α k and β k are the weight vectors of the rank-k matrix approximation model for all users and items, respectively.

α k i and β k j denote the weights of the rank-k model for the i-th user and j-th item, respectively.


In our setup the job of the discriminator is to analyze the local statistics of images.

after five convolutional layers with occasional stride we perform global average pooling.


stand for the preference relationships induced by r and r, respectively.

an equivalence class of the ratings r, denoted as [r], is defined as the set of ratings which are equivalent to r. That is, Now we give the definition of RDPS.


But in a short-term player's view, arm 2 behaves better than arm 1.

pulling arm 2 does not require any compensation, while pulling arm 1 needs 0.8 for compensation.


Therefore, we can compute a Bayesian predictive score for a frame, and then can compute a phoneme sequence score by using the Viterbi algorithm.

we can construct a VBEC framework for speech recognition by selecting an appropriate model structure and estimating posterior distributions with the VB approach, and then obtaining a recognition result based on the Bayesian prediction classification.


[18] addressed cross-domain disentangled representation with only supervision from single-domain data, empirical studies were performed to determine their network architecture (i.e., number of sharing layers across domains), which would limit its practical uses.

a unified disentangled representation model (like ours) for describing and manipulating multi-domains data would be desirable.


The first two have 5 human-created summaries per video and the last has one humancreated summary per video.

for the first two datasets, we follow the algorithm described in section 3.1 to create an oracle summary per video.


Denote by Q ⇤ the last query that the learner submits in the sub-interval I ⇤ in Phase 2, and by R ⇤ its response.

it follows by construction that either R ⇤ = 0 and the learner can produce the estimator by setting b X to the mid point of either , depending on the value of R ⇤ , and this guarantees of an additive error of at most ✏/2.


For denoising and inpainting we may exploit shifted versions of the data, which accelerates convergence substantially (discussed in detail below).

all denoising and inpainting results are based on efficient Gibbs sampling.


It takes the same time as the aggregation stage.

the total GST for a BCube(n,k) network is thus 2 * compared with the theoretical GST of 2 * (N −1) N * T F when running PS algorithm on a Fat-Tree network, BML algorithm on a BCube(n,k) network theoretically uses only 1 k of GST, with less network cost.


They only use the objective values and don't require gradients or higher derivatives of the objective function.

they are well suited for black box optimization problems.


Output the stationary distribution of matrix P . Rank Centrality aims to estimate rankings from pairwise comparison data.

to make use of M -wise comparison data for Rank Centrality, we apply a sample breaking method that converts M -wise data into pairwise data.


This is due to the fact that each of the k candidate vertices for each condition can be obtained by changing a single dimension with respect to the previous one.

the operation dominating the computational complexity is the sorting of the two k-vectors (β l S,1 , .


In each iteration π is learned to minimize sampled and inherently random losses of decisions.

convergence to a robust solution becomes unlikely.


A single input/output pair is insufficient for defining a program, since many (inequivalent) potential programs may be compatible with the pair.

following [1] the synthesis problem is defined by a set of input/output pairs.


However, training a single scholar model while referring to the recent copy of the network is equivalent to training a sequence of scholar models (H i ) N i=1 where the n-th scholar H n (n > 1) learns the current task T n and the knowledge of previous scholar H n−1 .

we describe our full training procedure as in Figure 1(a).


We can estimate an N -gram character-level language model p LM (•) from a separate text corpus.

our learning algorithm will work in a fully unsupervised manner, without any human labeling cost.


Our model setup consists of a NALU that is "configured by" an LSTM, that is, its parameters Ŵ, M, and Ĝ are learned functions of the LSTM output h t at each timestep.

the LSTM learns to control the NALU, dependent upon operations seen.


Intuitively, the expected number of oracle calls performed by a given AVG node s is proportional to: (1/ε 2 ) × (the product of the inverted squared gaps of the set of MAX nodes in the path from the root to s) × (the probability of reaching s by following a policy which always tries to reach s).

a near-optimal path with a larger number of small MAX node gaps can be considered difficult.


However, the PAC-Bayes-Bernstein inequality is expressed in terms of the true expected variance, which is rarely accessible in practice.

in order to apply the PAC-Bayes-Bernstein inequality we need an upper bound on the expected variance (or, more precisely, on the average of the expected variances of losses of each hypothesis h ∈ H weighted according to the randomized prediction rule ρ).


Unfortunately this null distribution takes the form of an infinite weighted sum of χ 2 random variables.

obtaining a consistent finite sample estimate of this threshold -that is, an estimate that converges to the true threshold in the infinite sample limit -is a significant challenge.


Unlike apprenticeship learning/IRL, we do not want to mimic the experts or infer their rewards, but we want to produce a policy with robustness guarantees by leveraging their data.

our results are applicable to settings where the training and testing environments are different (as discussed in the Introduction).


Based on an evolutionary argument, Friston suggests that high utility states are precisely those that have high probability, since otherwise organisms who find themselves frequently in low utility states are unlikely to survive.

adopting a control policy that minimizes a variational upper bound on surprise will lead to optimal behavior.


Note that steganography only aims to hide the presence of a message.

it is nearly always the case that the message is encrypted prior to embedding using a standard cryptographic scheme; the embedded message is therefore indistinguishable from a random string.


Then there exists a linear combination of K rows of F that will always have 0 at the j-th column index and it is not possible to generate any row of the given matrix A.

k is no less than 1 + M ax N o. of 0s in any column of F . Since, maximum value is always greater than average, A slight re-arrangement establishes the aforementioned lower bound.


First, it is not hard to reformulate the objective of the (k, z)-median problem as minimizing sup L≥0 p∈P d L (p, C) − zL , where d L is obtained from d by truncating all distances at L. By discretization, we can construct a set L of O log(∆n/ ) interesting values that the L under the superior operator can take.

our goal becomes to find a set C, that is simultaneously good for every k-median instance defined by d L , L ∈ L. Since now we are handling k-median instances (without outliers), we can use the communication-efficient algorithm of [2] to construct an -coreset Q L with weights w L for every L ∈ L. Roughly speaking, the coreset Q L is similar to the set P for the task of solving the k-median problem under metric d L . The size of each -coreset Q L is at most Φ, implying the communication cost stated in the theorem.


Only recently, similar results are starting to be proved for realistic neural network models .

a similar statistical model of the loss might also give useful insights for our empirical observations.


However, in these approaches, the inverse map is not incorporated in the estimation algorithm because it is not considered in the problem definition and the forward model is usually more complex, making inference and learning more difficult.

the key algorithmic problems are that of estimating the specialized domains and functions in an optimal way (taking into account the form of the specialized functions), and using the knowledge of the inverse function to formulate efficient infer-I( is a computer graphics rendering, in general called forward kinematics ence and learning algorithms.


Fixed points of the belief propagation message passing algorithm correspond to local minima of the KL divergence between a kind of trial function and the posterior distribution [17].

the belief propagation message passing algorithm can be regarded as an iterative algorithm based on an approximation of the posterior distribution, which is called the Bethe approximation.


For a paired sample drawn from empirical distribution π(x, z), its desirable joint distribution is well specified.

one can directly approximate the CE as where the approximation (≈) arises from the fact that p θ (x|z) is an approximation to π(x|z).


In both these models, the payoff vector, p t , is always observed at one of the k site nodes.

some communication is required to share the information about the payoff vectors among nodes.


We adopt this criteria since it is well known that the MLE asymptotically converges to N (✓ ⇤ , I 1 (⇥ ⇤ )/n).

constant 4 ⇡ 1.96 2 indicates an approximately 95% coverage.


RAPF was designed for HMMs, but GP-Vol is non-Markovian once f is marginalized out.

we design a new version of RAPF for non-Markovian systems and refer to it as the Regularized Auxiliary Particle Chain Filter (RAPCF), see Algorithm 1.


Moreover, we also require that HOOs run in parallel, perform the same number of function evaluations.

when we start running new instances, we first ensure to make these instances on par with already existing ones in terms of number of evaluations.


For k == K -1 : -1 : 1 Since the tempo trajectories can be integrated out online, we need to only store the links '¢k and quantization locations ci i ).

the random walk tempo prior can be replaced by a richer model as in Eq.


We can see from Equation 6 that this error signal is proportional to the fraction of p(π|X) log p(π|X) minus the fraction of p(π|X) for all feasible paths that go through symbol k at time t. Notice that these two fractions are zero when p(π|X) = 0, but p(π|X) log p(π|X) decreases more rapidly around zero and reaches its minimum when p(π|X) = 1/e.

paths with probability between 0 and 1/e, i.e.


The 2-D convolutional layers not only improve the recognition performance but also down-sample the input frames by two.

the down-sampling in the convolutional layers not only reduces the computational complexity of the recurrent layers in AM but also simplifies the decoding.


Since the size of each block could be much smaller than the size of the original problem, each block could be much easier to learn compared with the original problem.

efficient identification of blockwise structure could lead to substantial speedup in learning.


An optimal joint action in ' is an optimal Nash equilibrium of that state game.

the task of optimal coordination in a team Markov game boils down to having all the agents play an optimal Nash equilibrium in state games.


Note that we can rewrite the objective function of (8) as From here we can show that Q is positive semi-definite.

without constraints p i ≥ 0, ∀i, (9) is a linear-constrained convex quadratic programming problem.


Additionally, assume that the samples are distributed uniformly on the unit sphere.

if under these conditions it is equivalent to minimize Proof The proof is sketched in the supplementary material.


As discussed above, the intrinsic group additive structure G * can achieve the goodness of fit represented by the first two terms in (3) and the penalty on the structural complexity represented by the last term.

by properly choosing the tuning parameters, we expect that Ĝ is consistent in that the probability Ĝ = G * increases to one as n increases (see the Theory Section below).


The multinomial naive Bayes model assumes that the label Y is chosen from some prior distribution P (Y = •), the length N is drawn from some distribution P (N = •) independently of the label, and each word X i is drawn independently from some distribution P (W = •|Y ) over the dictionary.

we have: Since the length n of the document does not depend on the label and therefore does not play a significant role, we leave it out of our subsequent derivations.


In this scheme, even though W l still lacks a non-zero gradient, the W * l−1 of the previous layer can be learned using backpropagation to properly "rotate" X l−1 , such that it can be utilized by W l and the TS nonlinearity.

the representation learning here becomes indirect.


However, we can establish a regret bound (in the next subsection) for this method, showing it is theoretically sound.

it can be regarded as a principled variant of rounding.


The sparse PCA problem is highly nonconvex in nature.

though its global solution attains the optimal statistical rate of convergence, such solution is computationally intractable to obtain.


They are central tools for data analysis and of the simplest problems in a hierarchy of complex machine learning computational problems.

developing provably faster algorithms for these problems is often a first step towards deriving new theoretically motivated algorithms for large scale data analysis.


Figure 3b shows the results of the same analysis for sources that had been extracted with PCA, matching the numbers of non-zero elements of the estimated matrix C. For gait trajectories the source signals by PCA and ICA are virtually identical.

results in Figure 3 would be unchanged for ICA.


Many of these traditional methods synthesize new objects by borrowing parts from objects in existing CAD model libraries.

the synthesized objects look realistic, but not conceptually novel.


Therefore, we consider an equivalent MDP with an augmented state space, where each augmented state is defined by a pair (s, t) where s ∈ S and t, meaning state s in the t th horizon.

each augmented state will be visited at most once, which leads to the following set of distributions.


The "diffusion term", ⟨F 2 a ⟩, which moves p A away from p match A depends on p A but not on the magnitude of Ja 's.

within the order parameter set satisfying p A = p match A , the larger the magnitudes of J a 's are, the weaker is the repulsive effect.


CIFAR-100 has the same set of images as CIFAR-10 but with finer category distinction in the labels, with each "superclass" of 20 containing five finer divisions, and a 100 categories in total.

we expect (and observe) that Maximum-Entropy training provides stronger gains on CIFAR-100 as compared to CIFAR-10 across models (Table 2).


Instead of using the full pairwise combination of all in-and out-of-class images, we select triplets using elementary feature distances.

we refer to all the images available for training as the training set and the set of images used to train with respect to a given focal image as its learning set.


The equation (4) follows from the observation that ∇g where I is the identity matrix, and ∆x t = x t+1 − x t . Then equation ( 4) can be re-written as: (5) Now, since x t minimizes the convex function F t over the convex set K, a standard inequality of convex optimization (see [BV04]) states that for any point y ∈ K, we have ∇F t (x t ) • (y − x t ) ≥ 0.

for y = x t+1 , we get that ∇F t (x t ) • (x t+1 − x t ) ≥ 0.


Since we use the perplexity as the evaluation metric, we need to avoid probabilities equal to zero in the output of our models (which would result in infinite perplexity).

we always interpolate the probability distributions of the various models with the uniform distribution over the full vocabulary: This is a standard technique, which was previously used to compare language models trained on datasets with different vocabularies [9].


Second, by neglecting the relationships between latent parameters, we lose the ability to extrapolate trends in the data, meaning that in places where data is sparse we cannot expect good performance.

for a problem like time series forecasting, our approach will likely be unsuccessful.


The performance of a linear classifier on a complex task as IAM is quite limited.

we used polynomial feature spaces of degree one (d = 30), two (d = 495) and three (d = 5455), corresponding to polynomial kernels.


The results in column Min are computed by constrainedly minimizing V-ROUGE via the methods of [11], and the results in column Max are computed by maximizing V-ROUGE using the greedy algorithm.

the Max column is an approximate upper bound on our achievable performance.


Notice that under DS-PSRL, P T t=1 1 {A t }  log 2 (T ) always holds.

the first term can be bounded by H ] is related to how fast the posterior concentrates around the true parameter vector.


Given these sufficient statistics, the mean/moment parameters are given by the moments µ s;j := E ✓ I[X s = j] = P(X s = j; ✓), and µ st;jk , which precisely correspond to nodewise and pairwise marginals of the discrete graphical model.


Moreover, for any local optimisation method that computes a solution to (4), geodesic convexity ensures that this solution is globally optimal.

the key question to answer is: (i) does (4) have a solution?


The new formulation of inverted Wishart is consistent under marginalization.

similar to the way of deriving GPs from Gaussian distributions, we define a distribution of infinite-dimensional kernel functions, denoted by Σ ∼ IW ∞ (δ, Σ • ), such that any sub kernel matrix of size m × m follows Σ ∼ IW m (δ, Σ • ), where both Σ and Σ • are positive definite kernel functions.


the Mahalanobis distance parameterized by some covariance matrix Σ).

it has a unique minimizer [1] of the form Next, we note that equation ( 7) is strictly convex in Σ −1 .


In our p n 1 setup, we already have a rich enough feature space and it suffices to consider the linear kernel XX 1 .

in this paper we propose a randomized scheme to reduce the dimension of X and accelerate the computation of XX .


Our loss L α (X, M ) differs from common losses in allowing data points to be differently weighted.

to obtain the sample complexity of our formulation we need to bound the difference that a single sample can make to the loss.


Because of the study design, there is only one trial session (of 32 seconds) per stimulus for each subject.

there is the chance that the neural networks learned to identify the individual trials and not the stimuli based on artifacts in the recordings that only occurred sporadically throughout the experiment.


At each step in the algorithm, the weighted error of the previous model, using the new weights is exactly ¡ .

the new weak learner doing "better than random" on the re-weighted data means it can improve the previous weak learner's performance at the current fit, by achieving weighted classification error better than ¡ .


Now fix any P * ∈ M . Since P * ∈ M , it must have some density p * .

applying (8), and then (16), and then (8) again, we find that For i appearing in the second sum, with t j < i ≤ t j+1 , we have R i (P * , Pkj ) ≤ sup i ′ ≥tj+1 R i ′ (P * , Pkj ) = sup i ′ ≥tj+1 R i ′ (P * , Pδt j (x t j ) ) ≤ h(t j + 1), so that R i (P * , Pkj ) ≤ 1 t j + 1 • (t j + 1)h(t j + 1) ≤ 1 t j + 1 • ih(i) ≤ t j+1 t j + 1 h(i) ≤ αh(i), where the middle inequality follows because nh(n) is increasing (condition (b) of the theorem).


This is because there is an exchange of stability, as indicated by the arrows.

there is a tradeoff: w, and thus the mean firing rate of the memory neurons, can be increased by shifting the dashed line up or to the right , but eventually the background becomes destabilized.


By equivalence to the Barker acceptance function, this response rule defines a Markov chain with stationary distribution π(x) ∝ p(x|c) γ .

(6) using the weaker assumptions of Equation 5 as a model of human behavior, we can estimate the category distribution p(x|c) up to a constant exponent.


Functional scans from three subjects were taken while the subjects watched three video segments.

during the scan, subjects were exposed to rich stimuli including rapidly changing images of people, meaningful sounds such as dialog and music, and even emotional stimuli, all overlapping in time.


4 Also a regularizer can be designed to control overfitting to noisy observation, which is an issue in LDS that can result in an unstable latent dynamics estimate (Buesing et al., 2012a).

by controlling undesired overfitting to noisy samples one can also prevent unintended unstable model identification.


Returning to our grid-world example, at a particular state s, the reactive policy only needs to query the values of the states neighboring s in order to select the correct action.

the attention module in this case could return a ψ(s) vector with a subset of V * for these neighboring states.


In the REGULAR condition, the next trial is started from the marker position at the end of the previous trial.

in order to maximize the reward acquired in a long run, the subject has to select an action by taking into account both the immediate reward and the future reward expected from the subsequent state.


All the relevance metrics are based on utterance-pair discrepancy, i.e., the score assesses how close the system output is to the ground-truth response.

the MLE system output tends to obtain a high score despite being bland, because a MLE response by design is most "relevant" to any random response.


Moreover, the number of times a node k has been selected at a given iteration t > 0 follows a binomial distribution with parameters t and p k . Let us define m k (t) such that m k (0) = 0 and for t > 0: otherwise.

(11) For any k ∈ [n] and any t > 0, one has E[m k (t)] = t × p k × 1/p k = t. given that every node knows its degree and the total number of edges in the network, the iteration estimates are unbiased.


Our analysis indicates that given phaseless linear measurements, the regularized PhaseMax algorithm requires O(d(T L λ (x 0 ))) measurements for perfect signal reconstruction.


However, such an approach is destined to fail as Y can be any arbitrarily long and complex sequence, possibly consisting of several movements performed one after the other.

we might not have in the database examples that match sequence Y in its entirety.


This allows us to test whether an exploration strategy is truly capable of systematic exploration, rather than improving existing RL algorithms by adding more hyperparameters.

vIME is compared with heuristic exploration strategies on the following tasks with sparse rewards.


When the field is rotated, however, two things happen: the vector at r −1 x is moved to a new (rotated) position x, and each vector is itself rotated by a 3 × 3 rotation matrix ρ(r).

the rotation operator π(r) for vector fields is defined as [π(r)f ](x) := ρ(r)f (r −1 x).


From these results, transition and occupation probability γt ij and ζt ij can be obtained by using either a deterministic assignment via the Viterbi algorithm or a probabilistic assignment via the Forward-Backward algorithm.

q(Θ |O, m) and q(S, V |O, m) can be calculated iteratively that result in maximizing F m . Based on the discussion in section 3, a VB training algorithm for an acoustic model based on an HMM and Gaussian mixture model with a fixed model structure m is as follows: ----------------------------------- ------------------------------------τ denotes an iteration count.


We use one eigenvector corresponding to largest eigenvalue.

the approximate adjacency matrix becomes Let π * be the optimal permutation that minimizes ÂA i − π( ÂB j ) F . Note that here π applied on a matrix implies permutation of the basis.


This is because both functions are nonlinear and their ranges are not necessarily vector spaces, even though they are subsets of vector spaces.

we directly compare the ranges of sigsoftmax and softmax as the following theorem: Theorem 4.


Since we are using a non-saturating activation function, intermediate representations are not bounded from above, hence, they can explode even with a strong regularization over the connections, which is confirmed by preliminary experiments.

for stability reasons, we use a small fixed additional L2 penalty (10 −5 ) over both the connection weights and the unit activations, which resolves the explosion problem.


In such a context, each sample depends on all the factors but each gene within a sample usually depends only on a small number of factors.

it is more appropriate to model the factor loading matrix (A) with the IBP prior.


We will show that, for a specific budget and perturbation structure, the expected cost under the worst-case perturbation is equivalent to the CVaR of the cost.

we shall establish that, in this perspective, being risk sensitive is equivalent to being robust against model perturbations.


Solving (6) amounts to solving a semi-definite program; it cannot scale beyond a few hundred data points.

for efficient solution, we consider a different but equivalent formulation.


Define a closed convex function f as: There exists a convex function ).

when τ j ≡ 1 for 1 ≤ j ≤ p, the optimization problem (5) can be rewritten as: Problem ( 6) can be easily optimized via alternating optimization algorithm.


Since we use softattention, a read value is computed from all slots yet the main contribution comes from the one Algorithm 1 VMED Generation Require: Given p φ , r 1 0 , r with highest attention score.

we pick the maximum attention score in each read weight and normalize to become the mode weights: Armed with the prior, we follow a recurrent generative process by alternatively using the memory to compute the MoG and using latent variable z sampled from the MoG to update the memory and produce the output conditional distribution.


Most consumer-level 360 • cameras only record mono audio, and syncing an external spatial audio microphone can be expensive and technically challenging.

while most video platforms (e.g., YouTube and Facebook) support spatial audio, it is often ignored by content creators, and at the time of submission, a random polling of 1000 YouTube 360 • videos yielded less than 5% with spatial audio.


Consider the case when θ is a scalar, and hence η(θ) is a curve; the geometric curvature γ θ of η(θ) at point θ is defined to be the reciprocal of the radius of the circle that fits best to η(θ) locally at θ.

the curvature of a circle of radius r is a constant 1 r. In general, elementary calculus shows that γ ).


An even broader class of kernel parameterizations may be developed in the spectral domain: still assuming stationary white noise inputs, we may diagonalize C in the Fourier basis, that is, C( ω) = O t P ( ω)O, with O the (unitary) Fourier transform operator and P ( ω) the power spectral density.

comparing to the conditions above, if the spectral density may be written as where |.| denotes complex magnitude), for θ k > 0 and functions h k ( ω) such that sign(real(h k ( ω))) is constant in k for any ω, then the likelihood will be l.c.


Neither sampling-nor regression-based algorithms are satisfactory: while regression-based algorithms have better performances compared to the sampling-based algorithms, they are not scalable to high dimensions.

an important question is whether one can design an algorithm that can perform well on large datasets?


The term E x∼P θ [f k (x)] = x∈X f k (x)p(x; θ) is the expected feature values given the model.

when the gradient is zero, the model has perfectly matched the data in terms of the average values of the features.


This can be opposed to existing analysis of adversarial multiarmed bandits, where the regret bound depends on a predefined complexity of the underlying expert class (Beygelzimer et al., 2011).

the existing analysis of adversarial multiarmed bandits would either become looser if we add more side information or a-priori limit the usage of the side information through its internal structure.


[6] To informally see why, note that the effect of w t on x t+k can be written D k−1 w t (as in Equation 5).

moreover, the disturbance w t has a negligible influence on x t+k for large k. More precisely, it is possible to show that, under the assumption that ||w t || ≤ c 1 , the sequence on the right hand side of ( 5) is upper-bounded by a geometrically decreasing sequence, and thus its sum must also be bounded.


With more and more unlabeled data the random walk will fully mix, forgetting where it started, before it hits any label.

the probability of hitting +1 before −1 will become uniform across the entire graph, independent of the starting location x i , yielding a flat predictor.


If the dimensionality d of embedded parameter θ is large, computation of the gradient G θ and Hessian H θ (differ from g, H above) will be linear and quadratic w.r.t d, which may be unacceptable.

in the next section we attempt to reduce the computational complexity w.r.t d. In standard Newton's method, we need to compute the Hessian matrix and its inverse, which is intractable for limited computing resources.


1 Functions with higher-order terms are steadily gaining popularity in computer vision [31,33,1,16,17]; it is generally accepted that they correspond to better image models.

studying generalizations of roof duality to arbitrary pseudo-boolean functions is an important task.


Knowledge of the endpoints of each path imposes the constraints r 1,i .


While augmenting DDM with extra parameters gives it additional power in explaining subtleties in data, this also diminishes the normative interpretability of DDM fits by eliminating its formal relationship to the optimal SPRT procedure.

when the behavioral objectives change, e.g., in the GNG task, DDM cannot predict a priori what parameters ought to change and how much.


Consequently, we may not obtain positive definite kernels, built upon from the Wasserstein distance.

it may be necessary to approximate the Wasserstein distance to achieve positive definiteness for kernels, relied on Wasserstein geometry.


We proved that there cannot exist any approximation algorithm for the adversarial problem if the constraint on the observation set size must be exactly met, unless P = NP.

we presented an efficient approximation algorithm, SATURATE, which finds observation sets which are guaranteed to be least as informative as the optimal solution, and only logarithmically more expensive.


According to (3), the inducing variables can be seen as projections of the target function f (x) on the feature extraction function over the whole input space.

each of them summarises information about the behaviour of f (x) everywhere.


We define a set Θ = {i||x i | > θ}, and I Θ be the indicator function of the set Θ.

approximately solving an optimization problem of minimizing L2 distance between the original full-precision values and the quantized ternary values, we can get an sub-optimal threshold θ and the scaling factor α [21] calculated by We quantize both weights and activations in RNNs to ternary values using TTQ and the evaluation results are shown in Table 1(b).


non-trend category is in comparison to the size of the trend category.

for simplicity, we intentionally control for the class sizes by setting them equal.


One reason for this is the pixel-level L1 distance is not good for natural images.

learning the generative model on the code space helps.


6, we can see that as ν or n a tends to infinity, the predictive distribution tends to the one from a GP .

for small ν and n a , a T P would give less certain predictions than its corresponding GP .


. . z K−M (j)] T depends on the choice of the appended vectors.

where the last step uses the fact that This explicitly provides the vector z which completes the j-th column of Ã.


As the regressand at step 3, we use the deviation of the impression at iteration i from the baseline impression, i.e., α i − α 0 .

regularization penalizes large deviations from the baseline impressions, and a large ridge parameter prevents the impressions from wandering too far from the baseline.


The size of the environment is 91.2 by 94.4 meters, and the map is discretitized into cells of 0.4 by 0.4 meters.

the 2D state-space consists of 53808 states.


A single mapping discovered by a network would fail to elaborate the space of all bad functions, thus enforcing conditionality only on the relatively small set of bad functions it covers.

we propose to enforce conditionality using a self-regularization term on the generator, similar in nature to that used by Shrivastava et al.


Given this new family of CRFs, we additionally show that if covariates also follow node-conditional univariate exponential family distributions, then the functions over features in turn are precisely specified by the exponential family sufficient statistics.

our twin results definitively answer for the first time the key model specification question of specifying compatibility or feature functions for a broad family of CRF distributions.


Each page is labeled with the Yahoo's second level sub-categories from which the page is hyperlinked.

(the sub-categories are topics in our term.)


Besides, there is no analysis for generalization performance on deep learning with noisy labels.

we leave the generalization analysis as a future work.


As we proceed with the chain, update L −1 Y using Equations 3 and 4 when an insertion or a deletion proposal is accepted, respectively.

the average complexity of acquiring a sample from a distribution that is ǫ-close to P L is O(T 2 n log(n/ǫ)), where T is the average size of Y encountered during the progress of chain.


One challenge posed by the biological domain is that current systems for doing part-of-speech (POS) tagging or parsing do not perform as well on the biomedical narrative as on the newspaper corpora on which they were originally trained.

iE systems developed for biological corpora need to be robust to POS or parsing errors, or to give reasonable performance using shallower but more reliable information, such as chunking instead of parsing.


However, the requirement that θ k → ∞ will hold only during the first step using a constant α → ∞, assuming a bounded initial θ.

natural gradient based policy iteration using such a very large but constant step-size does not approach greedy value function based policy iteration after the first such iteration.


For the standard Gaussian ensemble, it is known [11] that the Lasso fails with probability one for all sequences such that n < (2 − ν)s log(p − s) for any arbitrarily small ν > 0.

corollary 1 shows that under suitable conditions on the regression coefficient matrix B * , 1 / 2 can provides a K-fold reduction in the number of samples required for exact support recovery.


[11] has shown, by examples, that each dimension of these vectors roughly represents concepts relevant to the task, e.g., 'desire to recommend the product', 'report of a faulty product', and so on.

an advantage of this representation is that there is no obvious noise between X 1 and X 2 since context X 2 is represented only by the concepts relevant to the task.


This paper follows the CAE setup to set the same settings in the proposed method.

three hidden layers (C = 5) and the regularized parameters = {10 −4 , 10 −6 , 10 −8 } are employed in the DHA method.


. . , α n ) is invariant to scaling, the gradient vector must therefore be the all zeros vector.

the point when all α i = 1 is an extremum or a saddle.


Fitting high-dimensional statistical models often requires the use of non-linear parameter estimation procedures.

it is generally impossible to obtain an exact characterization of the probability distribution of the parameter estimates.


The experimental results first indicate that the latent SMM can achieve state-of-the-art classification accuracy.

we show that the performance of the latent SMM is robust with respect to its own hyper-parameters, and the latent vectors for words in the latent SMM can be represented in a two dimensional space while achieving high classification performance.


This weakness of IMC can also be empirically seen in Section 5.

a more robust model is desired to better handle noisy features.


Our main result is, then, showing that every recurrent neural network computing the functions ft requires at least time nW /4).

we obtain almost tight time bounds for real computation in recurrent neural networks.


This is because most documents are far away from any given document.

for a document d a we can use the WCD to get a cheap neighbor ordering and only compute the NCA probabilities for the closest set of documents N a , based on the WCD.


Furthermore, in high-dimensional settings, even attempting to estimate q hierarchical MV-sets of a distribution might be too optimistic an objective due to the exponential growth in the search space, which may lead to overfitted estimates, especially when the sample is relatively small.

various methods were proposed for estimating q-quantiles in multivariate settings without an intermediate density estimation step [3,21,2,20].


One can check that in this case we have when tasks are maximally related).

if the norms of the noise terms E i are roughly the same, and so are the smallest non-zero singular values of the matrices H i , we get Hence, given enough tasks, the estimation error of the left singular subspace of H 1 in the multitask setting (i.e.


However, many other rich and interesting patterns are possible.

in the relational setting, even more so than in simpler tasks, the issue of feature construction is critical.


More specifically, analog VLSI neural networks perform their computation using the physical properties of transistors with orders of magnitude less power and die area than their digital counterparts.

they could enable large-scale real-time adaptive signal processing systems on a single die with minimal power dissipation.


8 as are given by, at a stationary point of the total weight on data and negative examples precisely balances out.


Given only a limited amount of training data, even sophisticated learning algorithms such as the popular kernel methods are not able to suppress noisy or superfluous dimensions of such high-dimensional data.

there is a natural need for feature selection when using present-day informative features and, particularly, curvature self-similarity.


. . , Y i−1 ), and 1 + 8 × 8 + 4 × 4 = 81 functions in the operator space, resulting in i × 81 possible options.

for B = 5 branches, the search space contains B!


Then ψ satisfies the hypotheses of Lemma 3.

the functions in C satisfy the hypothesis of Theorem 2.1, giving the query complexity lower bound.


On the other hand, multinomial opinions or equivalently Dirichlet distributions can be used to model a probability distribution for the class probabilities.

in this paper, we design and train neural networks to form their multinomial opinions for the classification of a given sample i as a Dirichlet distribution D(p i |α i ), where p i is a simplex representing class assignment probabilities.


The entire procedure is summarized in Algorithm 1 and implementation details are given in Appendix D. The main difficulty in analyzing the convergence of Algorithm 1 stems from the fact that we cannot solve the basis vector selection subproblem globally when Ω = • 1,2 or • 1,∞ .

we need to develop an analysis that can cope with the multiplicative approximation ν. Multiplicative approximations were also considered in [18] but the condition they require is too stringent (cf.


Since the GILBO is trained on generated samples, the dataset is limited only by the number of unique samples the generative model can produce.

it should not be possible for the encoder, e(z|x), to overfit to the training data.


Let T = S\{s in } and assume r(A T ) = n. By the matrix determinant lemma we have the acceptance ratio .

the transition probabilities can be computed in O(n 2 k) time.


The Pyramidal Width is defined as We have CWidth(A) ≥ PWidth(A) as the minimization in the definition (9) of CWidth(A) is only over the subset g-faces(cone(A)) of faces(conv(A)).

the decrease per iteration characterized in Theorem 3 is larger than what one could obtain with FW on the rescaled convex set τ A (see Section 4.1 for details about the rescaling).


This difficulty does not arise in the problem of sparse PCA detection, since there the learner must output a good sparse vector.

it is not clear whether the approach given in [Berthet and Rigollet, 2013] can be used to establish computational-statistical gaps in supervised learning problems.


Given the learned receptive fields (groups of features) we can subsequently apply an unsupervised learning method independently over each receptive field.

this method frees us to use any unsupervised learning algorithm to train the weights of the next layer.


we have demonstrated various models and regularizers in which decomposability is satisfied with interesting subspace collections V. We now show that decomposability has important consequences for the error ∆ = θ − θ * , where θ ∈ R p is any optimal solution of the regularized M -estimation procedure (1).


the optimization involves concave minimization, which in general is a difficult problem.


Recently, the global optimal solution of VBMF when the target matrix is square has been obtained in [15].

our contribution to VBMF can be regarded as an extension of the previous result to general rectangular matrices.


To circumvent this problem, we extend SDTP by introducing a composite structure for the output layer where o is the predicted class distribution on which the loss is computed and z is an auxiliary output vector that is meant to provide additional information about activations of the penultimate layer h L−1 .

the inverse computation g(h L ; λ L ) can be performed conditional on richer information from the input, not just on the relatively weak information available in the predicted and actual label.


The orbit is also unique for each object: the set of images obtained by rotating x only intersects with the set of images obtained by rotating x when x = gx.

an intuitive method of obtaining an invariant signature for an image, unique to each object, is just to check which orbit it belongs to.


Importantly, many of PAC-Bayesian inequalities hold for all posterior distributions ρ simultaneously (with high probability over a random draw of a training set).

pAC-Bayesian bounds can be used in two ways.


The natural extension is to search for those vectors a 1 and b 1 that maximize We interpret a 1 as the stimulus filter whose output is maximally correlated with the output of the "response filter" b 1 .

we are simultaneously searching for features of the stimulus that the neural system is selective for, and the patterns of activity that it uses to signal the presence or absence of this feature.


Now, crucially, oblivious algorithm are invariant under Scheme 1 (that is, when applied on a given oblivious algorithm, Scheme 1 produces another oblivious algorithm).

using [3, Theorem 2], we obtain the following.


∀y and ∀s ∈ {1, 2, ..., n − 1}, if I {T f (x (s) ) =y(s)} = 1 (i.e., ∃i 0 > s satisfying f (x y(i0) ) > f (x y(s) )), then e f (x y(s) ) < 1 2 n i=s e f (x y(s) ) .

we have − ln e f (x y(s) ) n i=s e f (x By further relaxing the inequality, we obtain the following result, We have the following inequalities by combining the results obtained in the previous subsections.


However, most HMMs used in speech recognition are 1st-order Markov processes but speech is a high-order or infinite-order Markov process.

a linear decision boundary in the likelihood-ratio score-space defined on 1st-order Markov model estimates is unlikely to be sufficient for recovering the optimal decision rule due to model incorrectness.


Note that any monotone transformation on S i (e.g., 2 Si − 1) will not change the ranking results.

the ranking results are not affected by any affine transformation on T (k), aT (k) + b, (a > 0), because For multiple classification, we consider the following common (e.g., [8,9]) surrogate loss function Algorithm 1 implements a boosting tree algorithm for learning class probabilities p i,k ; and we use basically the same implementation later for regression as well as multiple ordinal classification.


Analogously, if a financial transaction is flagged to be fraudulent, then the security teams want to know which activities or behaviors led to the flagging.

an explainable and interpretable system to reason about why certain decisions are made is critical to convince domain experts [27].


However, one cannot choose a scalar η t such that the secant equation holds with B t = (1/η t )I.

instead, one can find η t such that the residual of the secant equation, i.e., (1/η t )s t − y t 2 2 , is minimized, which leads to the following choice of η t : a typical iteration of the BB method for solving (4) is where η t is computed by (6).


To run Algorithm 1, we need to maximize the loss over (x, y) 2 F. Note that maximizing the hinge loss `(✓; x, y) is equivalent to minimizing yh✓, xi.

we can solve the following quadratic program (QP) for each y 2 {+1, 1} and take the one with higher loss: The results of Algorithm 1 are given in Figures 2a and 2b; here and elsewhere, we used a combination of CVXPY (Diamond and Boyd, 2016), YALMIP (Löfberg, 2004), SeDuMi (Sturm, 1999), andGurobi (Gurobi Optimization, Inc., 2016) to solve the optimization.


Since the leading contribution to π(x) comes from the value of x in the vicinity of √ Cσ 2 , we find the expression π(x) ≈ δ x − yβ K (Cσ 2 ) K 2 by using the power counting.

within the Parisi RSB scheme, one obtain a set of equations with the scale-invariant α c = β 2 L. This results in c g = √ 2 ln 2, as is mentioned before.


Our approach also relies on natural images statistics, but it takes the opposite direction: search for a kernel that will bring the unblurred distribution close to the observed distribution.

in addition to handling non uniform blurs, our approach avoids the need to estimate the unblurred image in every step.


Assume the singular values are sorted as τ 1 ≥ τ 2 ≥ ⋯ ≥ τ q , and let r be the largest index such that √ τ r > λ.

m has rank r. Note that where x 1∶k = (x 1 , .


Allowing the conditional probabilities in (4) to range over the uncertainty set ∆(q t (ξ t )) results in the robust data-driven dynamic programming (RDDP) formulation with terminal condition V r T +1 ≡ 0.

each RDDP recursion involves the solution of a robust optimization problem [4], which can be viewed as a game against 'nature' (or a malicious adversary): for every action u t chosen by the decision maker, nature selects the corresponding worst-case weight vector from within p ∈ ∆ (q t (ξ t )).


Let Π ∼ PoissonP(µ) and T be a probabilistic transition, then With a slight abuse of notation, we use T (θ) to denote an independent sample from T (θ, •).

we can derive a Gamma process and thus a Dirichlet process by applying the probabilistic transition to the location of each term, leading to the following: Theorems 1 and 2 are immediate consequences of the results in [9].


From Theorem 3, we know the break down point of E is β = β 1 β 2 β 3 = 0.125.

we know the estimator E is not very robust, and it may be not a good choice to use E(P flat ) to represent the temperature of the US in a certain day.


Fix y; for a given permutation π, define a corresponding ±1 sequence r π by r π i = y πi y i ; then, y πi = r π i y i .


For a stationary 1st-order Markov chain, the entropy rate which measures the uncertainty of the stochastic process X is given by: H . More details can be found in [2].

the entropy rate of the random walk X on our proposed graph G = (V, E) can be written as a set function: Intuitively, the maximization of the entropy rate will have two properties.


This is because for two computers that solve the same equation Mx i = r i , the remaining error at the slow worker is a deterministic multiple of the remaining error at the fast worker (see equation ( 3)).

taking a weighted average of the two computation results through replication does not reduce error as in independent fading channels.


functions in Γ 0 , up to some additive constant.

the right function class for f is to have the proximal map P f (x) = λ • x for some λ ∈ [0, 1] that may depend on x as well 3 .


Interestingly, it has recently been proved that for approximation with smooth radial kernels, the coefficient decay is nearly exponential [9].

for functions in L 2 (X ), most of the energy content comes from the initial coefficients, and we can hope to keep M ⌧ N for computationally efficient training.


In the first, there is some vertex u ∈ S such that u is not connected to vertices in both V (i) \S and V (j) \.

since the removal of S disconnects G, the removal of S must also disconnect H. S must contain a minimal separator of H. Since H is a (k − 1)-tree, all minimal separators of H must contain k − 1 vertices which must therefore be S \{u}.


The fact that blur changes the derivatives distributions also suggests that it might be captured as a kind of texture cue.

it's particularly interesting to try segmenting the image using texture affinities (e.g.


While mean-prediction is efficient, there is nothing in the algorithm that prevents trajectories from drifting away from the training data.

it is sometimes desirable to optimize a particular motion under the GPDM, which often reduces drift of the mean-prediction mo-  [4], and is a natural consequence of the Bayesian approach.


6 in [2]) apparently allows us to extend this to cases where X m and X reside in some infinite dimensional, separable Hilbert space (for example, if Θ is a subset of an infinite dimensional reproducing kernel Hilbert space in kernel clustering).

recalling that √ m( θ − θ 0 ) converges in distribution to a random normal vector Z, we have that uniformly for all x, a, b, Here we think of a((∂/∂θ)f θ0 (x)) ⊤ as the vector y to which we apply the theorem.


Therefore, there exist k 0 , h 0 , i 0 such that |λ i k0,h0 − λ * ∞ | < δ for any i > i 0 .

from the uniform continuity we also learn that the first summand is bounded above by , and from Equation (10), we get that the last summand is bounded above by 0.


A stable algorithm, i. e. {3-stable with a small {3, has the property that replacing one element in its learning set does not change much its outcome.

the empirical error, if thought as a random variable, should have a small variance.


Without loss of generality, we assume that this item is unique in all contexts.

the hardness of discriminating between items i and i * (y) is measured by a gap between the expected gains of the items: Our analysis is based on counting how many times the policies π t and π g choose a different item at step k. we define several variables that describe the state of our problem at this step.


Thus, the total (discounted) loss of this trajectory can be written as From ( 9), it is clear that the quantity in the parenthesis of ( 8) is the value function of the policy θ at state (x 0 , ν) in the augmented MDP M, i.e., V θ (x 0 , ν).

it is easy to show that (the second equality in Eq.


Moreover, we cannot easily employ existing pose similarity measures [13].

we represent the conditional probability using a exemplar-based likelihood function: where Z θ verb is the normalizer and d pose is the distance between the pose descriptor I i,p pose and its closest class representative vector r can be written similarly as in the weighted kernel k-means method [9]: The center vector µ k r is defined as , where π k r is the cluster of vectors assigned to µ k r , and w(b) is the weight for each point b, representing the likelihood that b belongs to the class of µ k r (as in equation ( 11)).


Before the calculation of GGV, we have to estimate the FIM of GMRF by decomposing the FIM of the full Gaussian.

we obtain one common FIM for all expectation parameters.


Linear regression Suppose x = x + " is generated by by adding noise with Var["] = 2 I d⇥d to the original feature vector x.

note that Var Applying these facts to (6) yields a simplified form for the quadratic noising penalty: we recover the well-known result that linear regression with additive feature noising is equivalent to ridge regression [2,9].


Considering the computational cost of Algorithm 1, due to the Markov chain assumption, our algorithm TLRisk can be implemented using dynamic programming.

in the worst case, the time complexity of TLRisk is in training, and O(|C||Y t |) for predicting an instance.


The first-order smooth optimization approach [17,18] has iteration complexity O( L/ε) for finding an ε-optimal solution.


The first terms depends on the number of tasks only via δ ≤ i a i .

if a i = 1 (uniform prior) the quantity δ ≤ K is bounded by the number of tasks.


[5] for submodular maximization problems.

our result for non-monotone submodular objectives represents the first non-trivial result in this model for such objectives for any constraint other than a single matroid constraint.


In addition, we assume w * to be the optimal solution to f (w).

then, the output of Algorithm 1 satisfies that: the best solution we can obtain is controlled by γLM 2σ .


An additional advantage to using ordinal relations is their natural robustness to sensor noise.

it would seem that local ordinal representations may be well suited for devising compact representations, robust against Figure 3: The absolute brightnesses and their relative magnitudes change under different lighting conditions but several pair-wise ordinal relationships stay invariant.


If one views the lattice as forming a simple model for geographic factors, the hierarchical model can similarly be interpreted as forming a "topic hierarchy" of occupations, with individuals at the leaves.

for example, the occupations of "banker" and "stock broker" may belong to the same small sub-tree; since the target in one of Milgram's experiments was a stock broker, it might therefore be a good strategy to send the letter to a banker.


Model 1 serves both the correlated and uncorrelated design cases.

each row of the design matrix X is a draw from the vector M T Z, which has covariance c M T M for some constant c. the choice of M = I leads to an uncorrelated design.


The goal of an algorithm is, at each time step t, to propose one point x t where to sample the space X , so that its allocation of samples {n t (R)} R∈P (that is, the number of points sampled in each region) minimizes some objective function.

the algorithm is free to sample everywhere in each region, with the goal that the total number of points chosen in each region is optimal in some sense.


The NPDP algorithm uses the solution given by which always exists for any stochastic matrix λ.

the derivation has shown that the exact value function for the model in Section 2.1 has a Nadaraya-Watson kernel regression form, as shown in Eq.


Instead, each cell has its own set of labels.

if we maintain a global collection of cells, each with its own Q-regressor, we have a set of approximate Q-functions that give both a compact representation and the ability to generalize across state-action pairs for any set of policy consistent assumptions.


The reason is that in NAIVE the human agent overrides the AI, which is more costly than having the AI perform the same policy (as it happens to be for HUMAN).

simply assuming that the human has the correct model does not only lead to a larger error than knowing the human's model, but it can also be worse than simply adopting the human's erroneous model when making decisions.


We assign a finite prior mass B to self transitions for each state; this is the third hyperparameter in our model.

when first visited (via 8 in the HDP), its self-transition count is initialised to B .


First, due to the given incidence relations, the block y r has two components corresponding to the elements indexed by r and r + 1.

for any r ∈ g(y) = 1 n , y 2 2 > 4 3 n, which results in * < 3 2n 2 ≤ 7 N 2 for all n ≥ 3.


In addition, a variable can belong to at most one pair in P ′ .

any given variable is either in a single pair of P ′ or else in exactly one of the sets P , P + , or P − .


The first is that its loss landscape has nice properties with provably few bad locals.

we can have some guarantees on the performance of gradient descent.


Comparison with Picard-Nesterov Since the code acquired for Picard-Nesterov is implemented purely in Matlab, a computational time comparison might not be fair.

only the number of iterations required for convergence is reported, as both methods adopt the first order method.


Evaluating a single policy with a degree of certainty requires testing many subjects to reduce sampling variance due to individual differences, factors outside of experimental control (e.g., alertness), and imprecise measurement obtained from brief evaluations and discrete (e.g., correct or incorrect) responses.

exhaustive search over the set of distinguishable policies is not feasible.


In addition, we define for each state-action couple (s, a) the vector φ(s, a) ∈ R d such that φ(s, a) = (φ i (s, a)) d i=1 .

the action-value functions are characterized by a vector θ ∈ R d and noted Q θ :  θn ) n∈N respectively.


However, the proposed versatile convolution filters adopt a more complex approach to capture useful information from input images, i.e., a large filter consists of a series of smaller filters and each of them will employ on the input images to generate feature maps.

it is necessary to visualize and compare filters in original CNN and the network using the proposed versatile convolution filters for having an explicit illustration.


The main problem is that, even if P and R are factored, the optimal value function generally does not have a compact representation (nor does the optimal policy).

obtaining an exact solution appears to require a return to explicit representations.


More generally, by summing up the update equations for h consecutive timesteps and then minimizing the left and right sides' expected squared difference, we can minimize the h-step prediction error.

it may seem that we can directly solve for the parameters that minimize the lagged criterion of Eqn.


This amounts to determining all bits such that sign(r T G t1 u) = sign(r T G t u), or equivalently, (r T G t1 u)(r T G t u) ≤ 0.

expanding the update given in (3.2), we can write r T G t u as finding the bits that have changed sign is equivalent to finding all u such that (r We can use a second level of locality-sensitive hashing to approximately find all such u.


We measured computation time in terms of the number of full evaluations by the algorithm (i.e., the number of ads scored against a given page).

the true test of an algorithm was to quickly select the most promising T ads to fully score against the page, where T ∈ {100, 200, 300, 400, 500} was externally imposed and varied over the experiments.


We are interested in estimating µ 1 − µ 0 from the observed samples {(x i , y i )} n i=1 .

by a standard reduction argument [24], the fundamental limits of this estimation task are captured by a hypothesis testing problem, namely, where γ n denotes the signal strength that scales with n. we focus on studying the fundamental limits of γ n for solving this hypothesis testing problem.


Unfortunately, this method incurs a cost, since the size of the dictionary then depends on the size of the input signal.

it is advantageous to use a formulation based on convolutions rather than following the naive method outlined above.


If they do so at a rate then the constant will be Q .

the accelerated inexact proximal-gradient method will have a faster convergence rate than the exact basic proximalgradient method provided that Q < (1 − γ).


Because one of the primary goals of our regression analysis is to identify a subset of imaging markers which are highly correlated to the AD progression reflected by the cognitive changes over time.

we examine the imaging markers identified by the proposed methods with respect to the longitudinal changes encoded by the cognitive scores recorded at the four consecutive time points.


First, the distance to an optimum never decreases.

the value function, being a correction toward an optimum, never vanishes toward a 'neutral' state.


The prior of the transition parameter π z L ,z T is modeled by a Dirichlet process DP (α , β) which is a distribution over the other distribution β. α is a hyperparameter which controls the variability of π z L ,z T over different states across all nodes: as α increases, the state transitions become more regular.

the HDP makes use of a Dirichlet process prior to place a soft bias towards simpler models (in terms of the number of states and the regularity of state transitions) which explain the texture.


We will refer to t as the translation of the camera P. We can thus write For a set of unknown scene points U * j ,, j = 1, . . . , n, expressed in homogeneous coordinates (i.e., U * j is an element of the projective space P 3 ), we assume that noisy images of each U * j by some cameras P * i are observed.

we have at our disposal the measurements where e , = 1, 2, 3, stands for the unit vector of R 3 having one as the th coordinate and I j is the set of indices of cameras for which the point U * j is visible.


Our approach is based on having the loss decouple from the label generating model, the user feedback being a noisy version of the gradient of a surrogate convex loss associated with the model itself.

the algorithm is not directly dealing with the original loss when making exploration.


8 can effectively reduce the number of massive 1 × 1 convolution filters.

we directly applied the versatile v2 convolution filters with channel stride approach on it.


The batch learning algorithms MLE-SGLP and MLE in [27] each took about 1.5 hours.

our algorithm achieves the performance similar to batch learning algorithms with a run time close to that of parametric online learning algorithms.


Our approach, which we call a sample-based method, assumes a set of samples from each distribution and proposes an error measure of the disagreement defined directly in terms of the samples.

a second set of samples drawn from the model serves in place of a PDE or an analytic expression of the model's density.


If we choose a random stopping time T , then the above inequalities say that on average w T also solves this L 1 -regularization problem approximately.

we use the last solution w T instead of the aggregated solution ŵT in experiments.


Finally, the elements of x t store the coefficients that best reconstruct y t by linearly superposing harmonic templates of W. Note that only partials from the same source form harmonic relations.

the number of nonzero elements in x t indicates the number of periodic sources at time t, while the indices of nonzero elements indicate their fundamental frequencies.


The second type has a smaller amplitude, thus the excitation of nearby ganglion cells is most efficient at 10 • .

the effect depends on the spatial extend of the eye movements.


We obtain the HOG descriptor of a rectangle of w × h cells by stacking the enclosed cell descriptors (this is a 9 × w × h vector).

given the cell histograms, we can immediately obtain the HOG descriptors H(x, y) for all the cellaligned translations (x, y) of the dw × dh pixels rectangle.


But g(x) = f (Rx) R f Lip is 1-Lipschitz over a set K R of diameter 1.

for general functions over a general convex sets, our result trivially implies the rate for being able to optimize approximately-convex functions is Our algorithmic result matches the lower bound on well-conditioned bodies.


If M / ∈ Q, α nj = 0 for some j = n and some n. Then m T n l(η) = l j=1 α nj ˜ j (η), which cannot be equal to n (η) for all η.


Note that, to be consistent about when approximations are made and thus produce weights ŵL which still sum to Ẑ, we repeat the procedure for computing Ẑ exactly, including recomputing the running lower bound Z min .

this algorithm is guaranteed to sample each label L with probability pL ∈ [p L − , p L + ]: Proof: From our bounds on the error of the estimated probability of choosing label L has at most error 5 Computational Examples In this section, we compare the sampling methods discussed in this paper on three challenging one-dimensional examples, each involving products of mixtures of 100 Gaussians (see Figure 4).


The set W n−1 (s → * ) denotes walks that start at node s computed at the previous iteration.

the middle term W( * denotes a length-1 walk (called a hop) across an edge in E\S n . Finally, W(• Sn −→ t) denotes walks in S n that end at node t. the first term in (7) refers to previously computed walks starting at s, which hop across an edge in E\S n , and then finally propagate only in S n (ending at t).


We have where the second inequality holds since The first and third inequalities come from U t ⊆ S t and the definition of U t . Putting (10) into ( 9), we have It follows from the above lemma that, if w t converges to w * , we have D t w t = w * , and hence S t includes the support of w * .

moreover, it holds that , since w t is independent of x t and y t . to bound Lemma 4 ( [11]).


Sparse multinomial distributions arise in many statistical domains, including natural language processing and graphical models.

a number of approaches to parameter estimation for sparse multinomial distributions have been suggested [3].


These experimental conditions have been reported to exhibit highly similar neural activity patterns in GLM analyses of that dataset [4].

also local activity differences (in the striatum and visual cortex in this case) can be successfully captured by brain-network modelling.


So, if K is sufficiently large, by Chebyshev's inequality, with 99% probability we have that (γ) r > F r (p)/2 ≥ C n 1−r .

in the remainder, we can assume that , where we use that ≥ n −1/4 and m = Cn 2/3 / 4/3 .


However, p bma only starts to behave like p 2 when it catches up with p 1 at a sample size of about 310 000, when the codelength of p 2 drops below that of p 1 .

in the shaded area p bma behaves like p 1 while p 2 is making better predictions of those outcomes: since at n = 100 000, p 2 is 40 000 bits behind, and at n = 310 000, it has caught up, in between it must have outperformed p 1 by 40 000 bits!


If a single stimulus can possibly elicit millions of different responses on a neural layer, how can this plethora of data be effectively decoded by downstream networks?

neural information processing requires encoding mechanisms capable of producing as few responses as possible to a given stimulus while keeping these responses different between stimuli.


The disentanglement term in the training objective encourages the desired clean partition.

essential factors required for predicting y concentrate into e 1 and all other factors migrate to e 2 .


Note that, distinct from a traditional sentence generator, here, the argmax operation is used, rather than sampling from a multinomial distribution, as in the standard LSTM.

all randomness during the generation is clamped into the noise vector z.


All the models are explored sufficiently long so as to be able to estimate the optimal reward level in the true model, and to learn its policy.

nothing has to be known about the "wrong" models.


When initialized with the same starting point, DFacTo and its competing algorithms will converge to the same solution.

we only compare the CPU time per iteration of the different algorithms.


The first transition Figure 1: The Taxi task instance recorded during an execution is independent of the last instance recorded during a previous execution.

we do not allow updates across segments.


Due to the iterative nature of Manifold Sculpting, it's difficult to produce a valid complexity analysis.

we measured the scalability of Manifold Sculpting empirically and compared it with that of HLLE, L-Isomap, and LLE.


Notice that the relation ⊥ π is symmetric and reflexive.

we can group all the ground atoms corresponding to the transitive closure of this relation, yielding a "meta ground atom" such that if the meta atom is assigned to true (false), all the ground atoms in the transitive closure will be true (false).


On the other hand, the closer we get to the true parameter value, the more resolution we need.

the computationally optimal batch-size is a function of the residual estimation error.


It follows that the total complexity of computing k(G, G ) is When computing the kernel matrix i=1 of graphs with N > m + n + δ 2 , note that Algorithm 4 only needs to be run once for every graph G i .

the average complexity of computing one kernel value out of all K ij becomes 1 Classification experiments were made with the proposed GraphHopper kernel and several alternatives: The kernel PROP [16], the connected subgraph matching kernel CSM [14] and the shortest path kernel SP [17] all use continuous-valued attributes.


. . , γ N (t)), we have a parallel to the average trajectory γ δ has the same form as the geodesic but with randomly perturbed delays.


where g is the channel stride parameter and ĉ < c is the number of non-zero channels of secondary filters.

f i is the i-th unduplicated copy of primary filter f given the length ĉ and the stride g. a filter will be used n times simultaneously to generate more feature maps by introducing Fcn.


Figure 1 displays both 1 − ηN (ε) and the average CCR as a function of ε for different values of N . Note that 1 − ηN (ε) quantifies the probability (with respect to the training data) that P belongs to the Wasserstein ball of radius ε around the empirical distribution PN .


The distance between two nodes in the tree depicts the difference between the two corresponding image categories.

in the category hierarchy in ImageNet, each internal node near the bottom of the tree shows that the image categories of its subtree are highly correlated, whereas the internal node near the root represents relatively weaker correlations among the categories in its subtree.


An exponential filter predicts that ARR− will always create a stronger expectation for an R on the current trial than AAR− will, because the former includes an additional past repetition.

if the current trial is in fact a repetition, the model predicts a faster RT for ARR− compared to AAR− (i.e., RT ARRR < RT AARR ).


By the property of reversal Markov chains and lemma 1, we have that lim i→∞ P i r − P ∞ r → 0 and lim i→∞ ((1 − β)P + β P ) i r − P ∞ r → 0, such that the Marvov chain P and its reversal (1 − β)P + β P converge to the same value.

the norm (P i − ((1 − β)P + β P ) i )r p also converges to 0 in the limit.


This is reflected in the lower bound 1/q for β and the fact that the players' rate of convergence to Nash equilibrium increases for smaller β; in particular, (3.3) shows that Algorithm 1 enjoys a convergence bound which is just shy of O(exp(−n 1−1/q )).

if the noise process ξ is sub-Gaussian/sub-exponential (so q can be taken arbitrarily large), a near-constant step-size sequence (small β) yields an almost linear convergence rate.


For each summand, we take the subgradient Φ(x i , s i (w t )), where We note that computing s i (w t ) for each training example can be costly.

but from our definition of L output , we have that s i (w) = ⊥ for a background example independent of w. for a background example Φ(x i , s i (w t )) = 0.


While this work suggests good strategies for evidencebased model selection the MAP estimator is not truly Bayesian in spirit because it is not based on the concept of model averaging which is crucial to Bayesian reasoning.

the MAP estimator is generally not as robust as a real Bayesian estimator.


Clearly, there is a distinction between customers borrowing from an MFI and those borrowing from commercial banks.

we model the village side as non-corporate agents.


The key benefit of trace norm regularization is that efficient solution methods exist that work with a low rank factorization of the matrix variable while automatically ensuring positive semidefiniteness and still guaranteeing global optimality [10,14].

we conducted the main optimization in terms of a smaller matrix variable B such that BB = θ P AQ .


In contrast, methods such as DR, AP and the primal-dual hybrid gradient descent (PDHG) proposed in [29] used for SSL on hypergraphs [17], require performing a complete gradient descent and computing a total of R projections at each iteration.

from the perspective of iteration cost, RCD is significantly more efficient, especially when R is large and computing Π(•) is costly.


Each therapy sequence ends with the current (most recent) therapy -the one that determines the label of the sample and the sequence alignment is adapted such that the most recent therapies are always matched.

it also accounts for the problem of uneven representation of the different therapies in the clinical data.


While the human visual system has been exposed to a number of distortions during evolution and lifetime, we clearly had no exposure whatsoever to many of the exact image manipulations that we tested here.

our human data show that a high level of generalisation is, in principle, possible.


In [9], Pelletier defines an estimator of a PDF on a Riemannian manifold M by using the distances measured on M via its metric tensor, and obtains the same convergence rate as in (2), with N being replaced by the dimensionality of the Riemannian manifold.

if we know that the data lives on a Riemannian manifold M , the convergence rate of this estimator will be determined by the dimensionality of M , instead of the full dimensionality of the feature space on which the data may have been originally sampled.


As illustrated in the next section, controlled drift can often lead to boundedness of a stochastic process.

the intuition of minimizing drift (t) is to yield small kQ(t)k bounds.


We use a Laplace prior distribution over the weights in order to favor sparse solutions over those which explain the data equally well but require more weights different from zero (c.f.

tibshirani (1996)): prior factors have the form In our applications, we allowed the prior variance 2 ρ 2 k of the stimulus-dependent features to be different from the variance of the spike-history features.


In semi-supervised learning, a common prior is that the discriminant function f does not change rapidly around sampled points.

we expect the norm of the gradient at these locations is small.


We cannot typically evaluate the integrals required to compute F (y) and K(y, y ).

instead, at each time step we estimate these functions using a small set of sample points {x i } P i=1 which we refer to as a batch of size P . we introduce noise by sampling random data to make imperfect estimates of the gradient of the objective function.


Consequently, the number of columns with non-zero elements (nnzc(•)) for X 1 , X 2 and X 3 is very close to the total number of non-zero entries in the tensor.

as predicted by theory, DFacTo (ALS, GD) does not enjoy significant speedups when compared to GigaTensor, CPALS and CPOPT.


That is, the average maximum number of queriable positive symptoms is only about 2.

it is extremely hard to find the positive symptoms.


In this paper we focus on the three original axioms introduced by Kleinberg.

our approach aims at complementing their study of the axioms by replacing their consistency axiom with a stronger one.


Starting from the vector τ ∈ R k , τ j = ϕ j (X i ), in the above sampling strategy we have for a constant c k that may depend on k but is O(1) and bounded away from 0.

to attain the unbiasedness condition E The full sampling and inferential scheme are as follows: (i) given a data point X i , construct the vector τ = [ϕ j (X i )] k j=1 ; (ii) sample Z i according to strategy (19) using τ and the bound B = B 0 √ k(e α + 1)/c k (e α − 1).


To see this consider two random variables X and Y and let with equality holding only for θ = 0 and θ = 1.

the variance estimates collected by Batch Normalization during training do not represent the statistics observed during testing if the expected values of Θ 1 and Θ 2 are used in a deterministic inference scheme.


A commonly used approach is to select the regularization parameter from a set of candidate values, whose values, however, need to be pre-specified in advance.

it is essential to specify the effective interval for the values of λ.


For an Ising model, the dependency matrix is bounded by Hayes [8] proves this for the case of constant β and zero-field, but simple modifications to the proof can give this result.

to summarize, an Ising model can be guaranteed to be fast mixing if the spectral norm of the absolute value of interactions terms is less than one.


Empirically, a stochastic gradient algorithm seems to do fairly well at tracking the true Σ v,11 .

one simple adaptive control scheme would be to take the current estimate of Σ v at each time step t, apply a standard LQG solver giving this estimate (and A, B, C, Σ w ) to it to obtain the optimal steady-state Kalman filter and control gains, and use the values obtained as our proposed gains L t and K t for time t. This gives a simple method for adapting our controller and Kalman filter parameters to the varying noise parameters.


We represent such spatial (and size) variations of object instances with affine transformations of a unit bounding box b.

the where module is a conditional GAN, where the generator G l takes x and a random vector z l as input and outputs an affine transformation matrix A via a STN, i.e., G l (x, z l ) = A.


FOVE cannot do lifted inference on this MLN and resorts to grounding.

it could only execute up to the domain size of 18; after that it consistently ran out of memory.


We use the symbols 1) can be equivalently rewritten as the sum of the products of the elements of the cores: ( The representation of a tensor A via the explicit enumeration of all its elements requires to store the tensor is stored in the TT-format.

the TT-format is very efficient in terms of memory if the ranks are small.


But increasing k will bring more communication and calculating time.

we should better select a moderate k. For some distributions, especially for the distributions over high-dimensional space, l (j) (k) is less sensitive to k, then we can choose a relatively smaller k to save communication time.


While we find that both approaches can assign most neurons correctly, only the nonlinear version of S3ID allows correct recovery for every neuron.

the flexibility of S3ID allows more accurate reconstruction and prediction of correlations in data which violates the assumptions of linear Gaussian dynamics.


It is interesting to note that some of the previous methods [1,4] take the full graph information into account.

the above results indicate that our method is statistically competitive (in fact better than most of the other methods) even though the full graph structure is discarded.


Interestingly, we prove that the higher-level clusters are associated with longer-term diffusive transitions on the graph, amounting to smoother and more global similarity functions on the data mani-fold.

the cluster hierarchy exposes the observed affinity structure gradually in different resolutions, which is somehow similar to the wavelet method that analyzes signals in different bandwidths.


Determining the starting point θ 0 is analogous to having an initial pretraining phase.

the proposed Wasserstein procedure can also be seen as finetuning a standard RBM, and forcing the finetuning not to deviate too much from the pretrained solution.


this means that by finding the eigensystem (Ai, «Pi) of Tk we can also determine the regularization operator P via [8] (8) The eigensystem (Ai, «Pi) tells us which functions are considered "simple" in terms of the operator P. in order to determine the regularization properties of dot product kernels we have to find their eigenfunctions and eigenvalues.


These two formulations model different setups: if the system, more specifically the distribution of uncertain parameters, evolves with time, then the non-stationary model is more appropriate; while if the system is static, then the stationary model is preferable.

for any given policy, the worst expected performance under the non-stationary model provides a lower bound to that of the stationary model since CS ⊆ C∞ S . one can use the non-stationary model to approximate the stationary model, when the latter is intractable (e.g., in the finite horizon case; see Nilim and El Ghaoui [5]).


However, it was reported that no clear performance degradation was observed in experiments [14].

simpleVBMF would be a practically useful approach.


The idea of using nearest neighbors for unsupervised learning has been explored in manifold learning [1,7], but unlike these approaches and more recent work on representation learning [2, 13], we do not seek to model the full data distribution.

even with relatively few parameters, the transformations we learn naturally extrapolate off the training distribution and can be applied to novel types of points (e.g., new types of images).


Such a procedure is called a "consistent estimator."

recoverability is the sole property of G and Q, not of the data available, or of any routine chosen to analyze or process the data.


In practice, feature vectors may be very high-dimensional, and which components are 'important' will depend on the specific properties of the shapes being matched.

we introduce a parameter, θ, which controls the relative importances of the various feature components.


We will say that a buyer lies whenever a t = 0 while p t < v. At the beginning of the game, the algorithm A used by the seller to set prices is announced to the buyer.

the buyer plays strategically against this algorithm.


Deep learning for autonomous driving may be one prominent example: one would like to achieve robust classification performance in the presence of confetti, despite not having had any confetti exposure during training time.

from a machine learning perspective, general noise robustness can be used as a highly relevant example of lifelong machine learning [28] requiring generalisation that does not rely on the standard assumption of independent, identically distributed (i.i.d.)


However, assumptions A1 through A4 can be verified to hold for a matrix of known rank using its observed entries.

it is possible to verify that these assumptions hold for a given Ω and A and provide a certificate guaranteeing exact recovery before matrix completion is attempted.


Note that MMD does not optimize an upper-bound to the negative log likelihood directly, but it guarantees the mutual information to be maximized and we can expect a high log likelihood performance [35].

to translate this upper-bound, L 1 , to an implementable loss function in the model, we use the following empirical data distribution approximation: the upper bound can be approximated as: Since v y1 = f (x, v y1 ) and v y1 ∼ N (0, I), the implementable upper-bound, L, can be approximated as follows: As illustrated in Figure 2b, we train the Y → X → Y cycle starting from an image y ∈ Y.


A difference between the neural network and biochemical models is also highlighted: global coupling of rate equations through enzyme saturation can lead to global feedback regulation, thus allowing a simple network without explicit mutual inhibition to perform the winner-take-all computation.

the full complexity of the cell is not necessary for biochemical computation: a wide range of functional behaviors can be achieved with a small set of biochemical components.


This paper also indicates that such skewness comes from the Bellman equation passing the dispersion of transition dynamics to the state/action values.

as long as a value is estimated by applying the Bellman equation to the observations of transition, it can suffer from the skewness problem, regardless of the algorithm being used.


In Appendix B we describe a variant that uses only O(sn) operations.

we can solve (3) in O(min{n, s} log(n)) operations.


Moreover, v σ T P (I) (I) does not depend on D(I, a).

we can prune D(I, a) from iteration T 0 until iteration T 1 so long as In the worst case, this allows us to skip only R(I,a) U (I,a)−L(I) iterations.


g(S 1 ) + g(S 2 ) ≥ κg(S where the last inequality follows from Lemma 8.


By [21, Theorem 3.1], nV b (h 2 ) and nV (h 2 , Z) have the same limiting distribution, i.e., ϕ(nV b (h 2 ), nV (h 2 , Z)) → 0 in probability under certain assumptions.

it suffices to check these assumptions hold: Assumption A2.


Moreover, when ω 1 = ω 2 , the objective value is independent of their common value ω = ω 1 = ω 2 .

we should move probability mass out of resources ω 1 , ω 2 to some common region, which contain points that receive probability mass from both resources.


Since we have assumed that each u * i and the state x are jointly Gaussian, we may write any u * i as an affine function of x i plus independent Gaussian noise.

the minimum mean squared estimator is given by the conditional expectation we have found a closed form expression for the best regressor πi to predict u * i from only x i in the joint Gaussian case with squared error distortion.


Considering these results, it may be difficult for a method such as SimHash to cluster states appropriately using only raw pixels.

rather than using SimHash, we propose to use an autoencoder (AE) to learn meaningful hash codes in one of its hidden layers as a more advanced LSH method.


There is a distribution P over k-arm instances of the stochastic multi-armed bandit problem such that any fair algorithm run on P experiences constant per-round regret for at least T = Ω k 3 ln 1 δ rounds.

we tightly characterize the optimal regret attainable by fair algorithms in the classic bandits setting, and formally separate it from the regret attainable by algorithms absent a fairness constraint.


An independent line of work [22], based on Gomory-Hu trees (non-linear), established that submodular functions represent nonnegative solutions of the optimization problem (5) with β (e) = δ e − 1.

an unrestricted solution of the optimization problem (5) ensures that β (e) ≤ δ e − 1.


(D) In the RANDOM condition, the next trial is started from random state.

the optimal behavior is to select a larger reward at each state.


This is due to the lack of a succinct characterization of influence functions as under the DIC and DLT models.

in the next section, we explore improper learning approaches with the goal of designing practical algorithms and establishing learnability under a broader class of diffusion models.


Second, since we trained all models for a constant number of epochs, smaller batch sizes entail more training iterations in total.

there is a significant difference in the number of iterations and the corresponding weight distance reached at the end of the initial learning phase.


Consider the affine sub-problem: and the convex sub-problem: We will use the Von-Neumann [5] successive projection lemma stating that P 1 P 2 P 1 P 2 ...P 1 (K) will converge onto the projection of K onto the intersection of the affine and conic subspaces described above 1 .

what remains to show is that the projections P 1 and P 2 can be solved efficiently (and in closed form).


The excite operator then resizes the aggregates, applies a sigmoid and multiplies the result with the input.

each output feature map is computed as y c = x σ(interp(ξ G (x) c )), where interp(•) denotes resizing to the original input size via nearest neighbour interpolation.


Part of the reason for this is that there are intrinsic limitations in what one can do in the local model.

many basic questions, that are well studied in the central model, have not been completely understood in the local model, yet.


a matrix entry M ij corresponds to the dot product SGNS is factorizing a matrix in which each row corresponds to a word w ∈ V W , each column corresponds to a context c ∈ V C , and each cell contains a quantity f (w, c) reflecting the strength of association between that particular word-context pair.


Moreover, inverting requires O(D 3 N 3 ) computational complexity prohibiting an efficient inference when N is large.

our solution is to model the noise covariance matrix with a Kronecker structure, i.e., = ⌃ N ⌦ ⌃ D , where ⌃ N is the noise covariance across neurons and ⌃ D is the noise covariance across odors.


Furthermore, for each black vertex with degree d, we connected it to d randomly selected white vertices.

we build two bipartite graphs by setting the proper parameters in order to control the average degrees of the black vertices to be 50 and 3000 respectively.


Similarly, the "inactive" set on θ * is defined as I θ * = I \ I θ * .

from a geometric perspective, if we know θ * , i.e., the projection of y λ onto F , the predictors in the inactive set on θ * can be discarded from the optimization.


The negative gradient will be mostly pointing in directions of high curvature, and a small enough learning rate have to be chosen in order to avoid divergence in the largest positive curvature direction.

the gradient step makes very little progress in small curvature directions, leading to the slow convergence often observed with first-order methods.


See Appendix for the proof and the general results for multi-dimensional parameters.

theorem 4.6 suggests that θf any consistent θf is first order efficient, in that its difference from the global MLE θmle is negligible compared to θmle − θ * asymptotically.


• Because our goal is to bound the 1 -risk, which is the first-order moment, it is natural to require higher-order moment conditions.

in Assumptions 2 and 3, we introduce second-order moment conditions on inputs and outputs.


Each "trial" taken by a reinforcement learning algorithm shall consist of a T -step sequence in the MDP.

our goal is to characterize the scaling of the sample complexity for various reinforcement learning approaches (i.e., how many trials they require in order to learn a near-optimal policy) for large numbers of agents n. in our bounds below, no serious attempt has been made to make our bounds tight in variables other than n. Below we show that if an RL algorithm uses only the global reward signal, then there exists a very simple MDP-one with horizon, T = 1, only one state/trivial dynamics, and two actions per agent-on which the learning algorithm will require Ω(n) trials to learn a good policy.


Even more evidently, when solving optimization problems, dealing with convex functions ensures that many algorithms will provably converge to the global minimizer [6].

it would be interesting to be able to characterize the set of priors for which the MMSE estimate can be expressed as a minimizer of a convex function.


However, the amount of decrease at the nth time step can be arbitrarily small when Ψ n is close to 1.

we need to show that Ψ n is eventually bounded away from 1, i.e.


First, although the computational complexity of the CRP is low, the complexity of the value iteration algorithm scales linearly with the number of states discovered.

tractability of value iteration is an issue in EB-CRP-PIG.


By setting a = 1/16 and equating the two sides of the equation we have ⌘ 1 4( 1) (note that this also implies a sequence of sets B T by the definition of the functions in Lemma 1).

the semi-distance satisfies =: 2s T . Applying Theorem 4 we have where the final inequality holds since M 2 and a = 1/16.


Thus, we need to compute P (C = 1|T ) : where R(T ) = P (T |C = 2) where is the prior probability of C = 1.

it is equivalent to take decisions by thresholding log R(T ) 1 ; we will elaborate on this in Sec.


The input is a 4k point cloud of a 1m × 1m floor block in the scenes, and each block contains up to 150 instances.


proved by contradiction) that the only possible true labeling of the unlabeled data that is consistent with Table 1 and with the errors of the classifiers is that all the examples are labeled '+'.

we conclude that the majority vote over the "A" rules has zero error, performing significantly better than any of the base rules.


Q-MKL therefore reduces to a form of SVM in which w 2 becomes w T Qw.

in such cases we can reduce the Q-MKL model to a simple QP, which we call Q-SVM .


Comparison with Sample Complexity of Proposed Method: For Ising models, under uniform sampling of observed nodes, we established that the sample complexity of the proposed method scales as n = Ω(∆ 2 ρ −2 (log p) 3 ) for regular graphs with degree ∆.

we nearly match the lower bound on sample complexity in (12).


criterion is fixed across conditions in the model; the differences in error rates arise because of a mismatch between the parameters a and b used to generate trials, and the parameters â and b estimated from the trial sequence.

although the criterion does not change across conditions, and the criterion is expressed in terms of accuracy (Equation 1), the block composition nonetheless affects the speed-accuracy trade off.


However, the key difference is that the evaluation of F (X) in J 2 (X, λ) and J 3 (X, λ 1 , λ 2 ) involves not only a single sample, as in J 1 (x; λ), but also all other samples.

not all the pursuit methods, such as basis pursuit and LARS/Homotopy methods, that are applicable to the standard sparse representation method can be directly applied to optimize J 2 (X, λ) and J 3 (X, λ 1 , λ 2 ).


The results are shown in   11 ⇡ 1, indicating that the dHDP has detected a change in the characteristics of the spikes), where the known neuron goes inactive.

the model is more likely to draw new local clusters at this point, reflecting the nonstationary data.


The resulting likelihood function depends only on the total number of errors, and not on the distances of these errors to the decision boundaries.

rMGPC is expected to be fairly robust when the data contain noise in the labels.


Because the MSE between the exact KL estimator θ * KL and the true parameter θ * is O(N −1 ) as shown in Liu and Ihler (2014), the MSE between θKL and the true To make the MSE between θKL and θ * equal O(N −1 ), as what is achieved by the simple linear averaging, we need draw dn N bootstrap data points in total, which is undesirable since N is often assumed to be very large by the assumption of distributed learning setting (one exception is when the data is distributed due to privacy constraint, in which case N may be relatively small).

it is a critical task to develop more accurate methods that can reduce the noise introduced by the bootstrap process.


Then ω(F ) ≥ Moreover, equality is satisfied if and only if F is an equiangular tight frame.

an ETF minimizes the correlation between its individual elements, making each submatrix S A S A as close to orthogonal as possible, which is promising in light of property (4).


However, the direct use of the classic algorithms can be very costly (Agrawal et al., 2013).

the FASTXML algorithm (Prabhu & Varma, 2014) tackles the problem in a slightly different way.


Furthermore, the same result is true for the case of constant baselines, with A r (y) replaced by an arbitrary constant baseline A r , and A * r (y) replaced by A * r , the optimum constant baseline defined in Theorem 4.

its performance depends on the random variables (∇µ u (y)/µ u (y)) 2 and J β (j); if they are nearly independent, E [J β ] is a good choice.


We can view these partially activated neurons as the proposed samples drawn from the prediction density P (X k+1 |X k ).

let n j k+1|k be the number of proposed samples in j-th sub-population, we have the prediction probability in equation 1 is represented by the expected number of neurons that receive recurrent inputs.


Note that |S| ≤ n always; otherwise the affine minimizer is 0 which either terminates the program or starts a minor cycle which decrements |S|.

the number of minor cycles in a major cycle ≤ n, and it suffices to bound the number of major cycles.


If η is scaling with p, this can be prohibitively large (exponential in λ 2 η).

to have low sample complexity, we must enforce λ = O(1/ √ η).


It has been suggested that different perceptual mechanisms are used to handle resolved and unresolved harmonics [2].

our model employs different methods to segregate resolved and unresolved harmonics of target speech.


Given such a large population, it is not possible to compute unique policy for each agent.

similar to previous work (Varakantham et al., 2012;Nguyen et al., 2017), our goal is to compute a homogenous policy π for all the agents.


For most gene-expression problems [1], a binary factor loadings matrix (A) is inappropriate.

we instead use the Hadamard (element-wise) product of a binary matrix Z and a matrix V of reals.


That is because if 2 → 3 and 3 → 1 exist in a graph G, we can get that r 2 > r 3 , r 3 > r 1 according to the correspondence between graphs and ratings as stated in Section 2.


However, their brute-force learning leads to inexact estimation due to a finite dataset.

an important question is, with a finite dataset, can we leverage a constrained end-to-end model to overcome the above deficiencies?


Recently, several algorithms have been proposed for maximizing the AUC value locally [4] or maximizing some approximations of the global AUC value [9,15], but, in general, these algorithms do not obtain AUC values significantly better than those obtained by an algorithm designed to minimize the error rates.

it is important to determine the relationship between the AUC values and the error rate.


The problem of submodular minimization cannot be approximated with a 1/2 o(1) additive approximation given poly(n) samples from any distribution D. Proof.

for any set S ✓ N , observe that the number g + b of elements in S that are either good or bad is the same for any two functions f i , f j 2 F M and for any F M . by Lemma 4, we obtain the indistinguishability condition.


In our experiments, we exploit the fact that value functions in RL are Lyapunov functions if the costs are strictly positive away from the origin.

this follows directly from the definition of the value function, where v we can obtain Lyapunov candidates as a by-product of approximate dynamic programming.


Let G i be the set of neurons whose incoming weight vectors lie over the line L i .


When ι is equal to 1/t, this FIM Fs,a (θ, ι) is equivalent to the FIM of the t time-steps joint distribution from the stationary state distribution d θ (s) on M(θ) [12].

this interpolation controlled by ι can be interpreted as a continuous interpolation with respect to the time-steps of the joint distribution, so that ι : 1 → 0 is inversely proportional to t : 1 → ∞.