D estimations of univariate Beta distributions without loss of accuracy.

even with independent features, the unsupervised identification of image clusters based on high-dimensional descriptors remains a hard problem due to the omnipresence of noisy, redundant and uninformative features [6] that degrade the accuracy of the modeling and prediction.


One further limitation of the thresholding model is that it does not take into account the fact that the likelihood of a new shot transition is dependent on how much time has elapsed since the previous one.

the statistical formulation can easily incorporate the shot duration models developed in section 2.


Such Random Fourier Features have been used to approximate different types of positive-definite shift-invariant kernels, including the Gaussian kernel, the Laplacian kernel, and the Cauchy kernel.

they have not yet been applied to polynomial kernels, because this class of kernels does not satisfy the positive-definiteness prerequisite for the application of Bochner's theorem.


Note that the new problem ( 6) is non-convex because X(G) does not form a convex set on X .

if we fix G except for G k , it becomes convex with respect to G k . Combining with the random projection, we obtain the following minimization problem: We solve this by the ADMM method for each k = 1, . . . , K. Let g k 2 R I k R k 1 R k be the vectorization of G k , and W k 0 2 R D1⇥D2 be a matrix for the randomly projected matrix.


It is important to note that the worst case iteration bound in Lemma 3.2 does not seem so appealing for our purposes, since it depends polynomially on 1/✏, and in our online algorithms naturally we will need to take ✏ = O(T c ) for some c > 0, which seems to contradict our goal of achieving poly-logarithmic in T oracle complexity, at least on average.

as Lemma 3.2 shows, the more iterations Algorithm 2 performs, the closer it brings its final iterate to the set CH(↵K).


We devote Section 3 to their analysis, which is of independent interest.


Furthermore, we cannot determine directionality by the strategy of the power PC theory, as we do not necessarily have the correct t:JJ sign during the short run.

changing the classification of C i from generative to preventive (or vice versa) requires only removing from (adding to) the estimate (i) the Vi term; and (ii) all terms in which Vi was the only generative factor.


Moreover, the algorithm is essentially optimal in the general case, as the input has size Ω n n k when parent sets of size at most k are allowed.

in practice one often assumes that a node can have at most, say, 2 or 3 parents.


Often low-level pixel operations (such as downsampling an image) can be performed in parallel by dedicated hardware (on a video capture board, for example).

searching each subimage patch of the whole image to test whether it is the object in question takes time that is proportional to the number of pixels.


For most of the instances that CODE2INV fails to solve, we observe that the loop invariant can be expressed in a compact disjunctive normal form (DNF) representation, which is more suited for the decision tree learning approach with hand-crafted features.

cODE2INV is designed to produce loop invariants in the conjunctive normal form (CNF).


It is a special case of K-fold cross-validation where the training blocks are mutually independent since they do not overlap.

this independence does not modify the structure of e in the sense that γ is not null.


1 we highlighted the fact that for any value of n, it is always possible to design an environment which leads MV-LCB to suffer a constant regret.

the previous bound for ExpExp is distribution independent and indicates the regret is still a decreasing function of n even in the worst case.


In particular, in their model, the improved decoding accuracy after combining input cues (i.e., the decreased uncertainty given by Eq.3) depends on the linear response nature of neurons, a feature in accordance with the statistics of Poisson spiking train.

it is unclear how well this result can be extended to non-Poisson statistics.


The resulting learning rule is similar to previous models of the experimentally observed STDP as [9,13,16].

this mechanism will effectively serve learning only if the postsynaptic spikes co-occur with the presentation of a rare pattern.


Because there are no position or trajectory tracking control in RunBot, it is impossible to control its walking speed directly or explicitly.

knowing that runBot's walking gait is determined by only two parameters, Θ ES,h and G M,h (Fig.


Lu [17] solved the dual problem (2), which is a smooth problem, by Nesterov's algorithm, and improved the iteration complexity to O(1/ √ ϵ).

since the practical performance of this algorithm was not attractive, Lu gave a variant (VSM) of it that exhibited better performance.


Provably accurate solutions for PCA have been in use for a long time.

to the best of our knowledge, all existing theoretical guarantees for it assume that the data and the corrupting noise are mutually independent, or at least uncorrelated.


With these problem settings in place, our goal is to estimate the CT-VF in an RKHS and improve policies.

since the output V (x) is unobservable and the so-called double-sampling problem exists when approximating VFs (see e.g., [38,16]), kernel-based supervised learning and its analysis cannot be directly applied to VF approximation in general.


A crucial step in probabilistic inference is to compute the log partition function of the distribution based on these potential functions for a given graph structure.

computing the log partition function either exactly or approximately is NP-hard in general [2,17].


Instead, works [8,10,12,9,20] propose to integrate CRF into the networks, so as to enable end-to-end training of the joint model.

this integration may lead to a dramatic increase of parameters and computing complexity that the model usually needs many iterations of mean-field inference or a recurrent scheme [12] to optimize the CNN-CRF models.


Most related is the work of [AR15] who analyze a version of the variational inference updates when documents are long.

they require strong assumptions on both the warm start, and the amount of "non-overlapping" of the topics in the topic-word matrix.


In this section we compare the performance of GroupMiR with GenMiR++ [8], which is one of the popular methods for predicting miRNA-mRNA interactions.

unlike our method it does not use grouping of mRNAs and attempts to predict each one separately.


1 follows from applying this theorem to marginal inference.

notice that this does not enforce the constraint that µ ≥ 0.


As we added more views that were available for the Caltech-101 datasets, we found that the performance of the pairwise approach consistently went up as we added the third and the fourth view.

the performance of the centroid-based approach slightly got worse upon adding the third view (possibly due to the view being noisy which affected the learned U * ); however addition of the fourth view brought the performance almost close to that of the pairwise case.


There are several approaches to solving this problem, such as generic convex optimization solvers (e.g., QCQP solver) as well as gradient descent using iterative projections [10].

generic convex optimization solvers are too slow to be applicable to this problem, and gradient descent using iterative projections often shows slow convergence.


Fast KDE techniques have guaranteed error bounds.

if these techiques are used inside iterative schemes based on orthogonalization of the Krylov subspace, there is a danger that the errors might grow over iterations.


Equation 1 can be explained as follows: if the m th feature is irrelevant to the cluster c (i.e., ω c,m = 0), then φ c,m will be sampled from a Dirichlet distribution with a uniform prior λ.

if ω c,m = 1, then φ c,m has a prior of λ + µ on that feature value which matches the prototype's feature value, and a prior of just λ on all the other possible feature values.


Due to our large sample size, s > n, even writing down G above requires ⌦(nd) time.

to form Z we do not need G itself: it suffices to compute for m = 1, ..., s the column z(⌘ m ) with z(⌘ m ) j = e 2⇡i⌘ T m aj .


Reinforcement learning [8] is widely studied because of its promise to automatically generate controllers for difficult tasks from attempts to do the task.

reinforcement learning requires a great deal of training data and computational resources, and sometimes fails to learn high dimensional tasks.


Because reconstruction error can be considered to be an auxiliary task, our analysis resembles this auxiliary-task analysis.

there are key differences, as the result by Liu et al.


First of all, MKL-LDA achieves a considerable performance gain of 14.6% over the best recognition rate by the seven KFD classifiers.

while KFD-Voting and KFD-SAMME try to combine the separately trained KFD classifiers, MKL-LDA jointly integrates the seven kernels into the learning process.


For link creation event, the triplet means that destination node u creates at time t a link to source node s, i.e., from time t on, node u starts following node s. To ease the exposition, we restrict ourselves to the case where links cannot be deleted and thus each (directed) link is created only once.

our model can be easily augmented to consider multiple link creations and deletions per node pair, as discussed in Section 8.


9 and 11 are valid for any transformation T . I v/.

we may expect some transformations to perform better than others, in terms of the variance of the resulting estimator.


The accuracy of our algorithm is slightly worse than SVMs'.

our algorithm is quite faster and easier to implement than previous implementations of SVMs, such as those given in [17,5] .


Traditionally, POMDP planning has focused on small problems that have few states or can be neatly factorised into a compact representation.

real-world problems are often large and messy, with enormous state spaces and probability distributions that cannot be conveniently factorised.


Since linear programming relaxations are used for solving the MILP problem, any MILP formulation can be used to provide approximate solutions and error estimates in an anytime fashion (i.e., the method can be stopped at any time during the computation with a feasible solution whose quality monotonically improves with time).

the MILP formulations (both ours and that of Parviainen et al.


This is by no means a comprehensive test, as it can be met by a null model that simply predicts average values of label occurrences without regard to the input features.

it is a surprisingly useful diagnostic, and changes in metrics such as this are often indicative of an issue that requires attention.


The approaches in [12,14,10] can be thought of as following a global sampling strategy, where multiple samples are drawn from q v (x) and then variance reduction is built a posteriori in a subsequent stage through the use of control variates.

local expectation gradients reduce variance by directly changing the sampling strategy, so that instead of working with a global set of samples drawn from q v (x), the strategy now is to exactly marginalize out the random variable that has the largest influence on a specific gradient of interest while using a single sample for the remaining random variables.


In that sense, it is a conservative gradient.

until now, we supposed we had access to the true distribution p. this is usually not the case and, in general, the distribution p is only known through the samples of the training set.


For synchronous SGD, if one machine fails, the entire training process is delayed; whereas for asynchronous SGD, if one machine in a model replica fails, the other model replicas continue processing their training data and updating the model parameters via the parameter servers.

the multiple forms of asynchronous processing in Downpour SGD introduce a great deal of additional stochasticity in the optimization procedure.


This model achieves the lower bound (7) for the training set.

the larger the set of projections, the more values the function g takes, and thus the problem of selecting the correct projection becomes more difficult.


For a comparable amount Θ(|V |) of queried labels, Theorem 4 offers the larger optimality factor |V | 1/3 .

we can get a constant optimality factor by increasing the query set size to O(|V | 3/2 ).


Let us also mention [22], where the Fused Lasso is applied in a similar yet different way to perform hot-spot detection.

this approach includes an additional penalty, penalizing departures from the overall mean of the observations, and should thus rather be considered as an outlier detection method.


Therefore, the prediction may succeed or fail.

the network will eventually learn to yield higher values when the FSM is at an accepting state than when it is not.


In particular, hLinUCB extends linear contextual bandit by inclusion of hidden features and operates under a UCB-like strategy.

it still treats click as direct feedback, but aims at learning more expressive features to describe the observed clicks.


Then, given possibly noisy observations and the prior distribution, we can do Bayesian posterior inference and construct acquisition functions [29,38,2] to search for the function optimizer.

in practice, we do not know the prior and it must be estimated.


We fixed the elements of w 1 and b 1 to zero to avoid the well-known indeterminacy of the softmax function.

the true posteriors of the segment labels can be written, by the Bayes rule, as where p(C t = τ ) is a prior distribution of the segment label τ , and Assume that the feature extractor has a universal approximation capacity (in the sense of well-known neural network theory), and that the amount of data is infinite, so that the MLR converges to the optimal classifier.


Singular values of a data in a matrix form provide insights on the structure of the data, the effective dimensionality, and the choice of hyper-parameters on higher-level data analysis tools.

in many practical applications such as collaborative filtering and network analysis, we only get a partial observation.


This bounded behavior is characteristic of DBM, and well captured by appropriately parameterized EXP.

like EXP, RL has an element of exponential discounting if we write RL has no analytic setting for its free parameter , and parameter fitting yields a discount behavior different from DBM and EXP (Fig.


Notably, self-taught learning [13] performs coding and classification in two separate stages.

in our method, the construction of a good data representation is combined with the training of a maximum margin classifier under a unified framework.


The above algorithms that satisfy the span condition all have known upper complexity bounds of O((n + κ) ln(1/ )), and hence for κ = O(n) we have a sharp convergence rate.

it turns out that the span assumption is an obstacle to faster convergence when n κ (at least for sufficiently high dimension).


Introducing second order indicator features shows improvement in 10 out of the 12 combinations and degrades performance in two of the combinations.

the overall effect is small, with an average improvement of 0.62 in F1 score.


Note that the sorting order is similar between algorithm 1 (User Average) and 2 (Item Average).

algorithm 8 (NLPMF) has opposite order, which will be weighted higher in different part of the data, compared to algorithm 1 and 2. hypothesis RM SE F EAT < RM SE F W LS with paired t-test.


As a result, we get that for any algorithm with any partial information feedback model (where b bits are extracted from each d-dimensional loss vector), it is impossible to get regret lower than Ω( (d/b)T ) for sufficiently large T . Without further assumptions on the feedback model, the bound is optimal up to log-factors, as shown by O( (d/b)T ) upper bounds for linear or coordinate measurements (where b is the number of measurements or coordinates seen 2 ) [2,19,25].

the lower bound extends beyond these specific settings, and include cases such as arbitrary nonlinear measurements of the loss vector, or receiving feedback signals of bounded size (although some setting-specific lower bounds may be stronger).


Such high sample complexity comes from the tensor concentration bound.

we find the problem of tensor eigendecomposition in MLR can be reduced to R K⇥K⇥K space such that the sample complexity and computational complexity are O(poly(K)).


Like boosting, each weak learner (i.e., each weak tree) contributes a small amount to the overall model.

our procedure is defined by a statistical model: a prior and a likelihood, while boosting is defined by an algorithm.


The epoch SG method uses averaging to obtain the O(1/k) rate even for non-smooth objectives [15].

the convergence rates of these averaging methods remain sublinear.


Hence, one may speculate that the system is designed in such a way that each synapse should have an equal influence on the postsynaptic neuron when it receives its optimal spike train.

this effect is most evident for a spiking frequency f = NIT of 10 Hz and vanishes for higher frequencies.


The RMC achieved similar performance when the difficulty of the task was increased by using 32-dimensional vectors, placing a greater demand on high-fidelity memory storage.

this performance was less robust with only a small number of seeds/model configurations demonstrating this performance, in contrast to the 16-dimensional vector case where most model configurations succeeded.


We assume that once an event occurs at node i under the influence of a particular parent j in a cascade, the same event will not happen again.

in Figure 2(b), node e is susceptible given its parent a, b, c and d. only node d is the first parent who infects node e. Because each parent could be equally likely to first influence node i, the likelihood is just a simple sum over the likelihoods of the mutually disjoint events that node i has survived from the influence of all the other parents except the first parent j, i.e., Event c did not occur at node i.


In order to overcome this difficulty, they suggest approximating problem (18) by which can be solved efficiently using the Chow-Liu algorithm [3].

our preliminary experiments (accuracies not reported) indicate that this approach does not work well for minimizing the potential function Φ(y).


The ideal test of our optimal MM is against the true optimal for the real world, which is intractable.

if we find that the theoretical value function for the optimal MM in the Gaussian world matches the realized value function in the real world, then we have strong, though not necessarily conclusive, evidence for two conclusions: (1) The Gaussian world is a good approximation to the real world, otherwise the realized and theoretical value functions would not coincide; (2) Since the two worlds are nearly the same, the optimal MM in the Gaussian world should closely match the true optimal.


( 3) is shared by other empirical potential models such as the EAM method [3], and by many size-extensive ML models like the BPNN method [4].

these approaches differ in the representation of E i .


Recent advances that exploit the capacity and flexibility in this approach have expanded what kinds of models can be trained.

as a proposal for the posterior, the capacity of the recognition network is limited, which can constrain the representational power of the generative model and increase the variance of Monte Carlo estimates.


To compare the generation ÎB1 with the target image I B , we adopt L1 distance as the generation loss of stage-I.

since we only have a condition image and a target pose as input, it is difficult for the model to generate what the background would look like if the target image has a different background from the condition image.


The idea of deriving the lower bound is the following: instead of ensuring that the algorithm performs well in the most adversarial environment θ , we consider a set of "bad" environments and make sure that the algorithm performs well on them, where each "bad" environment θ is the most adversarial one by only perturbing one coordinate θ i of θ.

in order to get meaningful finite-time lower bounds, we need to perturb θ more carefully than in the case of asymptotic lower bounds.


Specifically, for low tree-width graphs, this can be done via a variant of max-product [19].

when finding MAPs is not tractable, it is much less clear how to approximate the M best case.


For this reason, being able to make a trade-off between (a) and (b) to explicitly use the original content image I c at each layer is important.

the update of [11] involves a single parameter γ and is of the form Notice that here the original image I c is used only once at the beginning of the process, and details that have been lost at layer l + 1 have no chance to be recovered at layer l. We present in the experimental section the effect of our variant.


2 for NC, in which the same brain regions in the left and right hemisphere are still connected even at a very large NC.


A recent study on fully-observed VBMF showed that, under a stronger assumption that the two factorized matrices are column-wise independent, the global optimal solution can be analytically computed.

it was not clear how restrictive the column-wise independence assumption is.


Ideally, we would like to use RFs for each component in the SLM.

due to the nature of the SLM, there are difficulties.


Rejection learning [8,10] proposes a solution: allow models to reject (not make a prediction) when they are not confidently accurate.

this approach is inherently nonadaptive: both the model and the decision-maker act independently of one another.


The HRTFs used in our virtual acoustic environment were recorded at a constant distance, so that we could only test the model performance in estimating the azimuth and elevation of a sound source.

in principle, it should also be able to estimate the distance when the source is close.


The main benefit of this approach is that the problem dimension does not grow, and the constraints are enforced everywhere, not pointwise.

it is not obvious how these approaches can be scaled for an arbitrary set of linear operator constraints.


This can clearly be achieved by using the trivial partition Z c i = X i (which achieves d(c)).

in many cases it is also possible to achieve it while using much coarser partitionings.


The Nyström method (and related techniques) can be used to approximate the kernel matrix while permitting faster training [4].

these methods still suffer from the drawback that the model size after n examples is typically O(n).


In the previous section we have shown how to efficiently compute a distribution over symbolic option models H. Recall that the ultimate purpose of H is to compute the success probabilities of plans (see Section 2.2).

thus, the quality of H is determined by the accuracy of its predicted plan success probabilities, and efficiently learning H corresponds to selecting the sequence of observations which maximizes the expected accuracy of H. it is difficult to calculate the expected accuracy of H over all possible plans, so we define a proxy measure to optimize which is intended to represent the amount of uncertainty in H. In this section, we introduce our proxy measure, followed by an algorithm for finding the exploration policy which optimizes it.


As we described earlier, such a biased representation for the high variance components is observed in our model (Figure 3b).

the distribution of the spatial-frequency tuning of the analysis vectors does not correspond to this trend; instead, it is bell-shaped (Figure 4b).


In sum, we can already see that correlated inputs imply nonlinear integration in the postsynaptic neuron, and that the form of nonlinearity needs to be matched to the degree and sign of correlations between inputs.

the finding that supralinear interactions are only expected from anticorrelated inputs defeats biological intuition.


Naturally, the quality of a generative model must be measured in terms of the quality of its outputs.

we also put special emphasis on investigating whether the stochasticity inherent in our model operates as advertised.


We note here that similar findings have been reported previously in [3].

the previous study was based on a smaller number of participants and did not include eye tracking, but instead trained models with all visible objects in the first-person scene, a less realistic approach.


examples from group-1 into the negative class, the Preferred impact classifier only incurs the former type of misclassifications.

the outcomes of the Preferred impact classifier do not satisfy the preferred treatment criterion: group-1 would attain higher benefit if it used the classifier of group-0 (0.96 as compared to 0.86).


As we would expect the squared error in the Simple Monte Carlo estimate decreases as ¢ `£ where £ is the sample size.

for more than about 10 samples, the BMC estimate improves at a much higher rate.


The classical approach to establish EEG-based control is to set up a system that is controlled by a specific EEG feature which is known to be susceptible to conditioning and to let the subjects learn the voluntary control of that feature.

the Berlin Brain-Computer Interface (BBCI) uses well established motor competences in control paradigms and a machine learning approach to extract subject-specific discriminability patterns from high-dimensional features.


Although β2X2 is strongly associated with Z, it is not influential enough to be a proxy.

0.7β1X1 + β2X2 is a component that exceeds both the association and the influence thresholds, so it is a proxy and may be disallowed.


Ideally, the variance should start diverging from x = 0, which is what happens in NPN.

pBP and BDK are not sensitive enough to capture this dispersion change.


After pruning the parameter, Θ l [q] , with the smallest sensitivity, the parameter vector is updated via Θl = Θ l +δΘ l . With Lemma 3.1 and (6), we have that the layer-wise error for layer l is bounded by Note that first equality is obtained because of the fact that E(Z l ) = 0.

it is worth to mention that though we merely focus on layer l, the Hessian matrix is still a square matrix with size of m l−1 m l × m l−1 m l . we will show how to significantly reduce the computation of H −1 l for each layer in Section 3.4.


Hence it is always possible to activate only a single neuron at an asymptotically stable fixed point.

as will become clear from the following Theorem, not all nonnegative vectors can be realized as an asymptotically stable fixed point.


In practice, we minimize the sample divergence of the two distributions, instead of the population divergence as analyzed in the above theorem.

these standard divergences are known to not generalize in training GANs [3].


Although the overall approximation in the thought experiment looks simple, the argument is based on an over simplified assumption that θ i * ,j , θ i,j ≈ π 2 for i = j.

when W * has constant spectral norm, even when W is very close to W * , θ i,j * could be constantly far away from π 2 , which prevents us from applying this approximation directly.


To prove the second statement, suppose x1 = z1 + ȳ1 and x2 = z2 + ȳ2 are both optimal.

let ∆x = x1 − x2 and ∆z = z1 since f (x) satisfies CNSC-T , by (8), for some m > 0.


It turns out that this is possible as soon as the signal-to-noise ratio β becomes larger than C √ k log k (and in particular β can remain bounded as the problem dimensions increase).

we analyze several polynomial-time estimation algorithms, based on tensor unfolding, power iteration and message passing ideas from graphical models.


[21] proposed a boosted deep belief network for facial expression recognition, where each weak classifier is learned exclusively from an image patch.

weak classifiers are selected from an FC layer in the proposed IB-CNN and thus, learned from the whole face.


By constraining Ω, we effectively regularize the ML estimate by imposing a model.

as with any model-based approach, the key is to select a feasible set, Ω, which is as small as possible while still accurately modeling the behavior of the data.


The other algorithms have worse or no known convergence rates.

such margin-maximizing algorithms are of limited interest for a practitioner working with noisy real-world data sets, as overfitting is even more problematic for such algorithms than for the original AdaBoost algorithm [1,8].


However, it often assigns non-negligible probability mass to non-expert actions when the number of actions increases [3,8].

as the optimal policy of the proposed method can adjust its supporting set, it can model various expert's behavior from a uni-modal distribution to a multi-modal distribution.


We find that the entropy of gMedLDA and iMedLDA are smaller than those of gLDA and fLDA, especially for (relatively) large K. This implies that sampling methods for MedLDA can effectively concentrate the probability mass on just several topics thus discover more predictive topic representations.

fMedLDA yields the smallest entropy, which is mainly because the fully-factorized variational methods tend to get too compact results, e.g., sparse local optimums.


R such that given a feature vector x with an unknown label y, the corresponding classifier predicts ŷ = sign(✓(x)).

this mapping function also needs to be fair with respect to the values of a user sensitive attribute z 2 Z ✓ Z 0 (e.g., sex, race), which are drawn from an (unknown) distribution f (z) and may be dependent of the feature vectors and class labels, i.e., Given the above problem setting, we introduce the following quality metrics, which we will use to define and compare different fairness notions: I.


The canonical example for a Q-regular measure is the Lebesgue measure λ on R n . If d is the metric induced by the Euclidean norm, then Q = n and the bound (12) is tight with c 0 = C 0 , a dimensional constant.

for general sets S ⊂ R n , λ need not be locally Q-regular.


[7] observed empirically that Verlet integration is stable and suggested that the stability relates to the symplectic property of the Verlet integration.

in our proof, we found that the order condition of Verlet integration would suffice to achieve acceleration.


To overcome the bias towards linear behavior, more sophisticated activation functions have been designed (Clevert et al., 2015;He et al., 2015;Klambauer et al., 2017;Dauphin et al., 2017).

these still limit all non-linearity to sit in the activation function.


The quality of these suggestions is crucial to the overall success of a webshop.

suggesting the right items is a highly nontrivial task: (1) There are many items to choose from.


Steinwart and Christmann [2009] proved fast learning rates for regularized algorithms with ↵-mixing stationary sequences and Modha and Masry [1998] gave guarantees for certain classes of models under the same assumptions.

stationarity and mixing are often not valid assumptions.


On the individual relations ("single"), the LFRM with random initialization also does well, beating the IRM initialization on both datasets.

the random initialization does poorly at inferring the global features due to the coupling of features and the weights for each of the relations.


Remark 2 In nonmetric MDS we require that dij ~ f(8ij ) for some monotonic function f. If the kernel function r is monotonically decreasing then clearly 1r is monotonically increasing.

there are valid isotropic kernel (covariance) functions which are non-monotonic (e.g.


Thus, the RLS-based ESN online learning algorithm is typically more expensive than standard techniques.

this drawback might be compensated by the following properties of RLS-ESN: • Simplicity of design and implementation; robust behavior with little need for learning parameter hand-tuning.


It is almost omnipotent: it knows the world θ * , the learner's hypothesis space Θ, and importantly how the learner learns given any training data.

1 it can only teach the learner by providing teaching (or, from the learner's perspective, training) examples.


We removed starting and ending silences before scoring in order to be as similar to [5] as possible.

to produce a more informative comparison between our results and results in the literature that do not remove starting and ending silences, we also present the phone error rate of our best model using the more common scoring strategy.


Vertex-exchangeable models such as the Stochastic Block model and its variants, explain the data by means of an underlying latent clustering structure.

such models yield dense graphs [7] and are less appropriate for predicting unseen interactions.


Chernoff bound ensures that the second-order error E S [ e 2 ] ≤ ε 2 is bounded.

first-order error terms are the true bottlenecks.


The one feature in the Birkhoff-based formulations that the permutahedron-based formulations do not have is the ability to average the solution over multiple vectors by choosing p > 1 columns in the matrix Y ∈ R n×p .

our experiments suggested that the best solutions were obtained for p = 1, so this consideration was not important in practice.


The method, however, cannot deal with arbitrary graphs, such as Markov chains with cycles.

while Hidden Markov Models (HMMs) [23,39] and dynamical systems [40,41] have been extensively studied for modeling sequential data, they have not been properly exploited in the context of subset selection.


This controller is statistically inefficient for solving Markov decision problems compared with the normative strategy of building a statistical forward model of the transitions and outcomes, and searching for the optimal action.

episodic control is computationally very straightforward, and therefore does not suffer from any excess uncertainty or noise arising from the severe calculational and search complexities of the forward model.


Our model for control is different in that it is from state-action pair to state-action pair.

rather than building a model for all stateaction pairs, we build only one state-action model that tracks the sequence of greedy actions.


An alternative to Gibbs sampling for Bayesian inference is to use asymptotic approximations to the intractable integrals based on Laplace's method [13,14].

the Laplace approximation requires twice differentiable loglikelihood functions, and cannot be applied to the lasso and elastic net models as they contain a non-differentiable term proportional to the sum of absolute values (i.e.


However, they did not focus on learning the input kernel.

our multiple operator-valued kernel learning formulation can be seen as a way of learning simultaneously input and output kernels, although we consider a linear combination of kernels that are fixed in advance.


In order to get accuracy on par with an oracle the models would have to mimic a function that incorporates: latent concepts, the difficulty of each exercise, the prior distributions of student knowledge and the increase in concept skill that happened after each exercise.

the BKT prediction degraded substantially as the number of hidden concepts increased as it doesn't have a mechanism to learn unlabelled concepts.


Similar to batch normalization, evolutional dropout can also address the internal covariate shift issue by adapting the sampling probabilities to the evolving distribution of layers' outputs.

different from batch normalization, evolutional dropout is a randomized technique, which enjoys many benefits as standard dropout including (i) the back-propagation is simple to implement (just multiplying the gradient of X l by the dropout mask to get the gradient of X l ); (ii) the inference (i.e., testing) remains the same 2 ; (iii) it is equivalent to a data-dependent regularizer with a clear mathematical explanation; (iv) it prevents units from co-adapting of neurons, which facilitate generalization.


We did not find one single set of hyperparameter for which the EKFAC optimization curve is below KFAC for all the epochs (and vice-versa).

doing a per-epoch model selection shows that the best EKFAC configuration usually outperforms the best KFAC for any chosen target epoch.


Similarly to recent trends, we rely on neural networks guided by cross-modal video analysis.

instead of only separating human speakers [44] or musical instruments [47], we aim to separate multiple unidentified types of sound sources.


As we will see in the next section, the constraints may sometimes help resolve this issue as, often, the corresponding logic rules do not exhibit that kind of symmetry.

for cases where that symmetry exists, we can resolve it by assuming that most of the function approximations have error rates better than chance (i.e., < 0.5).


( 2) using the set of linear monomials of observables as basis functions.

because the above algorithm works with operations directly in the functional space, the Koopman operator defined in Eq.


Much larger experiments need to be performed to establish the utility of the color change model for particular applications.

since the color change model represents a compact description of lighting changes, including nonlinearities, we are optimistic about these applications.


In recent work on the NormalHedge algorithm [4] the authors use a potential function which is very similar to our function g(x) (see (2) below), getting strong regret guarantees to the -quantile of best experts.

the use of the function g(x) seems to be quite different from ours, as is the focus of the paper [4].


In order to further improve the model accuracy, researchers always build deeper internal recurrence by increasing the number of recurrent sequence length, hidden units, or layers.

these methods usually introduce large models resulting in practical hardness, especially on embedded devices due to inferior resources for computation and memory.


With a group structure, one can employ a variety of methods including the group Lasso [19][20][21] and group orthogonal matching pursuit [22].

these methods are designed for either AWGN or logistic outputs.


If the mapping π is a one-to-one mapping, then there is no weight sharing and it corresponds to standard feedforward networks.

weight sharing exists if π is a many-to-one mapping.


This model seems entirely vulnerable to the issue of similar speech fragments but despite our expectations it was competitive on the original test set, reaching 18.7% phoneme error rate (PER).

its performance degraded quickly with longer, concatenated utterances.


Without loss of generality assume that Ψr⊗I + I⊗Ψc = RR for some matrix R. We then qualify that p(Z|Ψr, for all Z and E. For fixed values of Ψ r , Ψ c , and Γ we optimize this quadratic bound to obtain revised estimates for Z and E, noting that exact equality in (15) is possible via the closed-form solution In large practical problems, ( 16) may become expensive to compute directly because of the high dimensional inverse involved.

we may still find the optimum efficiently by an ADMM procedure described in [23].


The complexity is O(T N) per iteration, because updating any of the first T term approximations requires N operations.

some of the EP update operations require to compute integrals which do not have a closed form expression.


the distortion measure d. The bound can also be applied to a generator G ′ obtained by minimizing, e.g., some fdivergence [26] between P X and P G(Z) .

if W d (P X , P G ′ (Z) ) > W d (P X , P G ⋆ (Z) ) (which will generally be the case in practice) then the distortion obtained by using G ′ will asymptotically be larger than that obtained for G ⋆ .


There is a large and well-developed set of competing models that aim to account for accuracy, response time distributions and (sometimes) confidence in decision-making.

besides the effective application of hierarchical Bayesian methods to models that assume response times follow a Weibull distribution [8], most of the inference remains frequentist.


Note that even though F might be infinite dimensional the subspace where the data lies is maximally T -dimensional.

the data typically forms an even smaller subspace in F (cf.


A natural criterion or summary statistic often used to measure the ranking quality of a classifier is the area under an ROC curve (AUC) [8].

1 the objective function optimized by most classification algorithms is the error rate and not the AUC.


Although we approximate Z i by a Gaussian Zi , we are still free to choose a method of computing its mean μi and covariance matrix Σi , which also influences the computation of J = ∂( μi )/∂µ \i .

even if μi and Σi are general functions of µ \i and Σ \i , the derivatives ∂ μi /∂µ \i and ∂ Σi /∂Σ \i must equal the corresponding partial derivatives in (11), and ∂ μi /∂Σ \i and ∂ Σi /∂µ \i must be set to 0.


Now note that if X is an optimal solution to B then d(j, X) ≤ 1 for all points j ∈ C. So there exists a (deterministic) distribution achieving r j = 1.

suppose that A generates a solution X ∈ F k where E[d(j, X)] ≤ cr j ; the set X can also be regarded as a solution to the set cover instance.


Generative classifiers learn a model of the joint probability, p(x, y), of the inputs x and the label y, and make their predictions by using Bayes rule to calculate p(y|x), and then picking the most likely label y.

discriminative classifiers model the posterior p(y|x) directly.


Typically, significant discriminative itemset mining aims to find all feature subsets S for which a statistical association test rejects the null hypothesis, namely Z S ⊥ ⊥ Y , after a rigorous correction for multiple hypothesis testing.

for any feature subset such that Z S ⊥ ⊥ Y but Z S ⊥ ⊥ Y | C, the association between Z S and Y is exclusively mediated by the covariate C, which acts in this case as a confounder creating spurious associations.


[19] achieved vision based autonomous hover and landing.

autonomous flight achievements in other flight regimes have been very limited.


It can achieve the best approximation ratio for many subset selection problems without noise [6,18].

its performance for noisy subset selection was not theoretically analyzed until recently.


For PCA, the latent space looks deceptively good, showing non-intersecting loops.

(1) individual loops do not collect together as they should (for LELVM they do); (2) worse still, the mapping from 2D to pose space yields a poor observation model.


Likewise, a sensitivity analysis between the query variable and the target variables can be performed (Laskey, 1995;Kjrulff & van der Gaag, 2000).

both suffer from a quadratic blowup with respect to the number of queries and targets.


Such methods enjoy a linear convergence rate when the cost function F is µ-strongly-convex and each f i has L-Lipschitz continuous gradients, that is, to achieve an output x which satisfies F (x) − F (x ) ≤ δ, the total number of stochastic gradient evaluations needed is O n + L/µ log 1 δ . Nesterov's acceleration [15,16,17] has also been successfully applied to construct variance-reduced methods which have an accelerated linear-convergence rate [18,19,20,21,22,23,24,25]: It is worth noting that this convergence rate has been shown to be worst-case optimal [21].

all of these algorithms need explicit knowledge of the strong-convexity parameter µ.


When the model is correctly specified, « ÇÄË is an unbiased estimator of « £ .

for misspecified models, « ÇÄË is generally biased even asymptotically if AE Ç Ô ´½µ.


Environment design [Zhang et al., 2009, Zhang andParkes, 2008] is a related problem, where a first agent seeks to modulate the behavior of a second agent.

the interaction between agents occurs through finding a good modification of the second agent's reward function: the AI observes a human performing a task, and uses inverse reinforcement learning [Ng et al., 2000] to estimate the human's reward function.


Furthermore, all tasks are positively correlated (green color) in MRN, implying that all tasks can better reinforce each other.

some of the tasks (D and C) are still negatively correlated (red color) in MTRL, implying these tasks should be drawn far apart and cannot improve with each other.


The VB-based MF method (VBMF) was shown to perform well in experiments, and its theoretical properties have been investigated [15].

the optimization problem of VBMF is non-convex.


Despite ongoing research in RNN architectures, the good old LSTM [4] has been shown to outperform more recent variants [45] on standard language modelling datasets.

such networks do not perform well in NLR tasks such as question answering [16].


Generally, one seeks to optimize the discrimination performance of a classifier.

we are working in a domain where overall discrimination performance is not as critical as performance at a particular point on the ROC curve, and we are not interested in the remainder of the ROC curve.


These methods are all robust to sample-outliers.

some methods were proposed to deal with the intra-sample-outliers (or feature-noises), such as [12,15].


The rationale behind this scheme is that the initial model Â(0) is obtained essentially from an uninformed, ad hoc characterizer, for which one has to expect a large model variation and thus (on the average) a poor Â(0) .

the characterizer Ĉr(1) obtained from the reversed Â(0) is not uninformed any longer but shaped by a reasonable reverse model.


But it still causes underfitting or overfitting if the hyperparameters are badly learned offline.

we propose in this paper an online marginalized particle filter to simultaneously learn the hyperprameters and hidden function values.


In the first row, the baseline model wrongly detects a table.

the classification of the scene into the bathroom class might decrease the probability of such an object class, in favor to classes seen more often in these scenes.


However, the biggest drawback of their model is that the body part priors and the graph structure are explicitly hand crafted.

we learn the prior model and implicitly the structure of the spatial model.


When MFs are provided as input, they can be viewed as a kind of "labeled feature" studied in [7].

instead of a generalized expectation criteria, we use the prior to generate noisy labels for examples.


In the task of image query vs. text database, CRH performs comparably to CMSSH, which is better than CVH.

in the other task, CRH achieves the best performance.


Theorems 1 and 2 require the assumption ∑ n i=1 w 2 i = O(n −2c ).

for some algorithms, the value of c may not be available.


Moreover, for both graphs, CO2 is connected to the solar radiation factor GLO through CH4.

for subregion 33, which corresponds to the north part of Arizona, the estimated graph is quite different.


In fact, it is known that this affinity is efficient to compute for mixtures of Gaussians, multinomials and even hidden Markov models (Jebara et al., 2004).

this permits the affinity metric to efficiently pull together parameters θ m and θ n . for mixture models, there is the presence of hidden variables h in addition to observed variables.


This is similar to the retrospective space-time scan statistic of [16], which also uses time as a third dimension.

that algorithm searches over cylinders rather than hyper-rectangles, and thus cannot detect spatially elongated clusters.


A cascade model typically allows multiple infected nodes to infect their neighbors in parallel, so that infection can happen simultaneously in many parts of the graph.

iNVITE contains a single surfer that is responsible for all the infection via a random walk.


As we will see below, the latent count vectors {m n } N n=1 (which are functions of V and U) provide sufficient statistics for the model parameters.

each element of m n (if the corresponding element in y n is one) is drawn from a truncated Poisson distribution V l,: denotes the l th row of V and Thus we can also write if y ln = 0 then m ln = 0 with probability one (Eq.


In fact, Sun and Luo [SL15] showed that after this initialization the problem is effectively strongly-convex, hence many different optimization procedures can be analyzed by standard techniques from convex optimization.

in practice people typically use a random initialization, which still leads to robust and fast convergence.


Among multimodal neural network models, the long-term recurrent convolutional net [5] is related to our objective because their framework explicitly models the relations between sequential inputs and outputs.

the model is applied to a video description task of creating a sentence for a given short video clip and does not address the generation of multiple sequential sentences.


For regression or classification problems, the latent function values are evaluated at the observed inputs and the likelihood factorizes according to p(y|f ) = N i=1 p(y i |f i ).

for other type of applications, such as modelling latent functions in ordinary differential equations, the above factorization is not applicable.


This technique is quite general, in the sense that we can always augment Ω A with a no-op action and use this technique based on r t to push the no-op actions to the end of a T -length action sequence for a pre-chosen upper bound T .  Executing a plan in a noisy environment may not always result in the goal state being reached.

in the instances where a goal state is indeed reached, the executed action sequence can be used to bootstrap the learning of an optimal policy π(a | s, g), which represents the probability for action a in state s when the goal state to be reached is g. We define optimality in terms of reaching the goal using the shortest path.


In [10] it was mentioned that the computation of Ψ, which makes use of the random matrix A = (A p,i ) p≤P,i≤F , has a complexity O(F P N ).

in the multi-resolution schemes described here, provided that the mother function has compact support (such as the hat functions or the Daubechie wavelets), we can significantly speed up the computation of the matrix Ψ by using a tree-based lazy expansion, i.e.


Consequently, P 1 must be able to do (weakly) better if it knows µ 2 compared to if it just assumes that µ 2 = µ 1 .

this does not tell us how much (if any) improvement we can obtain.


Given that spiking data is discrete and therefore non-Gaussian, this might not seem surprising.

it is important to note that the Gaussian model has free parameters for the singleneuron variability, whereas the conditional variance of the Poisson model is constrained to equal the mean.


In addition to the frame rate down-sampling at the convolutional layer, we apply down-sampling in the recurrent layers also.

word piece model training, especially for AM, demands much more data because there are an increased number of labels.


It can be seen that the preprocessing step indeed helps remove certain outliers with large magnitudes.

when the fraction of outliers increases to λ = 0.5, more outliers with smaller magnitudes than the pre-defined threshold enter the remained samples and increase the error of "LR+P" to be larger than 1.


These approaches include detailed models of the relationship between fluorescence measurements, calcium transients and spiking activity [6] as well as model-free information-theoretic approaches [7].

these studies do not attempt to infer functional connections between non-simultaneously observed neurons.


As shown by the results in Table 2, the proposed CVGPDS performed better than the HMM and VG-PDS.

the classification accuracies in Table 3 are lower than the state-of-the-art phoneme classification results [12][13].


To simplify our notations, we define the parameter µ = µ/τ , which scales the strong convexity parameter µ by 1 τ , where τ is the quadratic growth constant.

in general, the constant Γ which defines the step-size parameter γ t is set to L. we allow Γ to be an arbitrary constant greater than or equal to L to facilitate the introduction of the multi-stage ORDA in the later section.


They aim to jointly learn or incorporate pre-trained CNN potential functions, and then perform inference/prediction using the potentials.

our method here directly learns CNN message estimators for the message passing inference, rather than learning the potentials.


As with many other complex hierarchical Bayesian models, exact inference of the latent variables Í and Î in the BMF model is intractable (ie there is no efficient way to sample exactly from the posterior nor to compute its exact marginals).

as with many other non-parametric Bayesian models, we can employ Markov Chain Monte Carlo (MCMC) methods to create an iterative procedure which, if run for sufficiently long, will produce correct posterior samples.


Disparate impact is addressed by the principles of statistical parity and group fairness [Feldman et al., 2015], which seek similar outcomes for all groups.

individual fairness [Dwork et al., 2012] mandates that similar individuals be treated similarly irrespective of group membership.


w. Establishing this for point losses such as hinge loss is relatively straightforward.

the same becomes non-trivial for non-decomposable loss functions as L t is now the difference of two loss functions, both of which involve Ω (t) data points.


In a recent survey of stochastic algorithms for PCA (Arora et al., 2012), the incremental algorithm was found to perform extremely well in practice-it was the best, in fact, among the compared algorithms.

there exist cases in which it can get stuck at a suboptimal solution.


Then, F (U (t) ) − min In [20], single-output PNs were trained using GECO [26], a greedy algorithm with similar O τ 2 β ǫν 2 guarantees.

gECO is limited to learning infinite vectors (not matrices) and it does not constrain its iterates like we do.


Then, θlinear would correspond to different power means on ŝk , depending on the choice of parameterization, e.g., where only the linear average of ŝk (when θ = σ 2 ) matches the overall empirical variance ŝ and equals the global MLE.

θKL always corresponds to a linear average of ŝk , equaling the global MLE, regardless of the parameterization.


We note that a similar result was obtained in linear space [12].

we will see in the next section that by generalizing the analysis for gradient dominated functions in [25], we are able to greatly improve this dependence.


The success of deep learning models lies in their supervision from large scale annotated datasets [14].

detailed 3D mesh annotations are tedious and time consuming to obtain, thus, large scale annotation of 3D human shapes in realistic video input is currently unavailable.


Compared to running n independent bandit algorithms (which corresponds to A ⊗ being the identity matrix), the bound in the above theorem has an extra term (i,j)∈E u i − u j 2 , which we assume small according to our working assumption.

the bound has also a significantly smaller log determinant ln |M T | on the resulting matrix M T , due to the construction of φ t,k via A −1/2 ⊗ .


For supervised semantic segmentation tasks, cross-entropy is the most common used loss function.

in some previously mentioned models [15,20], another loss for similarity estimation had been applied, where similarity groundtruth are transformed from labels.


Our method shows the highest ARI (0.754) when the number of HMM states is 3, which shows that important dimensions are successfully discovered from the "sea" of nuisance dimensions.

the ARI of K-Means decreases monotonically as the number of HMM states increases, which shows the K-Means is not robust against nuisance dimensions.


These criteria allocate resources with the given task (classification or regression) in mind, yielding improved performance.

under canonical ML each density is trained separately to describe observations rather than optimize classification or regression.


Recently, much work has gone into providing theoretically optimal [5,4,13] LSH.

as noted in [2], these solutions require very sophisticated locality-sensitive hash functions and are mainly impractical.


The parameters t R and γ escap 0 are quite difficult to learn as they occur on the flat section of the trajectory.

the error we made is not crippling as the most important for clinicians is the dynamic along both phases.


These methods have been visited in the context of deep models [32,33,14].

previous methods focused on global encoding of local features, and did not consider the case of a graph representation.


Intuitively, model selection of importance estimation algorithms (such as KMM) is straightforward by cross validation (CV) over the performance of subsequent learning algorithms.

this is highly unreliable since the ordinary CV score is heavily biased under covariate shift-for unbiased estimation of the prediction performance of subsequent learning algorithms, the CV procedure itself needs to be importance-weighted [9].


It may seem surprising that CorEx should perfectly recover structure in diverse domains without using labeled data or prior knowledge.

the patterns discovered are "low-hanging fruit" from the right point of view.


Based on the theoretical investigation, we showed that this method works well for linearly separable problems.

it over-penalizes mis-classification error, leading to loose generalization bounds in the non-separable case.


The gradient is given in an explicit form: 4λW (W T W − I), and can be directly appended to the original gradient w.r.t.

the current weight W . SO (1) is flawed for an obvious reason: the columns of W could possibly be mutually orthogonal, if and only if W is undercomplete (m ≥ n).


Furthermore, they demonstrated that given a noisy tensor, the true low-rank tensor can be recovered under restricted strong convexity assumption [18].

all these algorithms [15] [20] and theoretical results [22] [21] reply on the assumption that the observation noise has a bounded variance σ 2 .


The empirical risk/loss associated with the binary space partitioned classifiers is given by: Our goal is to minimize the empirical error jointly over the family of functions g(•) ∈ G and f i (•) ∈ F . From the above equation, when the partitioning function g(•) is fixed, it is clear how one can view choice of classifiers f 0 (•) and f 1 (•) as ERM problems.

even when f 0 , f 1 are fixed, it is unclear how to view minimization over g ∈ G as an ERM.


They leverage the slowly changing aspect of gradients exponents to minimize the number of exponent updates.

to minimize overflows, they end up requiring conservatively large exponents, leaving the higher bits of mantissas unused and increasing mantissa width.


Studies show that self-reports correspond poorly to communication behavior as recorded by independent observers [3].

we have used wearable sensors and recent advances in speech processing techniques to automatically gather information about conversations: when they occurred, who was involved, and who was speaking when.


Related work: Several constant-time approximation algorithms are known for combinatorial optimization problems such as the max cut problem on dense graphs [8,13], constraint satisfaction problems [1,22], and the vertex cover problem [15,16,25].

as far as we know, no such algorithm is known for continuous optimization problems.


Furthermore, we notice that 1/µ o (s) ≥ τ o (s) ≥ |S o |, suggesting that "long" or "big" options are indeed more difficult to estimate.

∆ µ becomes smaller whenever the transition probabilities under policy π o are supported over a few states (B small) and the rewards are similar within the option (sp(r o ) small).


Like LLC, his "charting" method defines a quadratic cost function and finds the optimal coordination directly and efficiently.

charting is based on a cost function much closer in spirit to the original global coordination model and it instantiates one local model centred on each training point, so its scaling is the same as that of LLE and Isomap.


This is the exact behavior EA converges to for both environments.

while ERL is initially confined by the local minima's strong basin of attraction, it eventually breaks free from it by virtue of its RL components: temporally correlated exploration in the action space and policy gradient-based on experience batches sampled randomly from the replay buffer.


DBSCAN [8], a density-based clustering algorithm, can detect clusters of arbitrary shapes and sizes.

it has trouble with data which have widely varying densities.


Maximum a posteriori (MAP) estimates normally perform well if reasonable guesses are available for the priors of hyperparameters.

maximum likelihood estimates (MLE) may be sensitive to the randomness of the initial data, and normally require a larger number of evaluations to yield appropriate results.


Direct policy search methods have the potential to scale gracefully to complex, high-dimensional control tasks [12].

their effectiveness depends on discovering successful executions of the desired task, usually through random exploration.


Bertsekas proposes to go through the data cyclically with a specialized weighting that allows the method to achieve a linear convergence rate for strongly-convex quadratic functions [25].

the weighting is numerically unstable and the linear convergence rate treats full passes through the data as iterations.


This addresses the problem of learning from global rewards; as the critic is defined over the input from all the agents (count tables n in our case).

instead of computing policy gradients directly from the global critic, we compute gradients from a linear approximation to the global critic using first-order Taylor approximation.


These inequalities hold iff the spectral radius is less than 1.

the approximation is achieved only at the cost of inducing a nonlinear distortion of the objective function by a problem-dependent reweighting matrix involving P , which is a variable to be optimized.


To modify the MCU formulation to include these new variables, we replace all Euclidean distance measurements for the data points in the first dataset in (2) with the weighted distance This adds a linear function of the new weight variables to the existing distance constraints of (2).

if we had to define the neighborhood of a data point itself using this weighted distance, the formulation would become non-convex.


It is often said that some knowledge is needed to ask a good question, but critics of our approach will point out that the model begins with substantial domain knowledge and special purpose structures.

many aspects of our grammar are domain general rather than domain specific, including very general functions and programming constructs such as logical connectives, set operations, arithmetic, and mapping.


Hence, each model performs well good only in a local region but poorly elsewhere.

their expertise is complementary, so the overall performance of the ensemble outperforms other baselines.


Optimal Brain Damage [18] and Optimal Brain Surgeon [19] prune networks to reduce the number of connections based on the Hessian of the loss function and suggest that such pruning is more accurate than magnitude-based pruning such as weight decay.

second order derivative needs additional computation.


Since W (x, s) is quadratic in the Y c (x) values, it is often the case, especially for smaller ρ-values in later AL iterations, that D W (t; x, s) is zero over most of [0, w n min (x, s)], simplifying numerical integration.

this has deleterious impacts on search over (x, s), as we discuss in our supplement.


These methods cluster an initial batch of tasks, and upon observing each new task, use the clusters to rapidly learn the corresponding reward function.

they do not address how to update the clusters after observing a new task.


5) that they enable the SDS filter to exactly track the tiger communication world extremely efficiently.

in general, there is no guarantee that these operators will be enough to maintain small sets of trajectories and histories.


Depth bringing a geometric information, the product can be interpreted as a spatial selection.

design of this component has not been fully explored and further experiments should yield better results.


This type of models overwhelmingly rely on the Dirichlet-Multinomial conjugate pair, perhaps mainly because its formulation and estimation is straightforward and efficient.

the ease of parameter estimation may come at a cost: unnecessarily over-complicated latent structures and lack of robustness to limited training data.


If we treat each label independently then standard machine learning procedures could be used to train individual classifiers and this can even be extended to do active learning.

such procedures can be fairly expensive when the number of labels is huge.


To better understand this operator, first notice that in Stochastic Approximation methods such as Algorithm 1, the policy is improved using soft updates with decaying stepsizes.

as Theorem 1 states, monotonic improvement is not guaranteed below a certain stepsize value.


This result is obvious in the sense that smaller values of M produce more errors in the sparse approximation of the covariance matrix.

both noise variance estimates are still different from the true values.


The inference network formed by Stuhlmüller's method inverts the edges of the model as in Figure 1b.

an inference network that is able to represent the true posterior requires extra edges between the branches, as in Figure 1c.


In that work a solution to the dynamics of the on-line algorithm was obtained in closed form for ¢¡ ¤£ " § learning iterations and a simple solution to the asymptotic dynamics under the optimal learning rate decay was obtained.

it was noted there that modelling the dynamics on an ¢¡ ©£ " § timescale is not always appropriate, because the algorithm typically requires much longer in order to escape from a class of metastable states close to the initial conditions.


In the four scenarios, HMC performed well only in ring; in cases where modes are distant from each other, there is little gradient information for HMC to move between modes.

a-NICE-MC is able to freely move between the modes since the NICE proposal is parametrized by a flexible neural network.


Large datasets can enable training of more expressive models, thus leading to higher quality insights.

when the size of such datasets grows to billions of training examples and/or features, the training of even relatively simple models becomes prohibitively time consuming.


This represents a significant advance in our understanding of reinforcement learning with rich observations.


This problem has been studied extensively over decades, yet remains challenging since object appearances involve significant variations that are potentially originated from pose variations, scale changes, occlusion, background clutter, etc.

in spite of such challenges, the techniques based on Deep Neural Network (DNN) demonstrate impressive performance in the standard benchmark datasets such as PASCAL VOC [1].


In particular, the choice of hypothesis class H and learning objective has received a lot of attention (Calders & Verwer, 2010;Zemel et al., 2013;Fish et al., 2016).

data collection-the chosen set of predictive variables X, the sampling distribution p(X, A, Y ), and the training set size n-is an equally integral part of deploying fair machine learning systems in practice, and it should be guided to promote fairness.


In high-dimensional problems where d ≫ n, the sample loss is not strongly convex.

it has been shown by many researchers [3, 13, 1] that under suitable conditions, this objective does satisfy restricted forms of the LSC assumption, allowing us to appeal to a generalized form of Theorem 1.


E.g., in education, a student's rating for a course also depends on whether the course has an inclusive and welcoming learning environment.

addressing this type of bias may require additional information or external interventions beyond the provided rating data.


Patient risk has an inherent temporal aspect; it evolves over time as it is influenced by intrinsic and extrinsic factors.

it has no easily measurable time series.


In addition, Figure 2 shows that GDMF recommends more sparse items than DCPF and is also more efficient, since the static portions aggregate the dynamic portions.

gDMF recommends less than our mGDMF especially on items with sparsity = 0% (i.e., cold-start items).


The cycle-consistency property has been previously leveraged in CycleGAN [12], DiscoGAN [13] and DualGAN [14].

in [12,13,14], cycle-consistency, L Cycle (θ, φ), is implemented via k losses, for k = 1, 2, and real-valued data such as images.


All arguments presented in this paper also go through when considering the top-k problem for k > 1.

to keep the arguments clear and concise, we chose to focus on the best-arm problem only.


The paper [2] contains an analysis of an approximate SVD in combination with an iterative recovery algorithm.

[2] only uses an approximate tail projection, and as a result the approximation ratio c T must be very close to 1 in order to achieve a good sample complexity.


Stochastic Expectation Maximization (sEM) reduces the cost of E-step by stochastic approximation.

sEM has a slower asymptotic convergence rate than batch EM, and requires a decreasing sequence of step sizes, which is difficult to tune.


The GP adapter uses Gaussian process regression to transform the irregular time series data into a uniform representation, allowing sparse and irregularly sampled data to be fed into any black-box classifier learnable using gradient descent while preserving uncertainty.

the 30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.


Conceptually, parameter optimization for the adversarial method forces the adversary to "behave like" certain properties of the training data sample, making labels easier to predict within the minimax prediction game.

a key bottleneck for these methods has been their reliance on zero-sum game solvers for inference, which are computationally expensive relative to inference in other prediction methods, such as SVMs.


It also implies that BALANCE-H and BALANCE-C are also NP-hard.

we will see later that we can obtain approximation guarantees for these maximization problems.


Now, Theorem 1 involves the expected conditional entropy over the data distribution.

during training we only have sample access to the data distribution, which we can use as a surrogate.


[6] advocate the use of the most uncertain sample with respect to the current classifier.

unlike our setting, in active learning the labels of all the samples are not known when the samples are chosen.


(We note that the lower bound from [7] is proved using a construction in which the class probability of positive examples is less than the noise rate; the essence of that proof is to show that in that situation it is impossible to balance the distribution given access to noisy examples.)

having a weak ranker provides enough leverage to yield a two-sided weak classifier without needing any rebalancing.


Among them are techniques that utilize a 3-D CAD model for supervision [44], employ fully convolutional feature learning [5], learn filters with geometrically consistent responses across different object instances [28], learn networks using dense equivariant image labelling [37], exploit local self-similarity within a fully convolutional network [18,19], and estimate correspondences using object proposals [7,8,38].

none of these methods is able to handle non-rigid geometric variations, and most of them are formulated with handcrafted regularization.


For any fixed target accuracy, some choices of yield faster convergence than the joint subspace scheme.

for any fixed optimization eventually hits a noise floor due to the perturbation, while the joint subspace scheme continues to improve.


The key insight is the fact that minimizing per-information set counterfactual regret results in minimizing overall regret.

the vanilla form presented by Zinkevich and colleagues requires the entire game tree to be traversed on each iteration.


When the parameter is well estimated, the greedy algorithm may find the best arm in little time and then leads to small regrets.

the exploration/exploitation tradeoff is not correctly handled by the ǫ-greedy approach causing a large variability in the regret.


[28] considers a more general connectivity structure and obtains sharper bounds for the recovery rates than in this paper.

in [28] the clusters V i cannot overlap and, hence, the results of this paper and [28] are incomparable.


Specifically, the joint distribution is given by the product Probabilistic model in the frequency domain.

rather than employing this product form directly, we work in the frequency domain and use the DFf to write where ii~ is defined in (1).


So, the preprocessed data is If we define E 2 (X) = median(X) and E(P flat ) = E 2 (X), then the breakdown point of E 2 is β 2 = 0.5, and the breakdown points of E is β = β 1 β 2 = 0.225.

if we use another E 2 , then E can be more robust.


In the former work, a meta-loss function is minimized to learn initial hyperparameter configurations.

their method requires manually setting a kernel bandwidth to combine the predictive means of the past models, and an ad hoc procedure for the uncertainty which ignores the predictive variances of the past models.


If the regret scales linearly with the time horizon, it means that the learner does not approach the performance of the best action, that is, the learner fails to learn the problem.

sublinear regret indicates that the disadvantage of the learner compared to the best fixed strategy fades with time.


Therefore, much work has been dedicated to robustifying PCA [12,2,24,6].

all of these methods work in batch mode and cannot handle sequentially revealed samples in the online learning framework.


In principle, MDNs can be applied to multi-dimensional data.

the number of parameters that the network has to output grows quadratically with the number of targets, unless the targets are assumed independent.


Due to limited variation in the pose and large variation in lighting, the mixture of gaussians is unable to extract the mean images.

mixture of product analyzers is able to capture the distributions well.


The strength of the events framework for task specification lies in both its intuitive interpretation and flexibility: though we can obtain similar behavior in standard reinforcement learning, it may require considerable reward tuning and changes to the overall problem statement, including the dynamics.

events provides a single unified framework where the problem parameters remain unchanged, and we simply ask the appropriate queries.


In the following, we assume the class F to be a class of d dimensional spherical Gaussians with identity covariance matrix.

the true model is an equiprobable mixture of perturbed versions of these individual class densities.


If used for PBVI, an efficient direct backup computation of the optimal α-function for belief state b i should be used in line 17 that is linear in the number of observations [5,11] and which obviates the need for lines 23-25.

for an alternate version of approximate value iteration that will often produce more accurate values for belief states other than those in B, one may instead retain the full cross-sum backup of line 17, but omit lines 23-25 -this yields an approximate VI approach (using discretized observations relevant only to a chosen set of belief states B if continuous observations are present) that is not restricted to alpha-functions only optimal at B, hence allowing greater flexibility in approximating the value function over all belief states.


In this problem, a face test sample y is assumed to be represented as a linear combination of training faces in the dictionary X, y = Xβ where β is the coefficient vector used for classification.

it is often the case that the face is occluded by unwanted objects such as glasses, hats etc.


Let us first consider the naive approach of concatenating (q, d) into a single vector and using f (q, d) = w [q, d] as a linear ranking model.

this clearly does not learn anything useful as it would result in the same document ordering for any query, given fixed parameters w. considering a polynomial model: where Φ k (•) is a feature map that considers all possible k-degree terms: does render a useful discriminative model.


. . , S p , the backfitting algorithm is a first order algorithm that guarantees to converge [5] and achieves the minimax rate of convergence as if only estimating a univariate function.

for sparse learning problems, since the true index set is unknown, the backfitting algorithm no longer works due to the uncontrolled estimation variance.


Conceptually, these proposed kernels are defined as instances of convolution kernels [3,11], which provides the concept of kernels over discrete structures.

unfortunately, experiments have shown that in some cases there is a critical issue with convolution kernels in NLP tasks [2,1,10].


Existing defense techniques either harden a DNN model so that it becomes less vulnerable to adversarial samples [5,6,[12][13][14] or detect such samples during operation [15][16][17].

many these techniques work by additionally training on adversarial samples [5,6,12], and hence require prior knowledge of possible attacks.


We use the bootstrap to construct tight data-driven confidence sets.

only some metrics are sufficiently "regular" to be amenable to bootstrap inference, which guides our choice of a suitable metric on trees.


In this paper, we mainly focus on the scenario in which c ≫ k, i.e., we use b-bit minwise hashing or VW for the purpose of data reduction.

in some cases, we care about c ≪ k, because VW is also an excellent tool for compact indexing.


Using the Bayes rule we obtain the posterior distribution p(✓ | D) = p(D | ✓)p(✓)/p(D).

computing posterior distribution using the Bayes rule usually involves computation of intractable integrals, so we need to use approximation techniques.


In comparison, non-parametric approaches are, by design, more flexible and versatile.

in the context of model-free RL with continuous state spaces, the convergence behaviors and finite-sample analysis of non-parametric approaches are less understood.


Similar to our framework, the DCC method achieves the misclassification rate no larger than twice the misclassification rate of the optimum classifier.

computation of the DCC classifier requires solving a non-separable non-smooth optimization problem which is computationally demanding, while the proposed algorithm results in a least squares optimization problem with a closed form solution.


The fundamental philosophy behind triplet loss is the following: for an input (query) example, we desire to shorten the distances between its embedding vector and those of positive examples while enlarging the distances between that of negative examples.

during one update, the triplet loss only compares an example with one negative example while ignoring negative examples from the rest of the classes.


However, the time required for exact verification might cancel out the advantages gained by using an approximation algorithm or a heuristic.

testing whether G is at least close to a k-NN graph will suffice for many purposes.


[3] tested algorithms that learned to treat patients from semi-synthetic observational data.

they both discretized the dosage into three categories, whereas we treat the dosage as a continuous decision variable.


They are rewarded based on the minimum distance of any agent to the target (so only one agent needs to reach the target landmark).

a lone adversary also desires to reach the target landmark; the catch is that the adversary does not know which of the landmarks is the correct one.


storing and searching in such datasets), large datasets are a burden and require more computation.

for learning tasks the situation is radically different.


That is, the edge function describes only the interdependency between the neighbor output labels.

the potential function of CNF at each vertex is different from that of CRF.


More specifically, they measured that cortical pyramidal neurons are the most efficient with a SER 1.2 while pyramidal neurons from the CA1 hippocampus region have a SER 1.6.

inhibitory neurons were found to have less efficient action potentials with cerebellar Purkinje neurons having a SER 2 and cortical basket cell interneurons having a SER 2.


A linear algorithm will have difficulty because the mapped data is not linearly separable.

if we could recover the inverse monotonic function, then a linear decision boundary would perform well.


They tend to trace the object shapes too closely and leave object silhouettes giving away the object class.

our model learns to perform removal by jointly optimizing the mask generator and the in-painter for the removal task with only weak supervision from image-level labels.


The experiments of Section 4.2.1 show that the choice of prior mechanism can be absolutely essential for e cient exploration via randomized value functions.

since the underlying system is a small finite MDP we might observe similar performance through a tabular algorithm.


In this context, each decision is taken by an expert who is typically chosen uniformly at random from a pool of experts.

these decisions may be imperfect due to limited experience, implicit biases, or faulty probabilistic reasoning.


(2010) adapted the greedy coordinate descent (GCD) to solve this convolutional sparse coding problem.

for long signals, these techniques can be quite slow due the computation of the gradient (FISTA, ADMM, L-BFGS) or the choice of the best coordinate to update in GCD, which are operations that scale linearly in T . A way to alleviate this limitation is to use a locally greedy coordinate descent (LGCD) strategy, presented recently in Moreau et al.


1(c)-(e)), we see disjoint sparsity zeros out the unneeded features for the upper-level metric, showed as black squares in the figure (e).

the ToM+Sparsity features are sub-optimal and fit to some noise in the data (d).


Using the reinforcement learning algorithm, the network itself searches for an efficient structure without human assistance.

they only focused on performance without considering the number of parameters.


The most similar to our work, in terms of goals, is [29], where the agent is given multiple suboptimal trajectories, and infers a hidden desired trajectory using Expectation Maximization and Kalman Filtering.

their approach is less general, as it assumes a particular noise model in the expert data, whereas APID is able to handle demonstrations that are sub-optimal non-uniformly along the trajectory.


With increasing noise levels, the number of groups is overestimated.

groupMiR still does very well at a noise level of 0.4 and estimates for the higher noise level are also within a reasonable range.


samples will essentially marginalize over time, simple averaging will lose the temporal/sequential information.

the SRU computes a moving average over time which captures the average of the data seen so far, i.e., when computing µ from ϕ (as shown in Fig.


Alternatively, a straightforward independence chain Metropolis Hastings algorithm (see Supplementary Material) is defined by iteratively proposing A ′ ∼ q which is accepted with probability min{r(A ′ | A), 1} where A is a previous sample of a hierarchical partition and The tailoring of the proposal distribution q to this application based on normalized cuts dramatically aids in improving the acceptance rate relative to more naive tree proposals.

the acceptance rate tends to decrease as higher posterior probability partitions A are discovered, especially for trees with many levels and large input spaces X for which the search space is larger.


The computation time of EVB-ICM grows almost linearly with respect to the number of iterations, and it took 86.6 [sec] for 100 iterations on average.

the computation of our analytic-form solution took only 0.055 [sec] on average, including the single-parameter search for σ 2 .


SSL algorithms based on deep neural networks have recently proven successful on standard benchmark tasks.

we argue that these benchmarks fail to address many issues that SSL algorithms would face in real-world applications.


Therefore, the domain-specific code, v x1 , will be irrelevant in the loss minimization and the learned domain specific code could be meaningless.

with very large values of β, y g carries the domain specific information of the x as well.


Augmented Lagrangian Method (ALM) was invented early in 1969, and since then there have been several works developed Linear Program solver based on ALM [15,16,17].

the challenge of ALM is that it produces a series of bound-constrained quadratic problems that, in the traditional sense, are harder to solve than linear system produced by IPM or Simplex methods [17].


[22] proposed the local intrinsic dimensionality (LID) and empirically showed that the characteristics of test samples can be estimated effectively using the LID.

most prior works on this line typically do not evaluate both OOD and adversarial samples.


In order to label a table, we need a draw from P k , which is obtained by inserting a customer into the corresponding restaurant.

adopting the traditional hierarchical Pitman-Yor process is not straightforward in our case since the size of the context differs between levels of hierarchy, that is, a child restaurant in the hierarchy can have more than one parent restaurant to request a label from.


As a final point of comparison, the actual weight estimate obtained from solving (15) when Φ T Φ = I is equivalent to the non-negative garrote estimator that has been advocated for wavelet shrinkage [5,18].

pSfrag replacements  the benefits the typically non-factorial nature of h * (x 2 ) are most pronounced when m > n, meaning there are more features than the signal dimension y.


A loss in image space leads to averaging all likely locations of details, hence the reconstruction looks blurry.

exact locations of all fine details are not important for perceptual similarity of images.


Location model The location model (LM) [14,10,9] also uses the same factorization as above p(x, b) = p(x|b)p(b).

the distribution for the binary variables p(b) is not of Boltzmann type but a general multinomial distribution and therefore has more degrees of freedom.


K-means is a non-convex procedure and is usually hard to analyze directly.

when the centroids are well-separated, it is possible to come up with sufficient conditions for a node to be correctly clustered.


In this paper we used reinforcement learning to learn the policy.

the bounds offered by the algorithms could also serve directly as metrics to be optimized while learning a policy; this is a research direction we are interested in investigating in future works.


Moreover, the use of spatial-partitioning tree in the algorithm provides stability to the method by clamping the actual maximum error to be within a reasonable rank threshold unlike the distance-approximate method.

note that the proposed algorithm still benefits from the ability of the underlying tree data structure to bound distances.


2, as N s increases, the accuracy becomes lower in all cases because the proposal is a weaker approximation of the true distribution.

at the same time, the complexity decreases allowing us to trade-off accuracy with efficiency.


It still requires solving p linear programs with 2p linear constraints for each.

the estimator from ( 9) has a closed-form solution as long as T ⌫ (S) is invertible.


Standard classification images is performed with perturbing real images with white noise.

this approach may negatively bias the template by the choice of dataset.


Moreover, pruning techniques cannot be applied directly to the resulting hypersurfaces, and even second order measures do not exhibit standard quadratic forms to apply quadratic programming.

convex functions can be efficiently approximated by piecewise linear functions, making it possible to apply the techniques described in Section 3.3 with a bounded error, as long as the approximation of ρ is bounded.


Then, β 1 X 1 + β 2 X 2 would be a subprogram, but β 1 X 1 + β 3 X 3 would not be even though Ŷ could have been computed equivalently as We might attempt to address this by allowing any subset of the terms used in the model to define a subprogram, thus capturing the commutativity and associativity of addition.

this definition still excludes expressions such as β 1 X 1 + 0.5β 3 X 3 , which may be a stronger proxy than either β 1 X 1 or β 1 X 1 + β 3 X 3 .


The idea of combining unsupervised learning with reinforcement learning has already proven useful [1], but the unsupervised features were trained prior to the reinforcement training.

[12] has shown that different systems of this sort could learn concurrently.


Proposed labeling architecture is a feed-forward neural network that can be trained end-to-end.

the recursion makes the depth of the neural network too deep for an efficient joint training.


Whereas the use of the Hessian matrix for second-order methods often provides dramatic speed-ups, a Hessian of (N K) 2 parameters is usually too large to fit in memory (let alone invert) for N > 1000 trials.

we observe that the Hessian of our log-posterior is sparse: where C −1 is a sparse (banded) matrix, and ∂ 2 L/∂w 2 is a block-diagonal matrix.


This emphasizes the arbitrariness of our choice of noise level -the greater the noise, the longer the dominance of episodic control.

in complicated environments, even very small amounts of noise are near catastrophic for model-based control (see brown line in Fig.


It has been shown that adapting a dictionary of basis functions to the statistics of natural images so as to maximize sparsity in the coefficients results in a set of dictionary elements whose spatial properties match those of V1 (primary visual cortex) receptive fields [24].

the basis functions are learned under a probabilistic model where the probability density over the basis functions coefficients is factorial, whereas the sparse coefficients exhibit statistical dependencies [15,16].


2 It has repeatedly been shown that deep neural networks can achieve human-or super-human-level performance on certain supervised learning problems by leveraging large collections of labeled data.

these successes come at a cost: Creating these large datasets typically requires a great deal of human effort (to manually label examples), pain and/or risk (for medical datasets involving invasive tests) or financial expense (to hire labelers or build the infrastructure needed for domain-specific data collection).


Training the localized combination approach requires more computational effort than training the global approach due to the increased size of QP problem in the second step.

the blockdiagonal structure of the Hessian matrix in (OPT8) can be exploited to solve this problem much more efficiently.


For example, decreasing λ 1 improves convergence along all directions, while decreasing any other eigenvalue only speeds up convergence in that direction.

decreasing λ 1 below λ 2 does not help unless λ 2 is decreased as well.


In general, M M F is not necessarily smooth in h since the topological structure of steady-states may change as h varies.

in the next section we show that if there exists a stable and entry-wise non-negative steady-state, and if J is strongly-connected, then M M F (b + h) is both smooth and concave in h, regardless of the interaction strength.


In a time series context a trivial processing order exists, but it is more difficult to find an appropriate order for a general PGM.

in Section 6.3 we show numerically that while the processing order has a big impact on the performance of non-twisted SMC, the effect of the ordering is less severe for twisted SMC.


As can be seen easily, the optimal bandwidth allows the density ridges to capture the underlying structures in every dataset.

the under-smoothing and the over-smoothing does not capture the structure and have a higher risk.


Although we have not come across a previous study of one-sided Huber estimator in this context, we should note that it is related to the technique in [11], where samples are assumed to be nonnegative, and in the sample mean estimator summands are shrunk when they are above a certain threshold (this technique is called winsorizing).

their model and application are quite different than what we consider in this paper.


On one hand, this explains why NH-Boost.DT is faster (besides the reason that it does not require a numerical step).

this also implies that NH-Boost.DT tends to achieve larger margins, since zero weight is assigned to examples with large margin.


This suggests that, when propagation delays are small and there are only a few events, independent consideration of the phase shifts at each time-frequency bin is sufficient to obtain good source masks and location estimates.

as region size grows and more sources are added, the time delays cause different sources to be active at the same time, reducing the effectiveness of this baseline method.


A plausible approach is to observe that at t = 1, the observations are generated by a tree graphical model; thus in principle one could learn the parameters of the underlying tree using existing algorithms [22,21,25].

this approach does not directly produce the HMM parameters; it also does not work for biological sequences because we do not have multiple independent samples at t = 1; instead we have a single long sequence at the steady state, and the steady state distribution of observations is not generated by a latent tree.


Adaptive methods attempt to adjust an algorithm to the geometry of the data.

stochastic gradient descent and related variants use the `2 geometry inherent to the parameter space, and are equivalent to setting H k = I in the adaptive methods.


The solution of ( 7) indeed has typically few elements that are non-zero, and the computational complexity scales with the number of non-zero coefficients [6,7].

a gradient-based method will have a harder time identifying the support of the solution, and therefore the required computations will involve all the coefficients, which is computationally expensive.


The theoretical and experimental results based on benchmarks are still highly consistent.

unlike in Figure 1(b), in Figure 2 only the errors of ĝpu decrease with n u , and the errors of ĝnu just fluctuate randomly.


Because M is a left-stochastic linear map between finite-dimensional spaces, it is a continuous map from the convex and compact S to a convex and compact subset of S that we denote S M . The shadow Dirichlet has two parameters: the generating Dirichlet's parameter α ∈ R d + , and the d × d matrix M . Both α and M can be estimated from data.

as we show in the following subsections, the matrix M can be profitably used as a design parameter that is chosen based on application-specific knowledge or side-information to specify the restricted domain S M , and in that way impose dependency between the components of the random pmfs.


. . = r N = r. Let the recovered tensor A = G ×1 U 1× .

. . ×N U N , the root mean square error (RMSE) is a frequently used measure of the difference between the recovered tensor and the true one: RMSE := 1 √ I N ∥D − A∥ F . [25] analyzes the statistical performance of the convex tensor Schatten 1-norm minimization problem with the general linear operator X : R I1×...×IN → R m . our model ( 6) is non-convex for the LRTC problem with the operator P Ω .


If we take B to be zero, the risk becomes c 2 , which is obtained by estimating all of the means simply by zero.

letting B → ∞, we recover the minimax risk in Pinsker's theorem.


To summarize, the following problem is formulated to find the optimal control policy π: We have formulated the control problem as an optimization in (17).

when control policy π m is to be implemented, only x m is observed and there are still uncertainties in future {x m+1 , .


This motivates the idea of using dimensionality reduction methods to identify the features that affect the neural response [1][2][3][4][5][6][7][8][9].

many neurons in the early visual pathway pool signals from a small population of upstream neurons, each of which integrates and nolinearly transforms the light from a small region of visual space.


Like VEEGAN, all of these methods also use a discriminator D ω (z, x) on the joint (z, x) space.

the VEEGAN objective function O(θ, γ) provides significant benefits over the logistic regression loss over θ and γ that is used in ALI/BiGAN, or the KL-divergence used in LFVI.


The central guiding formulation involves learning a scoring function that recovers the configuration as the highest scoring assignment.

in a game theoretic setting, myopic strategic interactions among players lead to a Nash equilibrium or locally optimal configuration rather than highest scoring global configuration.


AltGrad, RP, ScaledASD, ALT-Impute and OMDL are very fast because they use the simple 2 loss.

their RMSEs are much higher than RMFNL and RMF algorithms.


This method was proposed by Avron and Boutsidis ( 2013), who showed it to be a promising method for column subset selection and its multiple applications.

its wider adoption has been hampered by the lack of polynomial time sampling algorithms.


A similar one to our Result I for ERM was established in [39].

their result requires the convexity condition of random loss functions, making it weaker than our result.


Lion 5 Jaguar Cat Whale 0.8 0.6 Input 1 Input 2 Cat Jaguar Lion Whale Cat Jaguar Cat Lion sampled subset sampled subset sampled subset target Deep learning models have shown impressive performances on classification tasks [17,10,11].

most of the efforts thus far have been made on improving the network architecture, while the predominant choice of the final classification function remained to be the basic softmax regression.


This is still impressive since it allows a dense subset of data points to be corrupted and yet guarantees recovery.

as we shall see below, these results cannot guarantee consistency while allowing k = α • n corruptions.


Figure 3 (supplement) also further emphasizes the utility of Bayesian marginalization versus optimization, even with vague priors.

we do not necessarily expect high fidelity images from any arbitrary generator sampled from the posterior over generators; in fact, such a generator would probably have less posterior probability than the DCGAN, which we show in Section 2 can be viewed as a maximum likelihood analogue of our approach.