It is natural to view imitation learning from the perspective of generative modeling.

unlike in image and audio modeling, in imitation the generation process is constrained by the environment and the agent's actions, with observations becoming accessible through interaction.


Then Note that for fixed , δ, when α → 0 the above bound approaches the familiar O ln( 1 /δ) 2 bound by DKW [12] and Massart [17] for α = 0.

when α → 1 the above bound tends to ∞, reflecting the fact that the multiplicative variant of Glivenko-Cantelli (α = 1) does not hold.


The results in Figure 3 show that, as before, even relatively low a values can still get close to the optimal solution.

semigradient updates, corresponding to a = 0, still find a suboptimal policy.


We note that it is possible to ensure that under some conditions that the dual support method will correctly recover each of the individual signed supports, without any incorrect exclusions.

as illustrated by Theorem 2, doing so requires additional assumptions on the size of the gap Our results thus far show under standard mutual incoherence or irrepresentability conditions, the block 1 / ∞ method produces consistent estimators for n = Ω(s log(p − s)).


In the numerical experiment section, we show that this setting is very good in two out of three experiments.

in some cases, better performances are obtained with a cross-validation of this parameter.


Note that conditioned on {Y d } D d=1 , both the latent matrix Z and the weighting matrices {B d } D d=1 are independent of the observation matrix Gibbs sampler and, in consequence, is the standard method of choice in the context of the standard linear-Gaussian IBP [8].

this algorithm suffers from a high computational cost (being complexity per iteration cubic with the number of data points N ), which is prohibitive when dealing with large databases.


Here, note that both the left and right hand side of the inequalities contain optimization variables.

the objective and constraints in the above problem are also nonconvex and thus we adopt a similar strategy as in the case of preferred impact classifiers.


The above moving-average baseline contributes to reducing the variance of gradient estimates.

it was shown [5,16] that the moving-average baseline is not optimal; the optimal baseline is, by definition, given as the minimizer of the variance of gradient estimates with respect to a baseline.


In conventional CRF approaches, the final prediction is the result of inference based on the learned potentials.

our ultimate goal is the final prediction (not the potentials themselves), so we propose to directly optimize the inference procedure for the final prediction.


We prove that we can use an exact optimization problem (Problem 1) to either identify a proxy if one exists, or definitively conclude that there is no proxy.

because this problem is not convex and in some cases may be intractable, we also present an approximate version of the problem (Problem 2) that sacrifices some precision.


we found that methods which compensate for the dependencies did not yield substantial improvements.


In sampling, the population data is unknown, and cost arises from a data collection process.

the data is known in subsampling and can be exploited to draw better samples.


Systems with this goal have been successfully applied to, for example, synthesizing intricate bitvector routines from specifications [18].

when learning from examples, there is often uncertainty over the correct program.


In the permuted setting, almost all performances are lower.

the DILATEDRNN models maintain very high evaluation accuracies.


The source of this variability is almost always assumed to be intrinsic to the brain.

when inferring a particular stimulus property, variability associated with other stimulus attributes also effectively act as noise.


As we can see, ZO-SVRG-Ave requires more queries than ZO-SGD to achieve the first significant drop in attack loss.

by fixing the total number of queries (10 7 ), ZO-SVRG-Ave eventually converges to a lower loss than ZO-SGD: the former reaches the average loss 4.81 with std 0.32 (computed from the last 100 attack losses), while the latter reaches 6.74 ± 0.46.


Alongside components such as residual blocks and long-short term memory networks, soft attention provides a rich neural network building block for controlling gradient flow and encoding inductive biases.

more so than these other components, which are often treated as black-boxes, researchers use intermediate attention decisions directly as a tool for model interpretability [43,1] or as a factor in final predictions [25,68].


One would expect that the bias distribution is reproduced.

we £nd that a peak exists at ω + Ω α − Ω β = −1.


These results were further boosted by recent work on extra-gradient variants of the algorithm [11,17], and the ergodic relaxation of [8] where the independence assumption on the gradient samples is relaxed and is replaced by a mixing distribution that converges in probability to a well-defined limit.

all these works focus exclusively on the algorithm's ergodic average (also known as timeaverage), a mode of convergence which is strictly weaker than the convergence of the algorithm's last iterate.


[2] for PageRank estimation; our algorithm borrows several ideas and techniques from that work.

the FAST-PPR algorithm relies heavily on the structure of PageRank -in particular, the fact that the PageRank walk has Geometric(α) length (and hence can be stopped and restarted due to the memoryless property).


[2011] focus on the online learning algorithms where they only need unbiased estimates of the gradient of the loss to provide guarantees for learning with noisy data.

they consider a much harder noise model where instances as well as labels are noisy.


This operation is difficult to implement efficiently for large-scale data with many features even if all features are sparse, although important progress was made recently so the complexity is logarithmic in the number of features [5].

the soft-regularization formulation ( 2) is efficient for a batch setting [8] so we pursue it here in an online setting where it has complexity independent of the number of features.


This is the case for the Bernoulli distribution, but not for all exponential family distributions.

for many distributions, such as diagonal covariance Gaussians, even though the score is nonlinear in U, it can be computed by applying the nonlinearity elementwise to U.


For large margin separable problems, we show that exponential decay of generalization error may be achieved with an appropriately chosen regularization parameter.

the bound depends on a quantity which characterizes the robustness of the distribution.


Note that L NDCG is less sensitive to errors in predictions for the less relevant documents in the list.

our selection of thresholds ⇣ i 's implies that the gap between scores of lowerranked documents is very small compared to the higher-ranked ones, and hence chances of making mistakes lower down the list is higher.


Figure 4a  STC model performs significantly better than the subunit model (Figure 4a; < r Rust >= 0.81, < r subunit >= 0.33; p ⌧ 0.005).

this is primarily due to over-fitting: Visual inspection of the STC kernels for most cells reveals very little structure.


High-dimensional time series have been incorporated in many applications of machine learning such as robotics (sensor data), computational biology (gene expression data), computer vision (video sequences), and graphics (motion capture data).

no previous work has considered the GP-based approach for speech recognition tasks that involve high-dimensional time series data.


text-to-image queries by bridging the semantic gap.

it also poses challenges associated with handling cross-view similarity.


For online learning in games, it was shown by Kalai and Vempala (2005) that an optimization oracle giving the best decision in hindsight is sufficient for attaining optimal regret.

in many non-convex settings, such an optimization oracle is either unavailable or NP-hard to compute.


MAML can achieve this because it uses a standard PG learning algorithm to make progress beyond its initialization, and therefore enjoys the generalization property of generic PG methods.

ePG evolves a loss function that trains agents to quickly reach goals sampled from negative x-axis, never seen during metatraining.


However, this is true only if the associated distribution is continuous: then the probability of observing the same X-value twice is zero.

in practice even when the feature space has continuous components, data-sets sometimes contain repetitions (e.g., Adult, see Table 1), if only for the reason that continuous features may be discretized or truncated.


By Proposition 5.2, we know that RNA+SGD will not converge to arbitrary precision because the noise is additive with a non-vanishing variance.

proposition 5.3 predicts an improvement of the convergence at the beginning of the process.


The reason for guessing "y >= 100" instead of "y <= 100" is because part of the proof goal is "y > 2".

this guess will be rejected by the theorem prover.


Recently, multi-agent reinforcement learning has been applied to generalize social dilemma problems to temporally and spatially extended Markov games.

this has not yet generated an agent that learns to cooperate in social dilemmas as humans do.


Since the feedback of actions 2 and 3 are the same, one may also use action 3 in the same manner.

the loss per observation is 1.2 and 1.3 for actions 2 and 3, respectively, and thus it is better to use action 2.


This makes L2-LSTD an attractive choice when the features are observed in the sparse basis.

if the features are sparse in some unknown basis (observation is not sparse), then the time complexity of any linear solver in the observation basis can be prohibitive.


In line with Proposition 1, most methods for ensuring algorithmic fairness reduce discrimination by trading off a difference in noise for one in bias or variance.

this trade-off is only motivated if the considered predictive model is close to Bayes optimal and no additional predictive variables may be measured.


[9], Feige and Ofek [8] on the spectrum of random graphs.

the results of [13], crucially rely on the regularity of Erdős-Rényi graphs and do not extend to sampling graphs with skewed degree distributions even for rank one matrices.


It is important to note that c ∞ ≥ c 2 , which implies then that as long as c ∞ → 0, then c 2 → 0, so that the conditions of Theorem 3(a) and (b) are equivalent.

note that if c 2 does not go to 0, then in fact, the method could fail to recover the correct support even if θ 1,∞ > 1 + δ.


However, even for constant rank r it guarantees correction of only a vanishing fraction o(m 1.5 ) m 2 of errors.

our main result, Theorem 2.4, states that even if r grows proportional to m/ log(m), non-vanishing fractions of errors are corrected with high probability.


However, such models are limited to unpersonalized settings that ignore context, such as recommending a global set of blogs to monitor [14].

methods that use feature-rich models typically either employ unsupervised training [24] or require fine-grained subtopic labels [25].


In the extended version of this paper [13], we show that there is no ESD source model for whose dependencies can be completely eliminated by divisive normalization.

divisive normalization provides a rough approximation to RG, which suggests that RG might provide a more principled justification for normalization-like nonlinear behaviors seen in biological sensory systems.


In the update steps used by ARS V1 and V2 each perturbation direction k is weighted by the difference of the rewards r(⇡ j,k,+ ) and r(⇡ j,k, ).

if r(⇡ j,k,+ ) > r(⇡ j,k, ), ARS pushes the policy weights M j in the direction of k . If r(⇡ j,k,+ ) < r(⇡ j,k, ), ARS pushes the policy weights M j in the direction of k . since r(⇡ j,k,+ ) and r(⇡ j,k, ) are noisy evaluations of the performance of the policies parametrized by M j + ⌫ k and M j ⌫ k , ARS V1 and V2 might push the weights M j in the direction k even when k is better, or vice versa.


To improve the speed of learning decoupled models, we consider the use of natural gradient descent [2].

in particular, we are interested in the update rule for natural parameters, which has empirically demonstrated impressive convergence performance over other choices of parametrizations [29] it is unclear what the natural parameters (5) for the general decoupled basis in (5) are and whether finite-dimensional natural parameters even exist for such a model.


In this case, the information about the underlying density is implicitly contained in unweighted kNN graphs, and, at least in principle, accessible by machine learning algorithms.

in most applications, k is chosen much, much smaller, typically on the order k ≈ log(n).


Since the classification result is determined by the majority vote from 3 nearest instances, the classification loss is defined using the distance to the 2nd nearest instance in each class (which is referred to as a TN; see Section 2 for details).

since 'nearest' instances are generally changed when feature metric/weights are updated, TNs must also be updated to be kept consistent with the learned feature metric/weights during the learning process.


In recent work, determining a good metric for nearest neighbor classification is believed to be crucial.

traditional generative analysis of this problem has simply ignored the metric issue with good reason, as we will see in section 2.2.


Others increased the training set one pairwise comparison at a time but did not keep the pairwise comparisons independent of each other [23].

we increased the training set one pairwise comparison at a time and kept all pairwise comparisons independent of each other.


Further, improved upper bounds which account for sensitive edges were introduced in [16].

there has been little work on finding a tight lower bound for the influence.


Sparse learning with convex regularization has been successfully applied to a wide range of applications including marker genes identification [19], face recognition [22], image restoration [2], text corpora understanding [9] and radar imaging [20].

it has been shown recently that many convex sparse learning formulations are inferior to their non-convex counterparts in both theory and practice [27,12,23,25,16,26,24,11].


We then predict the electrical load over all 24 ) As expected, the RMSE net achieves the lowest RMSE for its predictions.

the task net outperforms the RMSE net on task loss by 38.6%, and the cost-weighted RMSE on task loss by 8.6%.


Analogously, updating the parameters L i and T i in the M-step exhibits a speedup of This improvement is exponential in the number of other agents.

the E-step at level 0 exhibits complexity that is typically greater compared to the total complexity of the E-steps for |M | FSCs.


From this low-rank approximation to H the model parameters A, C as well as the covariances Q and R can be found by linear regression and by solving an algebraic Riccati equation; d is given simply by the empirical mean of the data.

this specific procedure works only for linear systems with Gaussian observations and innovations, and not for models which feature non-linear transformations or non-Gaussian observation models.


Clearly, assuming f H to exist, implies it belongs to a ball of radius R ρ,H .

our results do not require prior knowledge of R ρ,H and hold uniformly over all finite radii.


In our simulations, we just use a uniform distribution.

this distribution could reflect the frequency with which each word occurs in English.


Then, the root of the circuit for a factor evaluates to the value of the factor for that instantiation.

aCs can also represent products of factors.


For strongly-convex and smooth objectives, line search techniques, [4], ensure linear convergence rate, without the knowledge of the smoothness parameter.

line search is not "fully universal", in the sense that it holds no guarantees in the non-smooth case.


We show that the variance of L1 may or may not be larger than L2 in SM; there is no absolute winner.

if q and a follow a Gaussian distribution, then L1 induces smaller variances than L2 for large enough d; see Lemma 9 in SM.


Indeed, if we consider the simple problem of knowing the position of a hidden object, it is possible to solve this without even having seen the object (for instance if all the locations but one have been visited).

the reward of a POMDP cannot model this since it is only based on the current state and action.


Yet, we were still able to obtain orders-of-magnitude speed-ups when decomposing the learning problem.

our proposed decomposition techniques do not apply fully to Markov random fields (MRFs) as the partition function cannot be decomposed, even when the data is complete (evaluating the partition function is independent of the data).


[25] trained a bidirectional LSTM network to transform laser range sensor data into location estimates, but the network was not shown to generalize across environments.

our recurrent network implementation is fully neural and generalizes successfully across environments with very different shapes.


We have chosen the smallest capacitance available in the CMOS process to implement the unit cell for reducing circuits power and area.

it introduces large capacitor mismatch ratios up to 5% which limits the effective resolution to 10-bit or below for previous works reported in the literature [14,19,20].


We then have: is the Gibbs error rate (by definition), we see that the expected margin is just one minus twice the Gibbs error rate.

the error for the Q-weighted majority vote is given by Hence, for large enough β, the sigmoid loss (or tanh loss) of f Q should be very close to the error rate of the Q-weighted majority vote.


Unfortunately, in our setup, we do not know a good kernel in advance.

we will show that a remarkable property of our algorithm is that it enjoys, up to logarithmic factors, a mistake bound of O(1/ 2 ) per task, where is the largest possible margin (over the choice of the kernel) which is achieved on all tasks.


One way to circumvent this problem is to parameterize policies [10,11,14] and then perform optimization with sampling.

in this case one has to impose the structure of the policy a-priori, therefore restrict the possible optimal control solutions within the assumed parameterization.


Our proposed log partition function tracking strategy is applicable to any Gibbs distribution model that is undergoing relatively smooth changes in the partition function.

we concentrate on its application to the RBM since it has become a model of choice for learning unsupervised features for use in deep feed-forward architectures [9,1] as well as for modeling complex, high-dimensional distributions [27,24,12].


It is also interesting to note that the NN3-30K initial model in panel (b), which is trained with more data, presents robustness to a larger level of noise compared to NN2-10K in panel (a).

the retrained models behave rather similarly (blue curves) indicating the saving that can be achieved in the number of training samples via Net-Trim.


They then apply standard trajectory optimization methods on these learned embeddings [Lenz et al., 2015, Watter et al., 2015.

these methods still require a manually specified and hand-tuned cost function, which can become even more difficult in a latent representation.


(NUTS is not available, e.g., in Edward 1.)

eager execution has tradeoffs (not unique to our approach).


While our model presentation assumed latent factors, we could observe factors, such as knowing the journal of each article in CLEP.

our experiments strictly focus on the unsupervised setting to measure what the model can infer on its own.


Current text generation approaches [16] often fail to capture the diversity of such small portion of descriptions, and pure generation pipelines are biased towards generating plausible sentences that look natural by the language model but poor at finding visual groundings [17].

a desirable medical report usually has to not only describe normal and abnormal findings, but also support itself by visual evidences such as location and attributes of the detected findings appearing in the image.


Thus, high bias estimates of the policy gradient, corresponding to smaller λ values make fast initial progress.

after this initial phase, it is important to follow an unbiased gradient, and consequently the low-bias variants corresponding to larger λ values show better asymptotic performance.


In FWBQ, we first select design points {x FW i } n i=1 using the FW algorithm.

when computing the quadrature approximation, instead of using the usual FW weights {w FW i } n i=1 we use instead the weights {w BQ i } n i=1 provided by BQ.


This myopic C-DAC algorithm, by retaining context-sensitivity, was found to nevertheless reproduce critical fixation choice and duration patterns, such as the confirmation bias, seen in human behavior.

exact C-DAC was still better than myopic C-DAC at reproducing human data, leaving room for finding other approximations that explain brain computations even better.


Roughly speaking, when all H i (w)'s are "similar", one has that λ K max ( , and thus κ(w) ≈ κ(w) ≈ κ(w).

in many real applications, such uniformity doesn't simply exist.


Meanwhile, the Spectral algorithm also caused numerical problem in practical.

our method Low Rank uses the same latent structure, but achieved much more robust results.


1(a), where close lines are clustered together.

the EM method accurately clusters the lines into different subspaces (Fig.


Positive coefficients for sender outdegree, reciprocity, and transitivity in these plots imply that users with a high numbers of friends tend to make more friends, tend to reciprocate their relations, and tend to make friends with their friends' friends, respectively.

these coefficients decrease towards zero (the blue line) and enter a second phase where the network is structurally stable.


As marked by filled circles, only synapses in the path of hierarchical synchrony activity develop maximum weight strength.

weights with final minimum strength are indicated by empty circles.


The quality of our prices and value functions will depend on whether it is valid to assume a single price for each resource: if the prices stay constant then our approximate plan will translate perfectly to the physical world.

if we are unlucky, we may find that prices are different than we had planned when we need to buy or sell.


(2010) [9] gave an ℓ ∞,∞ -norm error bound ) as well as a sign consistency result between Ŵ and W . A direct comparison between these two bounds is difficult due to the use of different norms.

the worst-case estimate of the ℓ 2,1 -norm error bound of the algorithm in Jalali et al.


Furthermore, it is assumed that the observed sequence of events are generated adversarially.

in our setting, the buyer behaves self-interestedly, which is not necessarily the same as behaving adversarially, because the interaction between the buyer and seller is not zero-sum.


The resulting network  exhibits both a block diagonal structure, as well as a power-law degree distribution.

the other two models have binomial (i.e.


e. Similarly, the block in the label matrix corresponding to the training samples may not contain 1's and −1's exactly (constraint ( 5)).

the better V is chosen, the better this constraint will be met.


A single wrongly classified edge of the affinity graph leads to an incorrect merger of two segments, causing many pairs of image pixels to be wrongly assigned to the same segment.

the segmentation corresponding to affinity graph #2 has a perfect Rand index, even though there are misclassifications in the affinity graph.


In this formulation, it can be seen that the columns of W (and hence the tasks) are independent of each other.

the tasks are in general not independent.


We can see that SVRG HEB with some underestimated values of T 1 (e.g, 1000, 2000) converge very slowly.

the performance of SVRG QEB-RS is not affected too much by the initial value of T 1 , which is consistent with our theory showing the log dependence on the initial value of c. Moreover, SVRG QEB-RS with different values of T 1 perform always better than their counterparts of SVRG HEB .


hirshnlan et a1 found that the difference in the LF and H'F FA.Rs \¥as not affected b:y 1\1idazolam.


In addition, it means that we cannot use standard minimax theory for Holder classes to establish lower bounds for the estimation error over GTF classes.

as we will see next, we can construct lower bounds for GTF classes via another (more purely geometric) embedding strategy; interestingly, the resulting rates match the Holder rates, suggesting that, while GTF classes do not contain all (discretized) Holder functions, they do contain "enough" of these functions to admit the same lower bound rates.


that characterize the set of selected relevant documents, and use these features in formulating a new query (e.g., [5,6]) .

we cast the problem as a problem of estimation and the goal is to recover the unknown docum ent weights or relevance assessments.


gen1Q tends to make more redundant sentences than countQ because countQ carefully checks the similarity between questions.

gen1Q tends to make the questions more related to the image.


For instance, nurse is moved to to be equally male and female in the direction g. In addition, we find that gender-specific words have additional biases beyond g. For instance, grandmother and grandfather are both closer to wisdom than gal and guy are, which does not reflect a gender difference.

the fact that babysit is so much closer to grandmother than grandfather (more than for other gender pairs) is a gender bias specific to grandmother.


For every correct answer in the 3 gold standard questions, the reward will double.

if any of these questions are answered incorrectly, then the reward will become zero.


Evaluating these gradients is not a requirement of VAEs or standard normalizing flows.

we have shown that in the case of a fairly large generative network we are still able to evaluate gradients and backpropagate through the layers of the flow.


The KL-divergence KL(ψ∥θ) := x p(x; ψ) log p(x;ψ) p(x;θ) is arguably the optimal divergence for marginal inference because it strives to preserve the marginals of p(x; θ) and p(x; ψ).

projection in KL-divergence is intractable here because the evaluation of the gradient ∇ θ KL(ψ∥θ) requires the marginals of distribution ψ.


Improvements include guarantees of convergence, approximations that are provably more accurate, and bounds on the results of exact inference.

extending these methods to continuous-valued systems has lagged behind.


A common paradigm has been category-level recognition, where categories and the similarities of all their instances to other classes are jointly modeled.

large intra-class variability has recently spurred exemplar methods [15,11], which split this problem into simpler sub-tasks.


There is also a very recent paper, [16], which also focuses on regret minimization using approximated Gittins Indices.

in that paper, the time horizon is assumed to be known and fixed, which is different from the focus in this paper on finding a policy that has low regret over all sufficiently long horizons.


This result shows that any POMDP can be encoded as an HGMDP with deterministic dynamics, where the stochastic dynamics of the POMDP are captured via the stochastic agent policy in the HGMDP.

the HGMDPs resulting from the PSPACE-hardness reduction are quite pathological compared to those that are likely to arise in practice.


As the significance threshold σ decreases, the number and density of detections increases, allowing the support tree nodes to capture feasibility information for a large number of support points.

neither the full multiple tree algorithm nor the single-tree algorithm performed well.


Given the appearance vector, the likelihood of each facial feature belonging to a particular character can be found using logistic regression.

the relative locations of the facial features also offer important cues in distinguishing one character from the other (e.g.


It is not difficult to see that this also applies to the case of MSU(x, y, k).


For example, in computer vision, we may have many 'car' images from which we wish to learn a 'car' model.

the exact location of the cars may be unknown and can be modeled as latent variables.


See the long version of this paper [10] for details.

on such instances EC 2 obtains an O(log n) approximation.


It is clear from the graph that trajectories sampled from reversible chains spread much slower, with only 1/5 of the states reached in (a) and 1/3 in (b), and the trajectory in (b) does not escape from the current peak.

with vortices added, trajectories of the same length spread over all the states, and effectively explore both peaks of the stationary distribution in (b).


We did not test PBM-TS [27], which is another algorithm for PBM, mainly because that its regret bound has not been derived yet.

its regret appears to be asymptotically optimal when {κ * l } are known (Figure 1(a) in Lagrée et al.


This suggested that action onset and contents could be decoded from preparatory motor signals in the brain before the subject becomes aware of an intention to move and of the contents of the action.

the readiness potential was computed by averaging over 40 or more trials aligned to movement onset after the fact.


For this scheme to give the right answer, the bottom-up weights should be the transpose of the top-down weights ) P ¥ .

we later consider forms of adaptation that weaken this dependency.


We have to adjust the two learning rates such that the generator does not affect discriminator learning in a undesired way and perturb it too much.

even a larger learning rate for the generator than for the discriminator may ensure that the discriminator has low perturbations.


Hence H satisfies this weak-learning condition.

it can be shown (supplement) that AdaBoost.MH's weak-learning condition is the pair (C MH , B MH γ ), where C MH has non-(positive)negative entries on (in)correct labels, and where each row of the matrix B MH γ is the vector quick calculation shows that for any h ∈ H, and C ∈ C MH with −1 in the first column and zeroes elsewhere, C • 1 h − B MH γ = 1/2 − 1/k.


Recently, Caron & Fox (2014)  proposed the use of a different notion of exchangeability after Kallenberg (2005) and obtained a network model which permits edge-inhomogeneity, such as a power-law degree-distribution whilst retaining desirable statistical properties.

this model does not capture latent vertex traits such as block-structure.


Modern-day robots can be carefully hand-programmed or "scripted" to carry out many complex manipulation tasks, ranging from using tools to assemble complex machinery, to balancing a spinning top on the edge of a sword.

[15] autonomously grasping a previously unknown object still remains a challenging problem.


For attack mitigation, the RNN-based visual-foresight method [13] is practical, considering the predicted policy distance from the chosen policy.

it was reported [13] that such a scheme can be fooled with a series of likely state perturbations.


Our above-mentioned single-task algorithmic testing method can be applied to this multi-task problem by aggregating the deviations for all tasks and thus can only discover the scenario where the algorithm deviates from the true output in an average sense.

when it is important to get the assurance for each task or output, this approach may be insufficient.


Sparse Gaussian process (GP) regression [Titsias, 2009] and the Bayesian Gaussian process latent variable model (GPLVM, Titsias and Lawrence [2010]) have been applied in many tasks, such as regression, density estimation, data imputation, and dimensionality reduction.

the use of these models with big datasets has been limited by the scalability of the inference.


2 -path norm dependence on the number of hidden units in each layer is unavoidable.

it is not clear if a bound that only depends on the product of spectral norms is possible.


While many trackers simply use raw pixels as features, some attempts have used more informative features, such as Haar features, histogram features, and local binary patterns.

these features are all handcrafted offline but not tailor-made for the tracked object.


These images can be quite different, as in the frontal and profile views of a person.

if we continuously change the viewing angle, these 2D images will form a one-dimensional manifold in a very high dimensional image space.


Evolutionary Algorithms (EAs), a class of black box optimization techniques inspired by natural evolution, are well suited to address each of these three challenges.

eAs typically suffer from high sample complexity and struggle to solve problems that require optimization of a large number of parameters.


The bounds of Theorem 3 are similar in form to those of Theorem 2.

the bounds for the ensemble Nyström are tighter than those for any Nyström expert based on a single sample of size m even for a uniform weighting.


As to the expressiveness of such networks, it is easy to see that neural networks of depth 2 and sufficient size can express all functions from {0, 1} d to {0, 1}.

it is also possible to show that for this to happen, the size of the network must be exponential in d (e.g.


In practice, the segmentation converges without this auxiliary stopping condition to a state in which the error corrector fails confidence threshold everywhere.

it is hard to certify convergence since it is possible that the error corrector could give different outputs on slightly shifted windows.


With the parametric form of quadratic splines, the total length and the curvature along a real fragment can be computed by summing over each spline its length and its total curvature as where l i is the length and κ i (t) is the curvature function of the ith spline.

estimating these quantities for a virtual fragment is not trivial, as no information is given on how the virtual fragment should look like.


(b) has no path that visits both 1 and 2, and thus every censored list must be a prefix of a permutation.

(c) has a path that visits both 1 and 2, thus can generate (1,2,3,4), a full permutation.


Both h 1 and h 2 are consistent with examples in Z.

as H is well-structured, mgh(Z) must also be in the version space M(Z), and consequently in H. mgh(Z) = h 1 ∪ h 2 because it cannot include any element without h 1 ∪ h 2 and must include all elements within.


Our approach is substantially different form usual supervised learning paradigms in spiking networks since it does not target the spike times explicitly.

observing spike times may be misleading since there are many combinations that can produce the same output [15,16].


The LP tightness is an inevitable condition to guarantee the performance of BP and no combinatorial optimization instance has been known where BP would be used to solve problems without the LP tightness.

in the LP literature, it has been extensively studied how to enforce the LP tightness via solving multiple intermediate LPs that are systematically designed, e.g., via the cutting-plane method [21].


Now by using the classical chaining method (Theorem 1) to upper bound E[X W ] by upper bounding E[sup φ∈[0,2π) X φ ] and ignoring the algorithm, we get On the other hand, for every k ≥ −1 we have Therefore, based on the chained mutual information method (Theorem 4), we have Numerical values of the right side of (36) for different values of are given in Table 1 (CMI bound).

note that indeed the slow rate of that convergence and the existence of the 2 −k term makes the sum not only finite, but very small.


A classical approach is principal component analysis (PCA), which implicitly models data to live in a single low-dimensional subspace within the high-dimensional ambient space.

a more suitable model in many applications is the union of multiple low-dimensional subspaces.


One might deem that this make our algorithm impractical because after dozens of epoches, the large value of penalty factor might slow down the decrement of function value.


Over the past years, Convolutional Neural Networks (CNNs) and Recurrent Neural Networks (RNNs) have emerged as the state-of-the-art learning framework for action recognition [3,6,35,51].

the success of existing supervised learning methods is primarily driven by significant amount of manually labeled data, which is expensive and time-consuming to collect.


In order to learn the optimal policies in this framework and to facilitate the development of the paper, we follow generalized policy iteration (GPI) (Sutton and Barto [2018]).

the Bayesian framework developed in this paper is dependent on neither the exploration policy used nor the value function representation, and can be applied with on-policy and off-policy learning, value function approximation, traces, deep RL (Li [2017]), and other approaches.


Compared to the aforementioned ZO algorithms [5,14,24], the convergence performance of ZO-SVRG in (10) has an improved (linear rather than sub-linear) dependence on 1/T .

it suffers an additional error of order O(δ n /b) inherited from (Sχ m /(T γ)) in (5), which is also a consequence of the last error term in (4).


A popular class of non-parametric RL techniques is Gaussian process temporal difference (GPTD) learning [5], which provides a Bayesian representation of the expected discounted return.

all aforementioned methods involve approximate offline planning for MDPs with known dynamics or online learning by sustained direct interaction with the system or a simulator.


Point process models accurately describe the spiking responses of neurons in the early visual pathway to light, and of cortical neurons to injected currents.

they perform poorly both in higher visual areas and in auditory cortex, and often do not generalize well to stimuli whose statistics differ from those used for fitting.


Note that for an 2 norm bounded adversarial perturbation, the dual norm is also an 2 norm, so we can use traditional random projections [Vempala, 2005].

experiments for the 2 norm are explored further in Appendix H. for the remainder of this section we focus on the 1 case arising from ∞ perturbations.


The results that are included follow the framework established in section 2.

the intensity value appearing in equations in section 2 is replaced by a 3D vector of RGB values corresponding to each pixel.


The recently emerged parameter server architecture [15,18] has started to pave ways for a unified programming interface for data parallel algorithms, based on various parallellization models such as stale synchroneous parallelism (SSP) [15], eager SSP [5], and value-bound asynchronous parallelism [23], etc.

in addition to Big Data, modern large-scale ML problems have started to encounter the so-called Big Model challenge [8,1,17], in which models with millions if not billions of parameters and/or variables (such as in deep networks [6] or large-scale topic models [20]) must be estimated from big (or even modestly-sized) datasets.


A naive nonsymmetric splitting method has been applied for CCAdL for fair comparison in this article.

we point out that optimal design of splitting methods in ergodic SDE systems has been explored recently in the mathematics community [1,13,14].


The sum-min objective inherits the good properties of both -it is robust to a small number of additions/removals, and it tries to ensure that each point is far from the others.

it is trickier to optimize, as we mentioned earlier.


This formulation only offers L 2 distance metrics and therefore may not be robust to deviation of data based on other metrics (e.g., adversarial perturbations).

our work formulates a loss function that directly maximizes the margin at any layer, including input, hidden and output layers.


Synthetic DNA has emerged as a promising option, with theoretical information density of multiple orders of magnitude more than magnetic tapes [12,24,26,52].

significant biochemical and computational improvements are necessary to scale DNA storage systems to read/write exabytes of data within hours or even days.


The answer to this question is "probably not."

when changing the rules of the standard game it is possible.


The proposed algorithm quantifies information on the basis of a neural circuit, on units with response properties corresponding to neurons appearing in the primary visual cortex.

given an analogous representation corresponding to higher visual areas that encode form, depth, convexity etc.


In summary, given an object category our algorithm automatically finds the modes of the GIST distribution in images of that object.

these modes in the GIST distribution are only approximations to the modes of the view distribution.


the Horn and Schunck model is close to the optimal Gaussian model, robust models are better, intensity discontinuities make motion discontinuities more likely).

a learned GMM model with 64 components significantly outperforms the standard models used in computer vision, primarily because it explicitly distinguishes between flat patches and boundary patches and then uses a different form of nonlocal smoothness for the different cases.


Traditionally, variational inference is a coordinate ascent algorithm.


Forms of these terms are similar to the one defined for the object-reflectance higher order terms; these terms are further explained in the supplementary material.

given the above model, our optimization problem involves solving following joint energy function to get the most likely solution for (R, Z, L, O, A): this problem is very challenging since it consists of both the continuous variables (R, Z, L) and discrete variables (O, A).


In general, algorithms for these problems have complexity O(n 3 ).

in many cases, a rank d approximation to the decomposition is sufficient, see for example [9].


Hence, none of these methods allow for local updates to modular policies, which makes them undesirable from a biological standpoint, and impractical for policies for which this derivative is prohibitively difficult to compute.

by combining these methods with the CoMDP framework reviewed in Section 2.2 and by taking advantage of Property 1, the updates to the actor can be modified to satisfy the locality constraint.


Several algorithms have been designed for learning HMM parameters for chromatin modeling, including stochastic variational inference [16] and contrastive learning of two HMMs [26].

none of these methods extend directly to modeling multiple chromatin sequences simultaneously.


-FG vs. SG: Although the performance of SG methods can be catastrophic if the step size is not chosen carefully (e.g., the covertype data), with a carefully-chosen step-size the SG methods do substantially better than FG methods on the first few passes through the data (e.g., the rcv1 data).

fG methods are not sensitive to the step size and because of their steady progress we also see that FG methods slowly catch up to the SG methods and eventually (or will eventually) pass them (e.g., the protein data).


Minimum norm estimates produce spatially smooth solutions but are known to suffer from depth bias and smearing of nearby sources.

minimum current estimates lead to focal source estimates that may be scattered too much throughout the brain volume [9].


A common solution is to treat such problems as a set of independent classification tasks, dealing with each instance in isolation.

it is well-known that this approach fails to exploit significant amounts of correlation information [7].


Traditionally, the maximum causal entropy (MCE) framework [1,7] has been proposed to model stochastic behavior in demonstrations, where the optimal policy follows a softmax distribution.

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


Similarly, we suggest that the large and variable object set used here may provide a meaningful constraint on network structure, as the specific object geometries may be less important then having a wide spectrum of such geometries.

a key next priority is systematically building an appropriately large and variable set of objects, textures or other class boundaries that more realistically model the tasks that a rodent faces.


The result with 15 epochs is usable in practice and predicts only 4 characters incorrectly.

the spelling time is about half a minute per character.


Becasue P processors concurrently sample new topic assignments in parallel CGS, we don't necessarily sample from the correct posterior distribution.

we can view it as a stochastic optimization method that maximizes p(z|x, α, β).


As already mentioned, the problem with current methods of minimizing w 0 is that every basic feasible solution unavoidably becomes a local minimum.

what if we could somehow eliminate all or most of these extrema.


Split-merge MCMC algorithms, e.g., Restricted Gibbs split-merge (RGSM) [13] and Smart-Dumb/Dumb-Smart (SDDS) [36], use handdesigned MCMC moves to solve such issues.

in our framework, it's possible to deal with such relations with a proposal block including all of z, µ and v. doing so requires significant training and inference time (due to larger proposal network and larger proposal block), and the resulting proposal can not generalize to GMMs of different sizes.


The well-known fact that the Laplace mechanism with b ≥ ∆/ε is (ε, 0)-DP follows from this result by noting that δ M (ε) = 0 for any ε ≥ θ.

theorem 3 also provides more information: it shows that for ε < ∆/b the Laplace mechanism with noise parameter b satisfies (ε, δ)-DP with δ = δ M (ε).


It is possible that a larger sample size is needed in order to achieve perfect reconstructions of the ground truth.

the model, on average, has an distance of 7.29 from the actual truth, which is better than the best individual over all problems.


By interpreting γ GA and γ M A as optimizing different variational objectives, we gain additional insight into their behavior.

for geometric averages, the intermediate distribution γ GA (β) minimizes a weighted sum of KL divergences to the initial and target distributions: γ M A minimizes the weighted sum of KL divergences in the reverse direction: See the supplementary material for the derivations.


We believe that similar identifiability results can be derived in the multivariate case (|V | > 2) and for non-Gaussian noise distributions.

proving such results seems to be significantly harder as the calculations become very cumbersome, and we leave this as an open problem for future work.


The parameters B a may be shared, i.e., we may have B a == Bal for some a =1= al.

for a Bayesian network, the energy term simplifies into a sum over local terms: the entropy term is as intractable as the normalization in (3) that we try to prevent.


Following the analysis in Wright [12], we can ensure that the proximal linearization algorithm will converge for the negative log-likelihood loss function with the SLasso penalty.

solving group lasso with overlaps is not trivial due to the non-smoothness at the singular point.


In that case, the theoretical asymptotics we describe here may not hold.

approximate Bayesian-optimal settings can be recovered through parameter learning using expectation-maximization algorithms [32].


We add back the contribution of block A k to each patch in {y i } i and like before perform an SVD decomposition of these residuals.

we are now looking for a K-dimensional subspace where most of the variance in {y i } i lies and this is exactly achieved by considering the first K principal vectors returned by SVD.


In [23], a SVM classifier is used to decide if probing (a bound tightening technique) should be used at a node in B&B.

both prior methods handle a relatively simple setting where the model only predicts information about the current state, so that they can simply train by standard supervised learning.


[2013] study the problem of classification under class-conditional noise model.

they approach the problem from a different set of assumptions -the noise rates are not known, and the true distribution satisfies a certain "mutual irreducibility" property.


This problem is of obvious interest to applications for internet companies that recommend content (such as ads, search results, news stories, etc...) to users.

we believe this also may be of interest for other application domains within machine learning.


In this regard, R-CNN more appropriately does not up/downsample every pixel in the image but only in those regions which are likely to contain objects to an appropriate resolution.

r-CNN does not share the convolutional features for nearby proposals like Fast-RCNN, which makes it slow.


To be safe, an operator for which is small should be chosen.

one generally prefers to be large, so that the agent can exploit its knowledge by choosing actions with higher estimated action-values as frequently as possible.


Traditional methods use cross entropy loss which is prone to exposure bias [27,31] and do not necessarily optimize evaluation metrics such as CIDEr [33], ROUGE [20], BLEU [25] and METEOR [4].

reinforcement learning can directly use the evaluation metrics as reward and update model parameters via policy gradient.


The iterates of Algorithm 3 and Algorithm 4 satisfy the following relationships: We note that in Algorithm 4, only a single block coordinate of the vectors u (k) and v (k) are updated at each iteration, which cost O(N i ).

computing the partial gradient may still cost O(N ) in general.


In addition, previous population models based on discrete latent states [10,12] used point-estimation for parameter learning.

we present algorithms for full Bayesian inference over the parameters of our model, making it possible to identify states in smaller or noisier data [13].


Thus, in the ∞ -norm, our work is inferior, if we ignore their limited applicability and massive computational demands.

our algorithm mainly targets the 2 -norm, and in that sense, the guarded area is significantly larger.


Together, equations 7 and 9 implement a rapidly mixing blocked Gibbs sampler.

the computational cost of solving Equation 10 is larger than for a single Markov step in other sampling methods such as MALA.


For simplicity, we assume all the subsets have the same data size (N/d).

the traditional maximum likelihood estimator (MLE) provides a natural way for estimating the true parameter θ * based on the whole dataset X, directly calculating the global MLE is challenging due to the distributed partition of the dataset.


Gaussian process inference and learning requires evaluating (K +σ 2 I) −1 y and log |K +σ 2 I|, for an N × N covariance matrix K, a vector of N datapoints y, and noise variance σ 2 , as described in the supplementary material.

for this purpose, it is standard practice to take the Cholesky decomposition of (K + σ 2 I) which requires O(N 3 ) computations and O(N 2 ) storage, for a dataset of size N . many real world applications are engineered for grid structure, including spatial statistics, sensor arrays, image analysis, and time sampling.


This allows us to train models comparable to traditional keypoint extraction pipelines, i.e., with built-in invariance to rotation and scaling.

in practice, many of the images in our indoors and outdoors examples are upright, and the best-performing models are obtained by disabling these augmentations as well as the orientation ans scale estimation completely.


IHW has several limitations: first, binning the data into G groups can be difficult if the feature space X is multi-dimensional; second, the decision rule, restricted to be a constant for each group, is artificial for continuous features; and third, the asymptotic FDR control guarantee requires the number of groups going to infinity, which can be unrealistic.

neuralFDR uses a neural network to parametrize the decision rule which is much more general and fits the continuous features.


The stochastic gradient descent (SGD) method and all of its variants studied in this paper are online.

gD, AGD [23,24], and Katyusha [2] are offline methods because their gradient complexity depends on n (see Table 2 in appendix).


Alternatively, we can easily obtain information about purchasers of the product at the moment when the purchases are actually made.

we cannot know whether those who are buying the product have been exposed to the advertisement or not.


However, nearly 90% of the training set was retained as radial bases by the large ensemble-RBF.

the number of support vectors found by both SVMs was only about 20% of the training set.


Table 1 shows that Clump-Far outperforms other training sets for the low-capacity GCM.

clump-Far, Clump-Near, and Spread are all optimal for high-capacity GCM, reflecting the fact that for high-capacity GCM the symmetry of the inner-most training item pair about the true decision boundary θ * determines the learned model.


Our method optimizes more sophisticated objective than the other methods so it takes longer to finally converge.

both its robustness and accuracy performance surpasses that of the reference models in only three epochs and keeps growing in the last two.


In our experiments CFR + vastly outperforms its theoretical convergence rate (in fact, every CFR variant does significantly better than the theory predicts, but CFR + especially so).

cFR + is known to eventually reach a point where it slows down and performs worse than 1 T . In our experiments we start to see CFR + slowing down towards the end of Endgame 7.


Then, the maximization of the bound proceeds by performing stochastic gradient updates for the model weights W and the variational parameters V . The update for W reduces to a logistic regression type of update, based upon drawing a single sample from the full variational distribution.

obtaining effective and low variance stochastic gradients for the variational parameters V is considered to be a very highly challenging task and current advanced methods are based on covariates that employ neural networks as auxiliary models [10].


Standard computational strategies such as greedy feature selection [3] and generic convex optimization [4][5][6][7] can be used to implement simple sparse estimators.

sophisticated notions of structured sparsity have been recently developed that can encode combinatorial patterns over variable subsets [8].


Domain alignment approaches this problem by matching the source and target feature distributions, and has been used as a key component in many state-of-the-art domain adaptation methods.

matching the marginal feature distributions does not guarantee that the corresponding class conditional distributions will be aligned across the two domains.


Maximum entropy (or "maxent") models, which seek to explain dependencies in terms of low-order interactions between neurons, have enjoyed remarkable success in modeling such patterns, particularly for small groups of neurons.

these models are computationally intractable for large populations, and low-order maxent models have been shown to be inadequate for some datasets.


Generative state estimators based on probabilistic filters and smoothers are one of the most popular classes of state estimators for robots and autonomous vehicles.

generative models have limited capacity to handle rich sensory observations, such as camera images, since they must model the entire distribution over sensor readings.


. . , µ K . Clearly, this implies that we require that m > d * .

when m is much larger than d * , we might end-up with a projection on too large a space.


For the special case of Gaussian kernel, element-wise kernel product would be same as simple feature concatenation if both kernels use same width parameter σ.

in our experiments, we use different width parameters for different views so the performances of kernel product may not be directly comparable to feature concatenation.


Batch Normalization is quite effective at accelerating and improving the training of deep models.

its effectiveness diminishes when the training minibatches are small, or do not consist of independent samples.


For rotation matrices it is computed as the Frobenius norm of the difference between ZT Z and Gaussian noise variance Mean 3D error [mm] GP GP + trsf Cstr GP [12] Cstr GP [12]    Gaussian noise variance Mean 3D error [mm] GP GP + trsf Cstr GP [12] Cstr GP [12]   Note that the global transformations estimated with the PnP method yield poor reconstructions.

our approach performs best among those that use these transformations.


The learned marginal pdf, shown in Figure 1d), is similar to a Gamma(4.15,0.00045) distribution.

in using a rolling window to retrain the parameters of g, we do not assume that the marginals of σ t are stationary; we have a time changing warping function.


It would be very interesting to relax this assumption and account for experts with different prediction abilities.

this entails a number of non trivial challenges and is left for future work.


Since the candidate decision trees are trained separately and independently without necessary information exchange, they may have non-negligible bias, which will lead to accuracy drop at the end.

we can clearly characterize the theoretical properties of PV-tree, and use it in an appropriate setting so as to avoid observable accuracy drop.


It can also be applied to the matrix recovery problem, and in the noiseless case, sufficient conditions for exact recovery are known [24].

the Nyström method requires sampling full columns and rows of the original matrix, a sampling scheme which may not be possible in many of our applications of interest.


From Theorem 1, we see that both RPF and EE encode the spectral information of graphs.


The standard approach is to let the server add the noise to the averaged gradients (e.g., see [14,2] and references within).

the above only works under a restrictive assumption that the clients can trust the server.


Then inf where B r V is an r V ball with the same center with U such that Vol n (B r V ) = V. Moreover, T (x) = r U r V x, for x ∈ B r V is the optimal transport map (up to a measure zero set), so that Complementarily, when 0 < V < Vol n (U ), the infimum inf W : Voln(W )=V W 2 (P U , P W ) = 0, is not attained by any set.

inf W : Voln(W )≥V W 2 (P U , P W ) = 0 by taking W = U .


In linear discriminative prediction we select the y that has the greatest value for the inner product Φ(x, y), w . Linear discrimination has been widely studied in the binary and multiclass setting [6,4].

the case of structured labels has only recently been considered [2,12,3,13].


For example one can use them also to compute the spike train which produces the largest peak of the postsynaptic membrane voltage.

in the following we will focus on the question which temporal pattern of a spike train produces the largest sum 'E~=l A• Uk .


The algorithm would always assign the individual to expert j = 1 and has a cumulative expected utility of 3T θ2,z 8 .

given the true thresholds, the algorithm would have always assigned the individual to expert j = 2 and would have a cumulative expected utility of T θ2,z 2 .


Being nonconvex in general, SP is essentially hard to solve from a computational perspective [1,3].

despite the hardness of SP in general, the objective function in the MLE formulation of SPNs has a special structure, i.e., it is the ratio of two posynomials, which makes the design of efficient optimization algorithms possible.


The simplest of the models, the feedforward model, does not consider the temporal structure in the task at all, and consists only of a feedforward convolutional network that takes in the observations o t and outputs a point estimate ŷt of the label y t . This approach is viable only if the label information can be directly inferred from o t , such as when tracking an object.

tasks that require long term memory, such as visual odometry, cannot be solved with a plain feedforward network.


If one can control the total-variation distance, one can obtain a bound like Theorem 4.2 by applying approximate maxinformation bounds for (e, d )-differential private mechanisms.

many algorithms do not control the total-variation distance to stationarity, or do so poorly.


The embedding GNN at layer l is a standard GNN module applied to these inputs: i.e., we take the adjacency matrix between the cluster nodes at layer l (from Equation 4) and the pooled features for the clusters (from Equation 3) and pass these matrices through a standard GNN to get new embeddings Z (l) for the cluster nodes.

the pooling GNN at layer l, uses the input cluster features X (l) and cluster adjacency matrix A (l) to generate an assignment matrix: where the softmax function is applied in a row-wise fashion.


[5], and another based on perceptron method [6], and propose alternating optimization algorithms to train the classifiers.

they only evaluate the proposed methods with a few small datasets (with no more than 1000 samples in each), and do not compare them to other widely used (nonlinear) classifiers (e.g., KNN, decision tree, SVM).


Most solutions proposed in these variants of GARCH involve: a) introducing nonlinear functional relationships for the evolution of volatility, and b) adding asymmetric effects in these functional relationships.

the GARCH variants do not fundamentally address the problem that the specific functional relationship of the volatility is unknown.


Since the latter models assume that a sample is generated from one component, they cannot represent "multiplicity."

pMM1 can represent 2 L − 1 multi-category classes with only L parameter vectors.


Somewhat similar in spirit to our approach is the work on supervised probabilistic PCA [25] that extends probabilistic PCA to the setting when we also have access to labels.

it assumes a fixed number of components and does not take into account sparsity of the projections.


Looking at equation ( 1) and ignoring the ridge term, it is clear that our feature network regularization penalty is identical to LLE except that the embedding is found for the feature weights rather than data instances.


In the case of the sparse decomposition, we have no guarantee that this will happen.

the robustness interpretation gives us a natural stopping criterion: if all the coefficients in |A i | are smaller than the noise level ρ (computed in the last section) then we must stop since the matrix is essentially indistinguishable from zero.


On the other hand, given a set of C labels, without any prior knowledge, a bag of M i instances could have maximum C Mi labeling vectors, which becomes a clustering problem.

we are more interested in situations when L i ≪ C Mi .


Subsequently, tree search is improved by using the neural network policy to guide search, increasing the strength of new plans.

standard deep Reinforcement Learning algorithms rely on a neural network not only to generalise plans, but to discover them too.


Sutton and McCallum [36] combined a parsing model with a semantic role labeling model into a unified probabilistic framework that solved both simultaneously.

it is hard to fit existing state-of-the-art classifiers into these technically-sound probabilistic representations because they require knowledge of the inner workings of the individual classifiers.


Moreover, training time may seem high in comparison to usual SVMs.

let us first remind that we jointly learn p regressors.


Therefore, ∆-GAN provides a natural way to perform semi-supervised learning, since the conditional GAN part and the BiGAN part can be used to account for paired and unpaired data, respectively.

when doing semi-supervised learning, there is also one potential problem that we need to be cautious about.


Note that UV-GAN mentioned above can be used to benefit face recognition in the same manner with DA-GAN, so it is also a data augmentation method.

our HF-PIM is trained to directly rotate the given profile to the frontal face, which can be directly used for face recognition.


In fact, in [8] it was shown that for most combinations of m and n the dimension of RBM n,m (as a manifold, possibly with singularities) equals either the number of parameters or 2 n − 1, whatever is smaller.

the geometry of RBM n,m is intricate, and even an RBM of dimension 2 n − 1 is not guaranteed to contain all visible distributions, see [20] for counterexamples.


Such models fail to incorporate the role of task demands and do not address the problem of resource contention.

our underlying premise is that much of routine human behavior can be understood in the framework of reward maximization.


On the other hand, detection challenges such as heavy occlusion, class ambiguities and tiny-size objects become more severe due to more categories within one image.

humans can still identity objects precisely under complex circumstances because of the remarkable reasoning ability resorting to commonsense  knowledge.


a nearly constant number of entries per row.

for arbitrary sampling distributions, the worst-case sample complexity lies between a lower bound of Ω(n 4 /3 ) [12] and an upper bound of O(n 3 /2 ) [13], i.e.


For the sake of simplicity in the analysis, we only report and analyze its synchronous version in this paper.

it can easily be implemented in an asynchronous fashion and our theoretical results can also be extended to this setting by following the same path as [8].


They use a hill-climbing algorithm to seek a model which maximizes a model fitness scoring function.

calculating the empirical density and comparing it to the model density is difficult, especially in high dimension.


(2010) that introduce the inducing variable formalism for computational efficiency in MOGP.

instead of marginalising the latent functions U to obtain a variational lower bound, we keep their presence in a way that allows us to apply stochastic variational inference as in Hensman et al.


Given these inputs and appropriate feature maps for superpixels and bounding boxes (defined later in detail), we want to recover latent variables z and y simultaneously by learning the two coupled classifiers in different feature spaces.

to constrain the two classifiers together, we need another indexing of superpixels detailed next.


The primal subgradient method uses the greedy algorithm in each iteration, which runs in O(n log n).

as we will see, its convergence is so slow to counteract any benefit that may arise from not using projections.


For non-stationary kernels we proceed as before and construct 8;j = (¢(Xi)-¢(Xj))T(¢(Xi)-¢(Xj)).

we can again show that the kernel MDS procedure operates on the matrix H K H. the distance 8ij in feature space is not a function of 8ij and so the relationship of equation 3 does not hold.


[21], who have shown that both sparsity of the representation and distinctiveness of the matched features increase monotonously with the depth of the network.

the correlation between model and human decision scores peaked at intermediate layers and decreased for deeper layers.


The low-rank structure of the precision matrix, i.e., L poses a great challenge for computation.

a typical way is to use nuclear-norm regularized estimator, or rank constrained estimator to estimate L. such kind of estimators involve singular value decomposition at each iteration, which is computationally very expensive.


We would like to identify the outliers, which can be arbitrary.

clearly an "outlier" point that lies in the true subspace is a meaningless concept.


Since the hyperparameters of a GP model can be adjusted by maximizing the marginal likelihood, GPR requires little effort and is easy and flexible to use.

the main limitation of GPR is that the computational complexity scales cubically with the training examples n. This drawback prevents GPR from applications which need large amounts of training data and require fast computation, e.g., online learning of inverse dynamics model for model-based robot control.


In general, solving µ(θ) = u corresponds to finding solutions to systems of multivariate polynomials, which is NP-hard [28].

µ(θ) often has additional structure which we can exploit.


One could simply constrain q ij + q ji ≥ for some positive .

multiplying all entriesof Q by some c > 0 does not affect the stationary distribution of the corresponding CTMC, so multiplication by 1/ gives a Q with the same selection probabilities.


At the end of each time step i, the expected sub-optimality of STRSAGA over T i is If we additionally assume that the condition number L/µ is bounded by a constant at each time, the above simplifies to Lemma 3 shows that the expected sub-optimality of STRSAGA over its effective sample set is not equal to n i the number of points so far).

our goal is to show that STRSAGA is risk-competitive to DYNASAGA(ρ) at each time step i; i.e., the expected sub-optimality of STRSAGA over S i is within a factor of H(t D i ).


A naive approach to this task is to first estimate training and test input densities separately and then estimate the importance by taking the ratio of the estimated densities.

this naive approach tends to perform poorly since density estimation is a hard task particularly in high dimensional cases.


# , the agents instantiate the state variables and then solve the cost network as in the previous section.

we note that the agents do not actually need to have access to all of the state variables: agent ¨only needs to observe the variables that are in the scope of its local ' function, thereby decreasing considerably the amount of information each agent needs to observe.


That is, let k * be the best single model defined as: then f k * = g. We are interested in an estimator f of g that achieves a small regret This paper considers a special class of model combination methods which we refer to as model averaging, with combined estimators of the form where ŵk ≥ 0 and k ŵk = 1.

a standard method for "model averaging" is model selection, where we choose the model k with the smallest least squares error: This corresponds to the choice of ŵk = 1 and ŵk = 0 when k = k. it is well known that the worst case regret this procedure can achieve is R( .


1, produces the following unconstrained problem: We use the term "reconstruction cost" for this smooth penalty because it corresponds to the reconstruction cost of a linear autoencoder, where the encoding weights and decoding weights are tied (i.e., the encoding step is W x (i) and the decoding step is The choice to swap the orthonormality constraint with a reconstruction penalty seems arbitrary at first.

we will show in the following section that these two forms of degeneracy control are, in fact, equivalent under certain conditions.


Going beyond the explanation of a single prediction, they both can be extended to explain the model as a complete entity by selecting a small number of representative individual predictions and their explanations.

explanations obtained through such approaches cannot describe the full mapping learned by an ML model.


The second assumption holds for the removing parameters α R because we know that we should move α R to 0.

for some of α A , the second assumption does not necessarily hold because we do not know the optimal α A beforehand.


Under the NB process and integrated to HDP [Teh et al., 2005], NB-HDP employed a Dirichlet process (DP) to model the rate measure of a Poisson process.

nB-HDP is constructed by fixing the probability parameter of NB distribution.


sparse locally tree-like graphs that arise, for instance, decoding modern error correcting codes [RU08] or in random constraint satisfaction problems [MM09].

in more structured instances -arising for example in computer vision -BP can be substantially improved by accounting for local dependencies within subsets of more than two variables.


In data mining community, many find clustering can be used to detect outliers, with often heuristic but effective procedures [17,10].

kernel based methods have been shown to be robust for many machine learning tasks.


The identification of the model parameters provides insight into the dynamical rules governing the time-evolution of the system under study.

especially in exploratory data analyses, it may be unclear what form this model should take.


As a trivial example, adding an arbitrary constant to the variables, λ x,y,v→α (ŷ v ) + c, does not change the objective value, hence the algorithm can generate non-decreasing unbounded sequences.

the beliefs generated by the algorithm are bounded and guaranteed to converge to the solution of the dual approximated structured prediction problem.


Suppose that we want to keep every activation unit with the probability of keep_prob, in dropout [1] the binary mask will be sampled with the Bernoulli distribution with mean 1 − keep_prob.

to account for the fact that every zero entry in the mask will be expanded by block_size 2 and the blocks will be fully contained in feature map, we need to adjust γ accordingly when we sample the initial binary mask.


This further suggests that people can not easily extract manifold structure from unlabeled data in order to learn, when there is no hint to do so.

most participants have given up the simple single vertical or horizontal decision boundary, because it contradicts with the four labeled points.


This formulation has the appeal of its roots on what are the only known full-functioning vision systems, and it has been shown to lead to interesting saliency behavior [7].

these solutions have one significant limitation: the lack of a clearly stated optimality criteria for saliency.


In non-mini-batch settings, AccProxSVRG only achieves the same rate as SVRG.

in mini-batch settings, AccProxSVRG significantly improves the minibatch efficiency of non-accelerated variance reduction methods, and surprisingly, AccProxSVRG essentially only needs size O( √ κ) mini-batches for achieving the optimal iteration complexity for strongly convex objectives, where κ is the condition number of the problem.


The work that closely resembles ours is level-k thinking [20] where level k agents respond to level k − 1 agents, and more closely cognitive hierarchy [18], in which responses are to distributions over levels {0, 1, . . . , k − 1}.

our goals and motivations are very different: we use the setup as a means to produce more general policies rather than to predict human behavior.


The first term might have some relation to F due the factor of V 2 7f.

the Q values weight this curvature of our policy and our metric is neglecting such weighting.


We emphasize that our k-SVD complexity (1.2) can be upper bounded by a quantity that only depends poly-logarithmically on 1/ε.

the worst-case 1-SVD complexity (1.3) of FW and Garber depends on ε −1/2 because the gap σ 1 − σ 2 can be as small as zero.


In other words, Õ(kn) samples should be enough to learn both inside A and inside B.

the regularization tradeoff parameter λ compounds the two problems.


wavelets with compact support), since the curve would have to cover the whole domain and thus l will be very large.

the situation may be much nicer when the basis is not localized, as in the case of a Fourier basis.


This is frequently given as a reason for using stochastic policies.

as we shortly show, there is no advantage to using stochastic (rather than deterministic) policies, when we are searching for non-stationary policies.


[32] addresses the first limitation of [15] by performing BPTT [31] to capture temporal effects.

similar to [15], the error gradient is computed for the continuous membrane voltage waveforms resulted from smoothing out all spikes, leading to inconsistency w.r.t the rate-coded loss function.


In Algorithm 1 we denote P r = |{p : r 2 p}|, we define (µ) as In Appendix D we show that the convergence rate of Algorithm 1 is O(1/ ✏), similar to subgradient in the dual.

algorithm 1 has several advantages over subgradient.


The KTD method achieved the second best performance for E samp , as a result of using a non-parametric representation.

the value tended to drop in sparsely-sampled regions, which lead to relatively high E unif and E max values.


If the learner fails to have w t increase quickly, then there is a u 1 against which the learner has high regret.

if the learner ever does make w t higher than a particular threshold, the adversary immediately punishes the learner with a subgradient g t = 2T , again resulting in high regret.


( 15) implies that span(L mAP ) = span(F mAP )).

as noted in Section 6 of [18], although the matrix F mAP is convenient from the consistency point of view (in the setup of [18]), it leads to the prediction problem max σ∈Sr (F mAP θ) σ , which is a quadratic assignment problem (QAP), and most QAPs are NP-hard.


For m = 2, we obtain a complete graph that does not convey enough information about the nodes.

for m = 3, the constructed hypergraph is a union of several connected components, each component representing a set of collinear points.


Recall that in a two-player game, σ * i is a best response to the opponent's average strategy, σT −i .

for two-player zero-sum games, we do know that the average profile σT converges to a Nash equilibrium.


Our third contribution demonstrates that under the ∞ metric, we can approximate any homogeneous density mixture within accuracy by combining O(1/ 2 ) shallow SPNs.

our proof requires the knowledge of the target density hence is not practical.


Also, we highlight the scalability of our approach: the number of iterations needed for convergence on these data sets ranges from 13 to 25, and each iteration completes in under one minute (see the right side of Figure 1).

for sampling methods, it is notoriously difficult to detect convergence, and generally a large number of iterations is required.


We could have additionally incorporated a loss function which further penalized similarity among the columns of α, such as maximizing pairwise distances between elements.

most reasonable distance measures -such as norms or Kullback-Leibler divergence -are convex, and maximizing a convex function is difficult.


The variational Bayesian (VB) approach is one of the best tractable approximations to the Bayesian estimation, and it was demonstrated to perform well in many applications.

its good performance was not fully understood theoretically.


They have recently gained prominence in several applications that rely on "diverse" subsets.

their applicability to large problems is still limited due to O(N 3 ) complexity of core tasks such as sampling and learning.


More precisely, when the data are well clustered around a few patterns, the accumulated updates (difference between the learned weights and the initialization) should be approximately low rank, which can be seen from checking the SGD updates.

when the difference is small compared to the initialization, the spectrum of the final weight matrix is dominated by that of the initialization and thus will tend to closer to that of a random matrix.


It is also easy to verify that F g satisfies ( 6) with respect to M g . To compute the corresponding polar, observe that the effective domain of M g is a lattice, hence (4) can be solved by combinatorial methods.

we can do better by exploiting problem structure in the LP.


But they also need to explore: to make decisions which do not seem optimal according to the algorithm's current point-predictions, in order to gather more information about less explored portions of the decision space.

in practice, decision systems often do not explicitly explore, for a number of reasons.


If we assume that the functions F t are all taken from a known finite set of functions F then we have linear loss over a |F| dimensional space.

this approach gives poor dependence on |F|.


One of the key strengths of I2As is being able to handle learned and thus potentially imperfect environment models.

for the Sokoban task, our learned environment models actually perform quite well when rolling out imagined trajectories.


Unlike BP and TRW, TRW-S is guaranteed to converge.

it is wellknown that TRW-S and other related algorithms [7,13,25] suffer from the following problems: (i) they are slower than algorithms based on efficient graph-cuts [22]; and (ii) they only provide a dual solution [10].


should be almost indistinguishable if H 0 holds.

if H 1 holds, then the classifier trained in Line 6 should be able to easily distinguish between the samples corresponding to different labels.


created their training set based on a mechanistic-level template-driven expert system.

our model is learned directly from real-world experimental data.


k , then a trace is given, even in this exploit case, because the teacher does not know V A and cannot distinguish this case from the "explain" case above.

this trace will still be helpful, because U T ≤ U A , so at least U T < V T − k (satisfying iii), and again by the simulation lemma, the trace will help us learn a parameter and there are a limited number of such mistakes, so (ii) holds.


Previous work has shown that visual information is important for training effective embedding models.

due to the lack of large training datasets of the same scale as the pure text dataset, the models are either trained on relatively small datasets (e.g.


A widely used heuristic is to choose several generalized eigenvectors from both ends of the eigenvalue spectrum.


In word embeddings, there are no clear individuals and no a priori defined classification problem.

similar issues arise, such as direct and indirect bias [24].


[2] exposed rats to a tone T immediately followed by a click train C. In a second phase, the tone was paired with a footshock US that either immediately followed tone offset (variant A), or occurred 5 sec after tone offset (variant B).

they found that when C and US both immediately follow T , little conditioned response is elicited by the presentation of C. when the US occurs 5 sec after tone offset, so that it occurs later than C (measured relative to T), then C does come to elicit a CR.


We can show that the correction to e' is typically by a factor of O(l/m) smaller than the leading term.

our approximation becomes worse with decreasing noise variance a 2 . a = 0 is a singular case for which (at least for some GPs with slowly decreasing eigenvalues) it can be shown that our approximation for eg decays to zero at the wrong rate.


A more accurate and plausible description of dendritic orientation at bifurcations might require a more complex model.

our simple choice yields surprisingly good results (see below).


Previous approaches for learning symbolic models from data [11,12] used random exploration.

real world data from high-level skills is very expensive to collect, so it is important to use a more data-efficient approach.


We show that this principle can indeed be employed by vertebrates to improve visual acuity in certain (hyperacuity) tasks.

eye movements have the reverse effect on detection tasks that require aliasing.


There is no significant difference between the LSTM and pool approaches (T = 10.0, p = 0.46).

graphSAGE-LSTM is significantly slower than GraphSAGE-pool (by a factor of ≈2×), perhaps giving the pooling-based aggregator a slight edge overall.


A popular geometry for PDs is the Wasserstein metric.

wasserstein distance is not negative definite.


Manifold regression can be interpreted as a vector-valued learning setting where outputs are constrained to be in C ⊆ R T and prediction errors are measured according to the geodesic distance.

note that the two problems are also significantly different since, 1) in MTL noise could make output examples y i lie close but not exactly on the constraint set C and moreover, 2) the loss functions used in MTL typically measure errors independently for each task (as in Eq.


Additional experiments on other stereo pairs with similar observations about the performances of the various algorithms can be found in [17].

we would again like to emphasize that unlike our method the range move algorithm provides no theoretical guarantees about the quality of the solution.


In [18] a classifier based on text features (such as whether the keyword appears in the URL) was used re-rank the images before bootstrapping the image model.

the text ranker was category-independent and thus unable to learn words predictive of a specific word sense.


However, it is hard to see how dendrites could be conduits to viscous fluid if their ends are sealed.

rail [11] has derived a similar law for branching dendrites by postulating impedance matching: the main purpose of Rail's law was to simplify calculations of dendritic conduction rather than to explain the actual branch caliber measurements.


In Figures 1c and 1d we fix m and vary n. Observe in Figure 1c how KBRL and KBSF have similar performances, and both improve as n → ∞.

since KBSF is using a model of fixed size, its computational cost depends only linearly on n, whereas KBRL's cost grows with n 3 .


αβ-swap and range move) do not provide any guarantees on the quality of the solution.

the α-expansion algorithm [4] (where each variable can either retain its label or get assigned the label l α at an iteration) provides a multiplicative bound of 2 for the Potts model and 2M for the truncated linear metric.


We point out the fact that the first diagonal corresponds to nondiscriminating scoring functions s 0 such that H s0 = G s0 .

searching for a scoring function s fulfilling this property is generally not sufficient in practice.


This dataset thus represents an ideal substrate to explore manifold learning (Moon et al., 2017a).

the data presents two distinct challenge due to 1. undersampling of cell types, and 2. dropout and artifacts associated with scRNA-seq (Kim et al., 2015).


This behavior is captured by our robustness statistics-the baseline neural net has lower adversarial frequency (so it has fewer adversarial examples with ρ(f, x * ) ≤ ) but also has worse adversarial severity (since its adversarial examples are on average closer to the original points x * ).

according to our metrics shown in Figure 3 (b), our neural net is substantially more robust than the baseline neural net.


Even adding the positive semi-definite constraint M 1 (µ) 0, for which TRW must be optimized using conditional gradient and semidefinite programming for the projection step, does not improve the accuracy by much.

both entropy approximations give significantly better pseudomarginals when used by our algorithm together with the cycle inequalities (see "TRW + Cycle" and "Logdet + Cycle" in the figures).


To summarise, given enough time AAOBF will find an optimal solution and can dominate AGS which is limited by the approximation inherent in BP.

with a limited time and difficult conditional summation problems AGS provides a better tradeoff in finding solutions quickly.


Now we expect that after c 1 d log d more levels, the size should go down further by a factor of 2 thereby giving us our desired result.

given the large number of nodes at this level and the fact that the success probability in Fact 2 is just greater than a constant bounded away from 1, it is not possible to argue that after c 1 d log d more levels the descendants of all these 2 c1d log d nodes will be of radius ∆/4 or less.


Furthermore, by combining projections in skewed-norm spaces and weighted proximal terms, it is possible to actually reduce the number of iterations given in Table 1.

for simplicity, we focus on the objective (2) and projections in skewed-norm spaces.


To address this problem, we propose PointCNN with denser connections (Figure 4b), where more representative points are kept in the X -Conv layers.

we aim to maintain the depth of the network, while keeping the receptive field growth rate, such that the deeper representative points "see" increasingly larger portions of the entire shape.


This paper concentrates on the link admission control problem.

the controllers we describe can be used as building block in optimal routing, as shown in [8] and [9].


Given a feature space, the SCM tries to find the smallest conjunction (or disjunction) of features that gives a small training error.

the SVM tries to find the maximum soft-margin separating hyperplane on all the features.


First notice, that as long as TrailBlazer encounters only AVG nodes, it behaves just like Monte-Carlo sampling -without the MAX nodes we would be just doing a simple averaging of trajectories.

when TrailBlazer encounters a MAX node it locally uses more samples around this MAX node, temporally moving away from a Monte-Carlo behavior.


Hence, they are suited to analog VLSI implementation, as the main barrier to the implementation of on-chip learning, the long term storage of precise analog weight values, can be rendered unimportant.

weight-independent STDP creates a highly unstable learning process that may hinder learning when only low levels of spike-timing correlations exist and neurons have few synapses.


Ideally, a speech recognition system should be trained only on data from the user it is intended for.

annotated data is expensive and difficult to obtain.


Recurrent Neural Networks (RNNs) have become the state-of-the-art choice for extracting patterns from temporal sequences.

current RNN models are ill-suited to process irregularly sampled data triggered by events generated in continuous time by sensors or other neurons.


The recursive grouping procedure is similar to the short quartet method from phylogenetics [15], which also guarantees efficient reconstruction in the context of evolutionary trees.

our method and analysis applies to considerably more general high-dimensional settings; for instance, our sample complexity bound is given in terms of natural correlation con-ditions that generalize the more restrictive effective depth conditions of previous works [15,21].


The authors interpreted their findings as suggestive that area V2 may serve as a place where different channels of visual information are integrated.

quantitative accounts of responses in area V2 are few in number.


The Common Spatial Patterns (CSP) algorithm computes filters that maximize the difference in band power between two conditions, thus it is tailored to extract the relevant information in motor imagery experiments.

cSP is highly sensitive to artifacts in the EEG data, i.e.


At first glance, keeping track of the I-covered edges may appear computationally inefficient.

at each step we only need to update this list locally; so the computational complexity of the algorithm is not drastically impacted by this procedure.


[39] reduce the co-attention method into two steps, self-attention for a question embedding and the question-conditioned attention for a visual embedding.

these co-attention approaches use separate attention distributions for each modality, neglecting the interaction between the modalities what we consider and model.


There are several methods one could use to create a model of the data, notably some form of parametric regression.

we chose to approximate the unknown boundary as a Gaussian Process (GP), as many forms of regression (e.g.


It has been observed that DCA and SDCA algorithms can have comparable and sometimes even better convergence speed than GD and SGD methods.

it lacks efforts in studying them in a distributed fashion and comparing to those SGDbased and ADMM-based distributed algorithms.


The learning curves converge to similar results with different reference sizes and comparison sizes.

learning with the large reference size and comparison set could potentially increase the computational cost.


[27] shares the idea of maximum distinctiveness (or "discriminability" used in [27]) criterion with our approach.

their focus is the setting of unsupervised clustering where ours is domain discovery.


The other extreme choice for the proposal, that has been considered in [10], is to apply Gibbs sampling where we iteratively draw samples from each posterior conditional density p(f i |f −i , y) with f −i = f \f i .

gibbs sampling can be extremely slow for densely discretized functions, as in the regression problem of Figure 1, where the posterior GP process is highly correlated.


In the context of text data, latent variables may assume the role of sentence representations that govern a lower-level generation process, thus facilitating controlled generation of text.

vAEs for text are notoriously hard to train when combined with powerful autoregressive decoders [5,18,47].


The task remains intractable despite the entropy approximation since the marginals p r (s r ) are required to arise from a valid joint distribution p(s).

if we require the marginals to be consistent only locally, we obtain a tractable approximation [34].


The computed utilitarian solution is optimal from the perspective of the system where the performance is additive.

as the utilitarian solution often discriminates against some agents, it is not desirable for many practical applications where agents have their own interests and fairness is expected.


To be robust against Byzantine machines, they average stochastic gradients within a single iteration and compare them across machines.

we average stochastic gradients (and other quantities) across iterations.


As the size of gridworld increases, it fails to find the location of the reward.

div-DQN consistently achieves 1.0 mean reward for all the gridworld sizes.


6 means that if we route a query point x through the tree to its leaf cell A(x), then its nearest neighbor is likely to be in the same cell, which is quite appealing when trees are used for nearest-neighbor search.

via Lemma 1, we can now show that such trees lead to inconsistent forests whenever we grow the trees deep and disable subsampling.


There are three parameters whose global estimation is desirable: x , Z , and Pr[X(u) = 0|Z(u) = 1].

for technical reasons, requiring that z be global makes it impossible to derive an EM algorithm.


Traditional machine learning relies on the availability of a large amount of labeled data to train a model in the same feature space.

labeled data are often scarce and expensive to obtain.


These will be addressed separately as follows.

by the definition of 2 -functional in (7) for the Finsler metric, we have we have that ⇢ Fin (A v1,v2 , V k )  1 always holds.


In the ∞ attack model where l = x− 1 and u = x+ 1, we have s j = W x− W j 1 and t j = W x+ W j 1 for j = 1,2,...m. From Figure 2a, we see that for a single ReLU unit taken in isolation, the LP is tighter than the SDP.

when we have multiple units, the SDP is tighter than the LP.


All these kernels can be computed efficiently via the construction of suffix-trees, as we will see in the following sections.


To learn w s 's as low-density separators, we are supposed to have certain high-density regions which w s 's separate.

accurate estimation of high-density regions is difficult.


This generalization inherits several properties: it is convex, has a rich duality theory and allows theoretically efficient solution algorithms based on iterating interior point methods to either follow a central path or decrease a potential function.

a limitation of SDPs is their computational complexity [1], which has restricted their application to small scale problems [6].


Both the components as well as their mixing proportions are unknown to us wherefore we treat them as latent variables in our model.

we may reasonably assume that each observed patch was only generated from a few components, so we employ a sparseness prior over the component weights.


As shown in the energy functions above, the momenta are sampled from zero-mean normal distributions, enabling us to use standard covariance estimation techniques from the literature.

since we are using discretized MCMC to obtain these samples, we have to address the variance arising from the Monte Carlo error, especially during the burn-in phase.


Considering that rare occurrences are also those that are most likely to provide new insight, we want a model that promotes recall over precision in the tail of the term frequency distribution.

well represented classes are easier to detect and correspond to massive, well-known mental processes.


Bennett [1] proposes a non-greedy multi-linear programming based approach for global tree optimization and shows that the method produces trees that have higher classification accuracy than standard greedy trees.

their method is limited to binary classification with 0-1 loss and has a high computation complexity, making it only applicable to trees with few nodes.


Recall that the generalization guarantees we described in the previous section rely on random selection of landmark pairs from a fixed distribution over the domain.

in practice, a totally random selection might require one to select a large number of landmarks, thereby leading to an inefficient classifier in terms of training as well as test times.


This refinement procedure may be similar to LLC [22].

in our case, we do not preset the number of active bases and determine by non-zero components of û.


In deriving these bounds, we have taken an approach similar to [17], who considers the problem of Gaussian process-modulated Poisson process estimation using variational inference.

our problem is made more complicated by the aggregation of labels, as standard lower bounds to the marginal likelihood used in [17] are also intractable in our model.


The other convolution methods performed best when employing multiple convolutional layers, suggesting that the networks are indeed learning a hierarchical representation of the data.

networks with more than four layers performed worse, which could be attributed to the relatively limited amount of labeled protein interface data.


In other words, usually the label assignment of a parent node is enforced to be compatible with the label assignment of its children by averaging.

in the proposed approach the parent node and all its children compete for the best possible label assignment.


In equation 1, v was set to exhibit zero mean and zero variance.

allowing some variance would give the model an observation noise term that could reflect the stochasticity of the reward signal.


This will limit interpolation complexity.

as the size of the support decreases, so will the capture range of the algorithm.


In these approaches, we are particularly interested in the similarity among objects.

the observation of similarity graphs often contain noise which sometimes mislead the learning algorithm, especially in unsupervised and semi-supervised learning.