability Pr(s | b t , A S ) can be treated as a normalization factor and is independent of hidden state 2 .

the transition probability function, is solely a function of the belief b t and is a stationary distribution over the belief space.


In our tests, we found that due to a design error, one of the input dimensions coupled to the other inputs.

we held that input fixed throughout the tests, effectively reducing the input to 7 dimensions.


The former reduces to the paramagnetic solution (f RS = f PARA ), which is unphysical for R < 1, while the latter yields π 1RSB (x) = δ(x), π1RSB (x) = δ(x) with m = β g (R)/β and β g obtained from the root of the equation enforcing the non-negative replica symmetric entropy with a free energy Since the target bit of the estimation in this model is J i1,••• ,iK and its estimator the product S i1 • • • S iK , a performance measure for the information corruption could be the per-bond energy e. According to the one-step RSB framework, the lowest free energy can be calculated from the probability distributions π 1RSB (x) and π1RSB (x) satisfying the saddle point equation ( 15) at the characteristic inverse temperature β g , when the replica symmetric entropy s RS disappears.


(The loss function often compares the predicted value with some label in a known way, in which case learning the value of the label for the round means that the whole loss function becomes known; hence the choice of the name.)

the learner naturally will choose to see the loss function after he provides his prediction; this provides feedback that the learner can use, to improve the predictor he produces.


On the other hand, if X j1 and X j2 belong to two different groups, then they do not interact with each other.

in terms of accommodating interactions, the group additive model can be considered lying in the middle between the original additive model and the functional ANOVA or higher order interaction models.


In between the starting and destination points, there are number of obstacles that the agent may pass through but should avoid for safety; each time the agent hits an obstacle it incurs a constraint cost of 1.

in the CMDP setting, the agent's goal is to reach the destination in the shortest possible number of moves, while hitting the obstacles at most d 0 times or less.


We will prove the following bound on average regret.

lim T →∞ R T /T ≤ 0, hence GIGA-WoLF has no-regret.


Repeating this procedure results in a chain of skills leading from any state in which the agent may start to the task's goal region as depicted in Figure 1 Note that although the options lie on a chain, the decision to execute each option is part of the agent's overall learning problem.

they may not necessarily be executed sequentially; in particular, if an agent has learned a better policy for some parts of the chain, it may learn to skip some options.


This further confirms that mixture-rank structure indeed exists in user-item rating matrices in recommender systems.

it is desirable to adopt mixture-rank matrix approximations rather than fixed-rank matrix approximations for recommendation tasks.


While fast computation techniques have been developed for graph kernels based on walks [12] and on limited-size subgraphs [11], it is unclear how to compute subtree kernels efficiently.

they have been applied to relatively small graphs representing chemical compounds [9] or handwritten digits [1], with approximately twenty nodes on average.


First, it receives and interprets the current goal given by the environment, and second, it encapsulates and memorises the features and structure of a single city region.

rather than using a map, or an external memory, we propose an architecture with two recurrent pathways that can effectively address a challenging navigation task in a single city as well as transfer to new cities or regions by training only a new locale-specific pathway.


Since the left hand side of the inner constraint is decreasing in ζ, one can efficiently compute the polar Ω • by a simple root finding search in ζ.

regularizers in the form of (20) can still be accommodated in an efficient GCG method in the form of Algorithm 1.


A symmetric observation system is equivalent to G t being an undirected graph or, more precisely, to a directed graph having, for every pair of nodes i� j ∈ V , either no arcs or length-two directed cycles.

from the point of view of the symmetry of the observation system, we also distinguish between the directed case (G t is a general directed graph) and the symmetric case (G t is an undirected graph for all t).


On further investigation, we found that the DeepSets model retrieves more relevant tags, which are not present in list of ground truth tags due to a limited 5 tag annotation.

this takes a toll on precision while gaining on recall, yet yielding improvement on F1.


This proxy task does not work for dense image prediction where high resolution imagery is critical for conveying multi-scale context information.

we propose to design the proxy dataset by (1) employing a smaller network backbone and (2) caching the feature maps produced by the network backbone on the training set and directly building a single DPC on top of it.


However, the previous variables are closely related to the new ones: when β = 0 the constraint associated with b disappears, and we recover q(x|µ); that is, p(x|µ, 0) = q(x|µ).


This is because its distance to F θ0,i,j is on the order of 1/ √ m, while its distance to any other boundary is on the order of m −1/4 .

if x does switch clusters, then it is highly likely to switch between cluster i and cluster j.


For r > 1, the global optima are nonisolated due to the rotational invariance property.

the analysis is more involved and challenging.


For the GGM this procedure reduces to PCA and the noise threshold depends on the eigenvalues of the covariance matrix, as expected.

we show that for simple graph structures, such as hierarchical or lattice graphs, as well as the random Erdös-Rényi graph, the noise threshold can possibly grow in the network size p. leveraging the structure of network interactions can enable extraction of high-dimensional patterns embedded in heavy noise.


Third, the SVM classification rate with respect to epochs was improved by increasing the data-space.

the input vector for the classifier was extended by combining data from the same epoch but from different electrodes.


( 8) and corresponds to the projection onto with a(x) to the solution of the standard vector-valued kernel ridge regression without constraints (we denoted Y ∈ R n×T the matrix with rows y i ).

nonlinear VVR consists in: 1) computing the unconstrained kernel ridge regression estimator b(x), 2) normalizing it by a(x) and 3) projecting it onto C. The example above shows that for specific loss functions the estimation of f (x) can be significantly simplified.


Our algorithm relies on image segmentation to mimic the layering of illumination mask.

using semantic segmentation algorithms instead of unsupervised one may improve the capability of learning exposures.


However, there is clearly a limit on the size of σ due to higher order terms in the Taylor expansion; very large σ will result in samples which do not represent the local gradient.

in the case of noisy measurements, there is some optimal sampling distance that is as large as possible without resulting in poor sampling of the local gradient.


Assume that the true generating distributions p(x) and p(y|x) are unknown.

we aim to estimate a conditional distribution p(y|x) from some set of hypotheses that achieves high conditional likelihood given the data-set D = {(x 1 , y 1 ), .


For a vector y, let y m:n = (y m , y m+1 , ..., y n ) denote the sequence of entries from m to n. Initialize λ (k) , r (k) = (0, 0) at the beginning of first period.

then, at each period t ≥ 1, the pair λ (k) , r (k) can be updated recursively as follows: where overall average payoff function rt (x) can be expressed as a sum of average payoff functions of individual goods.


In particular, according to the definition of information gain, those instances with larger gradients 1 (i.e., under-trained instances) will contribute more to the information gain.

when down sampling the data instances, in order to retain the accuracy of information gain estimation, we should better keep those instances with large gradients (e.g., larger than a pre-defined threshold, or among the top percentiles), and only randomly drop those instances with small gradients.


We conjecture that these results are due to the large change that the signal of one network stream undergoes after injecting a fusion signal from the other stream.

we developed a more subtle alternative solution based on additive interactions, as follows.


O(1/ǫ 2 ), in the number of calls to LMO; see [14] for the optimal complexity bounds for general non-smooth optimization with LMO.

when applying the algorithm to the variational reformulation of the problem of interest (3), we are able to get an ǫ-optimal solution within at most O(1/ǫ 2 ) LMO calls.


Under the PMRF or TPMRF models, both the log partition function and likelihood were computationally intractable to compute exactly.

2 approximating the log partition function of an LPMRF opens up the door for likelihood-based hyperparameter estimation and model evaluation that was not possible with PMRF.


This strategy obviously satisfies the condition in (8).

we have the following result: Corollary 2.


The modulation frequency spectrum is restricted to the range between 50-400 Hz and has a resolution of 15 channels.

the fundamental frequency of typical voiced speech is represented in the modulation spectrum.


If F achieves k error patterns on (y, c), then H achieves k classification patterns on y.

if H achieves k classification patterns on y, then F achieves k error patterns on (y, c) for any choice of c. σ(H, i) = σ (F, i).


. . , N r , respectively, at the end of phase i.

m e (i) and N e (i) are, respectively, the average payoff accumulated by expert e and the total number of phases this expert was followed on or before phase i.


If no such sets are found, they are declared as neighbors.

the method involves searching for separators for each node pair u, v ∈ V , by considering all sets S ⊂ V \ {u, v} satisfying |S| ≤ η.


Equivalently, (Vaf(a,x)}x -.J y Ef,:l Vaf (a,yi) = 0 or Because this equation holds for all a, we obtain px(a) = .J y Ef':l 8(yi -a), which is the ML solution.

the sample-based method can be viewed as an approximation to ML which gets more exact as the number of samples goes to infinity.


Conditional Entropy (CE) is an information-theoretic measure that quantifies the uncertainty of random variable x when conditioned on z (or the other way around), under joint distribution π(x, z): The uncertainty of x given z is linked with Intuitively, by controlling the uncertainty of q φ (z|x) and p θ (z|x), we can restrict the solutions of the ALI objective to joint distributions whose mappings result in better reconstruction ability.

we propose to use the CE in (4), denoted as L π CE (θ, φ) = H π (x|z) or H π (z|x) (depending on the task; see below), as a regularization term in our framework, termed ALI with Conditional Entropy (ALICE), and defined as the following minimax objective: L π CE (θ, φ) is dependent on the underlying distributions for the random variables, parametrized by (θ, φ), as made clearer below.


As mentioned previously, the multivariate t model is an approximation to the source model of natural sensory signals.

we would like to compare our analysis in the previous section with the actual dependency reduction performance of the DN transform on real natural sensory signal data.


Now, since 7 © acts transitively on W , for all ¥ $ ¥ 21 ¢ W there is 5 ¢ F7 © such that ¥ 45 i@ ¥ 61 .

we thus have that 7 £ we can write 7 £ I U¥ " w@ 7 £ I " since the normalizing constant for ¥ ¢ W does not in fact depend on ¥ .


The training set, consisting of labeled sequences, is partitioned into its positive and negative instances, and two HMM's, λ + and λ − , are trained using those sets, respectively.

λ + is trained to identify positively labeled sequences, and λ − -negatively labeled sequences.


Unfortunately, in our model, the coupling of the conditional and marginal distribution in the likelihood makes it difficult to derive analytical formulas for the posterior distribution.

note that if (X, Y ) ∼ N (µ, Σ) we introduce an approximation distribution for the datasets which decouples the marginal and conditional distributions as follows: where W is a p x × p y projection matrix and Λ is the precision matrix.


Thus, stimulus as well as noise correlations cause deviations of the conditional p(x B S ) from a single Gaussian.

the full conditional distribution p(x B S ) in general contain more information about the features which trigger this sub-population to evoke the specified response pattern, than the conditional mean, i.e.


As pointed out in the documentation of the dataset, Japanese food preferences are strongly correlated with geographical and regional information.

modeling user similarities may provide useful information during the elicitation process.


The key idea is to apply the pseudo-inverse expectation formula for standard volume sampling (see e.g., [11]) first on the subsampled estimator w ⇤ ⇡ , and then again on the full estimator w ⇤ .

in the first step, this formula states: where [k] d def = {S ✓ {1, . . . , k} : |S| = d} and ⇡ S denotes a subsequence of ⇡ indexed by the elements of set S. Note that since S is of size d, we can decompose the determinant: Whenever this determinant is non-zero, w ⇤ ⇡ S is the exact solution of a system of d linear equations: the rescaling of each equation by 1 p q⇡ i cancels out, and we can simply write w ⇤ ⇡ S = (X ⇡ S ) + y ⇡ S . (Note that this is not the case for sets larger than d whenever the optimum solution incurs positive loss.)


Note that the side information matrix W = (w i,j ) is provided where the w i,j s are independent random variables.

now assume the scenario when we use an algorithm ALG to assign x to one of the k clusters, V u , u = 1, . . . , k. given x, ALG takes as input the random variables w i,x s where i ∈ t V t , makes some queries involving x and outputs a cluster index, which is an assignment for x.


However, for DDM, the prior for Θ consists of degenerate Dirichlet distributions conditionally dependent on T , whereas the approximating distribution for the posterior is a fully factorized distribution with proper Dirichlets.

we have to make a further approximation to match the parameters of the two distributions.


6, which is insufficient to be used to estimate a full noise covariance with a Kronecker structure as well as a kernel matrix.

the cross-trial statistics should be considered for structured noise estimation.


But a was taken at time t, so it must have been in Γ 2 t which means it maximized the index.

for all optimal actions a * ∈ O(i; P, A), we have, on the event A t (i, a; ), U t (i, a * ) ≤ U t (i, a) < L * (i; P, A) − 2 .


The prior distribution over a n is a product of Bernoulli distributions with parameters ρ n = {ρ(u m , u n , θ)} N m=1 .

the conditional distribution is proportional to the ratio of the prior and posterior partition functions, we perform a joint update of a n and w n by collapsing out the weights to directly sample the binary entries of a n . We iterate over each entry, a m→n , and sample it from its conditional distribution given {a m →n } m =m .


It is important to note that this feedback operates only on the basis of the predicted solution ŷ, avoiding to make updates for every prediction the function has made.

the learning strategy is taking advantage of the recognition process, and concentrates on (i) assigning high scores for the correct phrases and (ii) making the incorrect competing phrases to score lower than the correct ones.


An imminent question now is how to make use of a combination of observational and interventional data (of the order of 100,000 cells / samples on 20,000 genes / variables) in the causal discovery process.

the development of efficient and consistent algorithms using both observational and interventional data that are implementable within genomics is now a crucial goal.


Although a full optimisation of the variational parameters gives the tightest bound, it turns out that the computational complexity of this optimisation is quite large for sigmoid belief networks.

we use the mean field solution for J-Li (equation 11) instead.


However, the Newton method does not treat saddle points appropriately; as argued below, saddle-points instead become attractive under the Newton dynamics.

given the proliferation of saddle points, not local minima, in high dimensional problems, the entire theoretical justification for quasi-Newton methods, i.e.


As there are γ-fraction of entries of the second type, the chance of getting more informative entries increases in γ.

the required sample complexity decreases in γ.


By this, we mean that n ∀i ∈ {1, . . . , n}, i.e., the total weight assigned to each boolean complement pair of classifiers is fixed to 1/n.

let For any quasi uniform Q, the output B Q (x) of the majority vote on any example x is given by the set of majority votes B Q over quasi uniform posteriors is isomorphic to the set of linear separators with real weights.


However, choosing an appropriate interpolation scheme is a challenge on its own and, possibly, requires a large number of grid points.

various extensions of convolutional layers even beyond the Euclidean space exist, e.g., for graphs [31,32] and 3d shapes [33].


While sampling methods have been used successfully for GBI [21,15], it is not easy to scale these for the robust BOCPD setting.

most work on BOCPD has focused on conjugate distributions [1,43,11] and approximations [44,34].


The idea is to group together all nodes of the same shell that are directly linked in the original graph and to represent them close one to another.

a shell is divided in many angular sectors, each containing a cluster of vertices.


Suppose a two-armed Bernoulli bandit has mean rewards 1 /2 + ε and 1 /2 respectively and the second arm is chosen at most F < ∞ times with probability at least 1 − δ, then one can easily show that in an alternative bandit with mean rewards 1 /2 + ε and 1 /2 + 2ε there is a non-zero probability that the second arm is played finitely often and in this bandit the expected regret will be linear.

sub-linear expected regret is only possible if each arm is pulled infinitely often almost surely.


When learning internal models, such uncertainty can result from an assumption that parameters of the world are constantly changing [16].

in the inference model for the trace conditioning experiment, the ISI duration is modeled with a probability distribution with some nonzero variance rather than an impulse function.


From the relation between B i ab and G i ab , it also follows that R (x,y) (B i ab ) ≤ 2R (x,y) (G i ab ) for any (x, y).


For a state q i , and given observation label l, we calculate the average number of visits (at q i ) whose corresponding label is l, as E [S t = q i | l t = l, O, λ] = 1≤t≤T,lt=l γ t (i).

l qi is chosen to be an l that maximized this quantity.


To discover a subspace representation shared by multi-view data, the unsupervised canonical correlation analysis (CCA) [17] and its kernelized version [1] ignore the widely available supervised information, such as image categories.

they could discover a subspace with weak predictive ability.


[2] For a closed convex set C and a point z 0 ∈ C, the normal cone to C at z 0 is Theorem 7 implies that θ * (λ) is known with λ ≥ λ max .

we can estimate θ * (λ) in terms of a known θ * (λ 0 ).


This leads to subsampling by a factor of 2 at each convolutional layer, resulting in reduced dataset size (Table 2).

the comparison between layers before and after convolutions is not entirely fair.


Take w x for example, we remove the irrelevant terms and get the following objective: where It is easy to realize that the objective function (2) can be expressed as a difference of two convex functions in different cases.

we can use CCCP to solve the nonconvex optimization problem iteratively with each iteration minimizing a convex upper bound of the original objective function.


recall that we have assumed that m > n, so the probability converges to 1 with increasing m or n. While Ã is not positive definite, the triangle inequality implies a bound on the positive definite projection Ã+ (11): similarly to the earlier result for B(t), the estimator (12) formed by substituting the positive semidefinite ρ( Ã+ ) into the GLasso objective enjoys the following error bound (Zhou et al., 2011).


In [8] it is shown that dim(F, P ) = O(kd).

by Lemma 2, when |S| ≥ O 1 2 (kd + log 1 δ ) , the weighted cost of S ∪ B approximates the k-median cost of P for any set of centers, then (S ∪ B, w) becomes an -coreset for P . The total communication cost is bounded by O(mn), since even in the most general case that every node only knows its neighbors, we can broadcast the local costs with O(mn) communication (see Algorithm 3).


But in doing so, it implicitly assumes that differences among the K networks arise from edge perturbations.

this approach does not take full advantage of the structure of the learning problem, which is that differences between the K networks are driven by nodes that differ across networks, rather than differences in individual edges.


For details on our asymptotic notation and proofs for this section, see Appendix D. Lemma 5.1.

the number of vertices and edges for both the multi-and binary graphs satisfy we can examine the asymptotic behavior of the random numbers of edges and vertices by examining the asymptotic behavior of their expectations, which are provided by Lemma 5.2.


More interestingly, despite the small number of images used for the car, the model correctly matches the orientation of the car to the synthesized images of the cup and truck.

the model can synthesize reasonable correspondences (given a latent point) even if the number of training examples used to learn kernel parameters is small.


When a good prior is available, convergence is very fast (almost immediate) in both domains.

the availability of model priors can help speed up convergence in many cases for HRL.


This does not allow the examples in the compression set to have a different label than their original label in S in .

we require a slight generalization of previous compression analysis, which allows setting arbitrary labels for examples that are assigned to the compression set.


Since computing the sum of synaptic weights is non-physiological, we draw on Theorem 1 and use the neuron's firing rate when responding to uncorrelated inputs as a proxy for kwk 1 .

in the offline phase, synapses receive inputs generated as in the online phase but without repeated patterns.


If the model made an error on the first trial, the mean deviation was larger-not smaller-on the second trial: 0.40 versus 0.32.

mLGA-Distrib fails to explain this phenomenon.


One exception is the important maximum a posteriori (MAP) inference, i.e., finding the set of items with the highest probability, which is NP-hard [25].

approximate inference methods with low computational complexity are preferred.


Indeed, if we marginalize the latent variable z, we get that the probability of y given x is a linear combination of softmax functions of linear functions of the input variables x.

the only difference with a two-layer neural network with softmax functions for the last layer is the fact that our last layer considers linear parameterization in the mean parameters rather than in the natural parameters of the multinomial variable.


We see that the MM method overfits when λ w is small.

mM requires a careful discrete search over the values of λ w , which is slow, since the quality of each such value must be estimated by cross-validation.


A similar result holds when the strict inequality is reversed.

we can compute both the Bethe optimum and partial solutions to the exact MAP inference problem simply by solving a max-flow problem over (G 2 ; φ G 2 , ψ G 2 ).


At each iteration of Algorithm 2, we draw total of t • m /16,δ/(4t 2 ) samples from t mixtures.

over t time steps, we draw a total of samples for learning f (1) , .


the posterior distribution, i.e., lim (2) In practice, the support of ⇡ almost surely contains {A, B} that are unstabilizable, which implies that (2) is infinite.

we shall consider averages over confidence regions w.r.t.


Although (reasonable) online algorithms will eventually learn a good or an optimal policy, there is no guarantee for their performance along the way (the performance of their intermediate policies), especially at the very beginning, when they perform a large amount of exploration.

in order to guarantee safety in online algorithms, it is important to control their exploration and make it more conservative.


First note that in the statements of Theorem 2.5.A and Theorem 2.5.B the only parameters that are assumed to grow are p and whichever other parameter is implied to grow because of ( 2) and (4).

importantly, n does not necessarily grow to infinity, if for example N, 1 σ grow appropriately with p. That means that Theorem 2.5.A and Theorem 2.5.B imply non-trivial guarantees for arbitrary sample size n. The proposition below shows that if σ is at most exponential in −(1 + ) (p+2n) 2 2n + (2 + p n ) log (RQ) for some > 0, then for appropriately chosen truncation level N the LBR algorithm recovers exactly the vector β * with n samples.


Inputs: • A concept class C and a probability measure ν defined over C. The algorithm: QBC assumes a prior ν over the class of linear classifiers.

the sample S induces a posterior over the class of linear classifiers which is the restriction of ν to V . the probability that QBC will query for the label of an instance x is exactly where ν|V is the restriction of ν to V . From (2) we see that there is no need to explicitly select two random hypotheses.


( 4), which can be performed efficiently due to the fact that sides of the same length along one dimension (horizontal or vertical) in the coordinates of face i will also be constant along the corresponding line when projected to the coordinate system of face j.

computing the side length ratios of two faces is not quadratic in the number of pixels but only in the number of horizontal or vertical lines.


However, because they do not use a convolutional architecture, features are shared only locally.

every hidden unit has to be learned 'from scratch' at each spatial location and the number of parameters in the fully-connected layers grows quadratically with population size.


This dimensionality is too large in practice for our learning.

we apply a linear dimensionality reduction x I = Pψ(f I ), where projection matrix P is obtained through PCA, so x I ∈ R n for n ≪ F (2r + 1).


For such graphs, the input to F would not contain all z i,j pairs but rather only features corresponding to valid edges of the graph, and we are only interested in invariances that preserve the graph structure, namely, the automorphisms of the graph.

the desired invariance is that σ(F(z)) = F(σ(z)), where σ is not an arbitrary permutation but an automorphism.


Therefore, for any x ∈ X , for which g(x) = 1, all the hypotheses in G agree, and in particular h * and ĥ agree.

applying Theorem 5.4 and the union bound we therefore know that with probability of at least 1−δ, Hanneke introduced, in his original work [5], an alternative definition of the disagreement coefficient θ, for which the supermum in ( 4) is taken with respect to any fixed ϵ > 0.


without segmenting the blurred areas there is no single blurring model p k that will describe the observed histogram.


This is mainly because gMedLDA alternately updates posterior distribution and Lagrangian multipliers.

it does Gibbs sampling for many times, which compensates for the influence that each Markov chain has not reached its stationarity.


When a digit is presented, two neurons fire among the three: one that fires with probability one, and one of the remaining two fires with uniform probability.

we expect three neurons to have strong probability of firing when the stimulus contains their preferred digit; however, one of the neurons does not spike due to competition with another neuron.


For a naive baseline model, we regard the relationship between each pair of nodes as the instance of independent Bernoulli distribution with Beta(1, 1) prior.

for a given pair of nodes, the link probability at each time equals to the expected probability from the posterior distribution given network data.


Both variational-EM and the wake-sleep algorithm work well when Q closely approximates P , but may fail to converge to a maximum of the likelihood if there is a significant mismatch.

the efficiency of these methods depends on choosing a good approximating distribution Q(Z|Y, φ) -one that closely matches P (Z|Y, θ).


The motivation of using the confusion matrix is that classifiers are not perfect and misclassification could happen.

the use of the confusion matrix incorporates the visual ambiguity into the word learning framework by providing an unbiased estimation of the true leaf node label for an image.


We describe how to use the shadow Dirichlet to model a random pmf that has bounded variation such that |θ k − θ l | ≤ k,l for any k, ∈ {1, 2, . . . , d} and k,l > 0.

to construct specified bounds on the variation, we first analyze the variation for a given M . For any d × d left stochastic matrix , so the difference between any two entries is to obtain a distribution over pmfs with bounded |θ k − θ | ≤ k,l for any k, components, it is sufficient to choose components of the matrix M such that |M kj − M lj | ≤ k,l for all j = 1, . . . , d because θ in (3) sums to 1.


A given node (h, i) is always picked at most once, but since its descendants may be picked afterwards, subsequent paths in the tree can go through it.

n) be the empirical average of the rewards received for the time-points when the path followed by the algorithm went through (h, i): where 0 < ρ < 1 and ν 1 > 0 are parameters of the algorithm (to be chosen later by the decision maker, see Assumption 1).


By considering a uniform goal distribution it is easy to verify that at any node in the tree there is an equal chance that the true goal is in the left or right sub-tree during any episode.

any policy will have a 0.5 chance of committing a misprediction at each step of an episode.


Unfortunately, we underestimated the runtime and memory requirements of MMMF on EachMovie.

we cannot report results on this data set using MMMF.


If the probability of a change occurring is α, the best guess will be 0.5 if that change occurs, or the mean p if the change does not occur.

the prediction made is actually y t+1 =1/2 α+(1-α)p. Note that we do not allow perfect knowledge of the probability of a changepoint, α.


This lower bound follows from the minimax lower bound proofs for regression in the next section.

an exponential improvement is possible using semi-supervised learning provided m grows exponentially in n. Let Y denote a continuous and bounded random variable.


In such settings, it is well known that the MLE, instead of approximating the true parameter, approximates a Kullback-Leibler projection of the data-generating distribution onto the fitted model class.

the MLE exhibits a desirable form of robustness to model mis-specification.


In the large sample limit, independent bootstrap samples yield similar CI-test results.

all the paths in T represent the same single CPDAG-the true underlying graph.


However, in practice, the discriminator class is usually implemented by a feedforward neural network, and it is known that a fixed neural network has limited capacity (e.g., it cannot implement the set of all the bounded continuous function).

one could ask what will happen if we restrict to a sub-class H ?


We show a new result relating the CVaR objective in (3) to the expected discounted-cost in presence of worst-case perturbations of the MDP parameters, where the perturbations are budgeted according to the "number of things that can go wrong".

by minimizing CVaR, the decision maker also guarantees robustness of the policy.


The resulting graph is an undirected graph called moral graph where each node is connected to its Markov blanket.

it contains spurious links to its spouses, to members of its district, to members of its children's district, and to parents of nodes in those districts, which we all call SD links (for Spouse/District).


We show that, under standard assumptions of Lambertian reflection and static illumination, the task can be posed as a convex minimization problem.

the solution, computed using efficient algorithms, is guaranteed to be globally optimal, for any number of independently moving objects, and any number of occlusion layers.


Unfortunately, it is argued by some researchers that Hessian-vector products are not general enough and may not be as simple to implement as evaluating gradients [9].

can we turn Hessian-free methods into first-order ones, without hurting their performance?


Here we do not want to predict if a finger is moving but which one.

at any time point t we could have one of K possible labels, such as "Index Finger" (y t = 1), "Ring Finger" (y t = 2), etc.


Specifically, under the restriction that θ ss < 0, we arrive at the following upper bound: by relating to the log-partition function of a node-conditional Gaussian distribution.

nodewise regressions according to the QPGM via the above variational upper bound on the partition function would behave similarly to that of a Gaussian graphical model.


It is important to realize that we generally expect a good sequential policy to do better, or no worse, than a batch policy with respect to performance per number of experiments.

the Sequential curve can be typically viewed as an upper performance bound and provides an indication of how much loss is incurred when moving to a batch setting in terms of efficiency per experiment.


In fact, the improvement in error compared to Loopy BP was not correlated with sub-model agreement for any K (maximum ρ = 0.0185, p < 0.05).

sub-model agreement does not explain the improvement over Loopy BP, indicating that submodel disagreement is not related to the difficulty in inference problems that causes Loopy BP to underperform relative to the ensembles (e.g., due to convergence failure.)


In the limit, where stimulus dynamics is extremely fast, and LGMD processes are kept far from equilibrium at each instant of the approach, α gets very small.

t max gets largely independent of l/|v|: The activity peak would cling to t max although we varied l/|v|.


A pair (ā, B) is optimal for (13) if and only if the optimality condition can be written as Substituting ā = a ⋆ , B = B ⋆ , and noting that a ⋆T w ⋆ /w ⋆T B ⋆ w ⋆ = 1, the optimality condition reduces to which is precisely (12).

we have shown that (a ⋆ , B ⋆ ) is optimal for (13), which in turn establishes (11).


These rules are usually constructed using data from clinical trials that are generally costly to run.

it is important to distribute the trial resources wisely so that the devised rule yields a good performance.


Let F j , j ∈ [d], denote the j th column of F . The multivariate distribution of F is O(m)-invariant in the sense that for a subset A ⊂ R m×d , P r (RA) = P r (A) for all R ∈ O(m).

p r (M , where the inequality is due to the fact that a positive semidefinite matrix necessarily has nonnegative diagonal, and the equality is due to the independence of the random variables F T j M F j , j ∈ [d].


If a task is completely distributed in the network, the contributions of all neurons to that task should be identical (full equipotentiality [2]).

we define the localization Lk of task k as a deviation from equipotentiality.


Third and most importantly, even with modern multi-electrode arrays, one is limited in the number of neurons which can be recorded simultaneously during an experiment.

when data are pooled over experiments involving different neurons, most pairwise noise correlation indices remain unknown.


Then, q opt,O ∈ O, it is clearly feasible, and F (q opt,O ) = F (q opt ).


Overall, subjects are thus always kept in the dark about the true outcome distribution of any one slot machine.


We first fix a value z \i for Z \i , and a value of R such that E holds, and consider the ratio of probabilities: Observe that this ratio of probabilities is equal to: which is in turn equal to: Observe that from Equation 13, Thus, the ratio of the probabilities is at most the ratio Pr(Z i ≥ γ)/ Pr(Z i ≥ γ + 2β) for γ = sup j =i z j + q(F (T, θ j , α 1 , r j ), V ) − q(F (T, θ i , α 1 , r i ), V ), which is at most e α2 by properties of the exponential distribution.

we have established that when R ∈ Σ k , for all j, Thus for any such R, we can write: Equation 11 now follows by integrating R over E. PROOF:(Of Theorem 1) The proof of Theorem 1 follows from a combination of Lemmas 1 and 2.


In order to find the causal direction, the authors of [GSKZ17] consider the particular case that a (i) = a (j) and either .

in this case, E|C , and we have C|E if and only if E . comparing the coefficients in both directions, the causal direction is identifiable if E . Note that if either of the variances does not change, this condition always holds.


[5] have shown that methods of multi-label learning which explicitly capture label dependency will usually achieve better prediction performance.

modeling the label dependency is one of the major challenges in multi-label classification problems.


When i is large, the noise in λ i (A)( ŨA ) :,i "outweighs" the signal.

we need to choose a d such that only the first d eigenvectors/eigenvalues of A are used to approximate Φ d . Here, we need to address the truncation error: the tail { √ λ i ψ i (x j )} i>d is thrown away.


VideoPosBL This baseline uses the intial detector to discover the top K video positives from the unlabeled video, then trains with all these examples without iterating.

it incorporates our idea of discovering video positives by scoring tracks and re-training, but does not use self-paced domain adaptation for learning weights.


This form arises due to the nature of the leaky integrate-and-fire model ( 1) and (2).

assuming the initial spike sequences are estimated reasonably accurately, one can obtain good initial estimates for the weights W ij and bias terms b IF,i by solving a standard classification problem.


Each iteration of CCCP approximates the convex part (negative joint entropy) by its tangent, that is ∂h(x 0 ), x in (6).

taking derivatives of the joint entropy with respect to p(y|x i ) and evaluating at parameters at iteration t − 1, denoted as θ t−1 and θ t−1 , yields Define similarly g y (x i ), g y (x i ), and g y (x i ) for the derivative with respect to p(y|x i ), p(y |x i ) and p(y |x i ), respectively.


Moreover, since the filter is drawn from the DSE kernel h(t) ∼ GP (0, K DSE ) it is, with high probability, temporally limited in extent and smoothly varying.

a relatively small number of samples N h can potentially enable accurate estimates of h(t).


Roughgarden [11], however, observed that there is often no degradation; specifically, the very same smoothness property that he identified as implying good welfare in Nash equilibrium also proves good welfare of coarse correlated equilibrium (equivalently: for outcomes from no-regret learners).

for a large family of static games, we can expect strategic behavior to lead to good outcomes.


It brings us not only robustness to overfitting but also numerous benefits such as improved prediction accuracy by Bayesian ensembling (Balan et al., 2015), active learning (Gal et al., 2016), and principled/safe exploration in reinforcement learning (Houthooft et al., 2016).

developing a Bayesian few-shot learning method is an important step towards robust meta-learning.


Perhaps, the key advantage of our approach is that it runs lifted inference as a pre-processing step, reducing the size of the theory and then applies advanced propositional inference algorithms to this theory without any modifications.

we do not have to explicitly lift (and efficiently implement) decades worth of research and advances on propositional inference algorithms, treating them as a black-box.


Then, the number of time steps that a 1 (t) = a 2 (t + 1) = 1 does not happen is upper bounded by , which is given by regret analysis.

the compensation we need to pay on arm 1 is upper bounded by P  In this section, we present experimental results for the three algorithms, i.e., the UCB policy, the modified "-greedy policy and the modified TS policy.


If we want to see into the future, we need to learn from representations extracted at different-level convolutional layers.

we apply a unified spatiotemporal memory pool and alter RNN connections as illustrated in Figure 1 (left).


This also allows some other interesting adaptations.

we note: is the leave-one-out template for subject k, which is indepedendent of g k . we replace (5) by: From ( 5) and ( 8) we observe that the multi-subject alignment problem reduces to a sequence of pairwise registrations, each of which registers one subject to an average of connectivity matrices.


We first contract the importance matrix of W with M along all dimensions except the D-th dimension, i.e.

the result S is a square matrix whose size is the same as the current dimension D. The element S ij represents the total value of masked elements wherein we mask the i-th slice in W with the j-th slice in M . the decrease of the objective function that we can gain from swapping the i-th slice and the j-th slice is G ij as shown in Equation (10), which can also be written in the matrix format (11), where L is the diagonal vector of S Then, we only need to find the maximum elements G ij in G and swap the i-th slice and j-th slice in D dimension of W . The algorithm is described as in Algorithm 1.


If violations occur, we propose to project the proposed gradient g to the closest gradient g (in squared 2 norm) satisfying all the constraints (7).

we are interested in: To solve (8) efficiently, recall the primal of a Quadratic Program (QP) with inequality constraints: where C ∈ R p×p , p ∈ R p , A ∈ R (t−1)×p , and b ∈ R t−1 .


Because EEG-data possess a very poor signal-to-noise ratio (SNR), identification of the target symbol from a single subtrial is usually not reliable enough to achieve a reasonable classification rate.

several subtrials have to be combined for classification, slowing down the transfer rate.


That is to say, the robot could understand relatively simple sentences in a systematic way, and could understand novel sentences.

our results can elucidate some important issues about the compositional semantic representation.


For fixed step size γ k = γ for all k ≥ 1, Theorem 2 shows that the bias is of order O(γ 2 + 1/(Kγ)).

if the number of iterations K is fixed then the choice of γ which minimizes this bound is Choosing this value for γ leads to the optimal rate for the bias of order O(K −2/3 ).


Ultimately, we obtain a new factor set F := F u ∪ {F ∩ V u : F ∈ F 0 }, and probability distribution Note that during the process, not only the number of variables is reduced, but also the number of factors and the size of each factor can never grow.

if sampling was feasible for the original distribution p, it will also be feasible for q, and potentially more efficient.


MTZ is able to zip more than two models by sequentially adding each network into the joint network, and the calculated Hessian matrices of the already zipped joint network can be reused.

mTZ is scalable in regards to both the depth of each network and the number of tasks to be zipped.


Obviously, every segmentation provides such a lower (albeit less tight) bound on the total evidence score.

if we find large enough contiguous regions in Q, with supporting regions in ref (i.e., high enough local evidence scores), and define R 0 to be the remainder of Q, then S = R 0 , R 1 , ..., R k can provide a reasonable lower bound on GES(Q|H ref ).


We use a soft-max with a temperature parameter that is increased slowly through training iterations.

specifically, we learn a vector w(n) ∈ R C for each location n of the multiplexing pattern, with the corresponding I(n) given during training as: where α t is a scalar factor that we increase with iteration number t. in addition to changes due to the SGD updates to w(n), the effective distribution of I(n) become "peakier" at every iteration because of the increasing α t , and as α t → ∞, I(n) becomes a zero-one vector.


requires object detection and classification.

for the VQA task, ReasonNet incorporates the following set of modules: 1. question-specific visual attention module, 2. object-specific visual attention module, 3. face-specific visual attention module, 4. object classification module, 5. scene classification module, 6. face analysis classification module.


Therefore the network has N = 3 identical units: two of which will be used for the inputs and one for the output, and K = 4 memories ⇠ µ i , which are the four lines of the truth table.

the energy ( 2) is equal to where the energy function is chosen to be a polynomial of degree n. For odd n, energy ( 7) is an odd function of each of its arguments, E n (x, y, z) = E n (x, y, z).


On the other hand, the Commentary and Web datasets are shuffled.

a local cache can hardly capture the relevant statistics to significantly improve upon the static model interpolated with the unigram distribution.


Now, all leaves in T δ are nodes that are exposed (i.e., made public) by the δ-sampling process, so in expectation T δ has at most δn leaves (and we can bound the probability that the actual number exceeds this by more than a small factor).

there will also be O(δn) branching nodes, and all other nodes in T δ must be single-child nodes.


Let α t = t. Algorithm 5 with update by option (A) is the case when the y-player uses OPTIMISTICFTL and the x-player adopts MIRRORDESCENT with γ t = 1 4L in Fenchel game.

w T is an O( 1T 2 )-approximate optimal solution of min x∈K f (x).


When the local error bound condition holds, one can show that dist( ⇤ µ , ⇤ ⇤ )  O(µ ).

one is able to choose µ smaller than " and get a better trade-off.


The curves are concave and lie below the identity, touching it at x = e −V0 .

we show in Section 3.3 that these properties extend to every odd K and every V 0 ∈ R. is indeed a lower bound on the marginal likelihood.


The target marginals r and c represent two images in a pair, and C ∈ IR m 2 ×m 2 is the matrix of 1 distances between pixel locations.

we aim to compute the earth mover's distance.


The speakers are grouped into 5 subsets which respectively include 30 similar speakers, and the subsets are named Isolet1, Isolet2, Isolet3, Isolet4, and Isolet5.

we naturally have 5 tasks with each task corresponding to a subset.


For example, we achieve a new "per-point" guarantee E[d(j, S)] ≤ (1 + 2/e)T ≈ 1.736T , while respecting the usual guarantee d(j, S) ≤ 3T with probability one.

while maintaining good global quality with probability one, we also provide superior stochastic guarantees for each user.


The issue with this approach is that in many cases performing experiments are expensive, unethical, impossible, or even undefined.

there is a high interest in finding causal relations from purely observational data.


All other positions within the funnel are visited less often.

f will tend to get most training signal from the robust transition.


Indeed, in reality, not all pairs of items can typically be compared, and the number of times each pair is compared is also very small.

while an important step is taken in [1], it stops short of achieving the desired goal.


For an agent moving in an environment, visual percepts of objects/cues undergo a combination of transformations comprising zooming and rotation in depth.

the question arises, how to construct a visual detector, which has a Gaussian like tuning with regard to the positions within the environment from which snapshots of a visual cue are taken.


But for harder problems where the prediction error is already high, a larger value of T introduces more noise which is detrimental to learning.

lower values of T are used in [20] for classification on larger datasets.


For this dataset, the learned tree was not significantly different from the given tree.

we expected the improvement from learning the tree to be marginal.


In several granule cells (but in none of the pyramidal cells) the greater noise in z was apparent upon visual inspection of the rendered structures.

we re-ran the analysis discarding the z coordinate (right columns).


. . , G N G }, and define the (1,α)-group norm by x 1,α := N G t=1 x Gt α .

if S G denotes the subset of groups where x Gt = 0, then the subgradient has the following form: where m t * α ≤ 1 for all t / ∈ S G . the group sparse norm is also decomposable with • Low-rank matrices: for the nuclear norm regularizer • * , which is defined to be the sum of singular values, the subgradient can be written as , where • 2 is the matrix 2 norm and U, V are the left/right singular vectors of X corresponding to non-zero singular values.


Note that by "max-sum-product" we mean a novel combination of max-sum and max-product as described in more detail in the proofs in the Appendix.

we see that architectures and layer types commonly used in today's DCNs can be derived from precise probabilistic assumptions that entirely determine their structure.


The intersection of this row and the diagonal line must, then, be the location of the large entry.

we have the following theorem, saying that if there is only one non-zero entry in every diagonal, HOLEX can recover the whole matrix.


This means that the penalization term in the objective function discourages the optimization process from selecting policies which are far from the behavioral policy.

the displayed behavior is very conservative, preventing the policy from reaching the optimum.


For the TORCS environment, a simple rule is leveraged to keep the car running and to prevent it from crushing.

we can use the rule as the optimal policy to generate expert trajectories.


The proof of this theorem is provided in Appendix F. Note that for monotone rules, a larger alphawealth reward at each rejection yields a possibly higher power, but never lower power, immediately implying statement (b).

we provide only a proof for statement (a) in Appendix F. For the reader interested in technical details, a key super-uniformity Lemma 1 and associated intuition for online FDR algorithms is provided in Section 3.2.


If X is discrete and Y is single-dimensional continuous and P X (x) > 0 for all x, for sufficiently large dataset, the k-nearest neighbors of sample (x i , y i ) will be located on the same x i with high probability.

our estimator recovers the discrete vs continuous estimator in [42].


Here we create five ordinal regression data sets described in Table 2, where all the data sets are originally regression and the output values are divided into five quantiles.

the overall task can be divided into four isolated classification tasks, each of which estimates a quantile.


For simplicity, let I (N ) (Z) be the estimate based on original samples {Z 1 , Z 2 , .

. . , Z N }, where Then a certain version of Efron-Stein inequality states that: Var we have sup Z1,...,Z N To upper bound the difference | ξ i (Z) − ξ i (Z \j ) | created by eliminating sample Z j for different i 's we consider three different cases: (1) i = j; (2) ρ k,i = 0; (3) ρ k,i > 0, and conclude that The detail of the case study is in Section.


Unfortunately, lossless abstraction often leads to games that are still too large to solve.