Due to the high variance of the score and its sensitivity to some implementation details [19], it is difficult to have a precise evaluation of Tetris controllers.

our brief tour d'horizon of the literature, and in particular the work by Szita and Lőrincz [18] (optimizing the "Bertsekas features" by CE), indicate that ADP algorithms, even with relatively good features, have performed extremely worse than the methods that directly search in the space of policies (such as CE and genetic algorithms).


Unfortunately, we do not have access to the true pair marginals, but if we would have estimates .

with the new parameters the estimates ¦ $# % will be changed as well, and this procedure should be iterated.


In general, an RBF kernel will lead to an effective criterion for measuring the dependence between random variables, especially in time-series applications.

we could also choose linear kernels for k and l, for instance, to obtain computational savings.


Hence, knowledge of the onset times can help separate the evoked sources.

the onset times, which are determined by the experimental design and available during data analysis, are ignored by ICA.


The SBA algorithm is defined up to permutations of the nodes, so the estimated graphon is not canonical.

this does not affect the consistency properties of the SBA algorithm, as the consistency is measured w.r.t.


Note that the above expression does not have exactly the same form as the distribution in Equation ( 7) and is not exchangeable since it depends on the order the data arrive.

if we consider only the left-ordered class of matrices generated by the stochastic process then we obtain the exchangeable distribution in Equation (7).


The parameter, which was initially designed to avoid precision issues in practical implementations, is often overlooked.

it has been observed that very small in some applications has also resulted in performance issues, indicating that it has a role to play in convergence of the algorithm.


The cross-entropy error function is commonly employed to train W out for a classification task.

this error function unfortunately cannot be directly applied to the present case.


Consequently, the reported variability in the learned network structures tends to be smaller than the uncertainty determined by local search (without this additional information).

we are mainly interested in the bias induced by the bootstrap here, which can be expected to be largely unaffected by the search strategy.


The former was first developed in [8], and has enabled the convolution including second-nearest-neighbor luminance data only using nearest neighbor interconnects, thus greatly reducing the interconnect complexity.

the fill factor was sacrificed due to the pixel parallel organization.


Conventional wisdom dictates that the number of bins is a key parameter for the coarse-graining strategy and should be carefully chosen for the balance of statistical noise and discretization error.

we will show in what follows that it is justifiable to increase the number of bins to infinity.


In the Gaussian process regression the observation model is commonly assumed to be Gaussian, which is convenient in computational perspective.

the drawback is that the predictive accuracy of the model can be significantly compromised if the observations are contaminated by outliers.


In general, the actual sequence of play may fail to converge altogether, even in simple, finite games [16,24].

there is a number of recent works establishing the convergence of play in potential games with finite action sets under different assumptions for the number of players involved (continuous or finite) and the quality of the available feedback (perfect, semi-bandit/imperfect, or bandit/payoff-based) [5,11,14,19].


Figure 2b shows that uniform convergence also does not preserve the property of being a global function: all points on the middle part of the limit function (blue curve) are spurious local minima.

it suggests that uniform convergence preserves a slightly weaker property than being a global function.


A general disadvantage of KRR, is that it can be difficult to know which aspects of X are relied on to perform the regression.

the kernel IB framework provides an intermediate representation, allowing one to visualize the features that jointly account for both X and Y (figs.


Note for P ⊂ R m , P is contained in the linear span of at most m points from P , and similarly the exact Carathéodory theorem states any point q ∈ Convex(P ) is expressible as a convex combination of at most m + 1 points from P . As the conic hull lies between the linear case (with all combinations) and the convex case (with non-negative combinations summing to one), it is not surprising an exact conic Carathéodory theorem holds.

the linear analog of the approximate convex Caratheodory theorem does not hold, and so the following conic result is not a priori obvious.


(The MSE results did not stay the same because the image completions happened over different subsets of the pixels.)

the performance of the Poon architecture dropped considerably due to the fact that it was no longer able to take advantage of strong correlations between neighboring pixels.


Young children demonstrate the ability to make inferences about the preferences of other agents based on their choices.

there exists no overarching account of what children are doing when they learn about preferences or how they use that knowledge.


Note that the I-norm term Ilwlll can be replaced by one half the 2-norm squared, ~llwll~, which is the usual margin maximization term for ordinary support vector machine classifiers [18,3].

this changes the linear program (19) to a quadratic program which typically takes longer time to solve.


We argue in Section 5.1 that the unbalanced Haar wavelets are a special instance of graph Laplacian eigenbasis when the underlying graph is hierarchical.

a lattice graph structure yields activations that are globally supported and smooth, and in this case the Laplacian eigenbasis corresponds to the Fourier transform (see Section 5.2).


The advantage of SDCA over AGD is that each iteration involves only a single dual vector and usually costs O(d).

each iteration of AGD requires Ω(nd) operations.


Finally, we point out that for many of the datasets we consider, there is no significant difference between the LP based algorithm, and the Local Search (and sometimes even the Greedy) heuristic in terms of the sum-min objective value.

as we noted, the heuristics do not have worst case guarantees.


Power Multiset Classification A brute-force approach based on the Combination Method in multilabel classification [22,17], is to transform the class set C into a set M (C) of all possible multisets, then train a multi-class classifier π that maps an input x to one of the elements in M (C).

the number of all possible multisets grows exponentially in the maximum size of a target multiset, 2 . rendering this approach infeasible in practice.


For any data matrix X that satisfies the SSC and SSS properties such that Using Lemma 12 (see the appendix), we can translate the above result to show that w T0 − w * 2 ≤ 0.95σ + , assuming k * ≤ k ≤ n 150 .

lemma 5 will be more useful in the following fine convergence analysis.


This gives us, For state s = s t , this simplifies to Substituting this back into Equation ( 6) we obtain, This gives us an explicit expression for our V estimates.

from an algorithmic perspective an incremental update rule is more convenient.


Methods for learning from demonstration (LfD) have shown success in acquiring behavior policies by imitating a user.

even for a single task, LfD may require numerous demonstrations.


When using only a few trajectories, the diagonal includes fluctuations that can have significant impacts on the resulting weights.

when using many trajectories (which we treat as giving ground truth), the diagonal tends to be relatively smooth and monotonically increasing until it plateaus (ignoring the final entry).


For smooth and strongly convex functions, we prove that this method enjoys the same fast convergence rate as those of SDCA and SAG.

our proof is significantly simpler and more intuitive.


Note the single-variable problem ( 14) has closed-form solution which in a naive implementation, takes O(nnz(A)) time due to the computation of ( 11) and ( 12).

in a clever implementation, one can maintain the relation ( 11), (12) as follows whenever a coordinate x j is updated by where a j = [a I j ; a E j ] denotes the jth column of A I and A E . Then the gradient and (generalized) second-derivative of jth coordinate can be computed in O(nnz(a j )) time.


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

adding diversity without improving semantic relevance may occasionally hurt these relevance scores.


Roughly speaking, the events ω ∈ Ω represent abstract states of nature, i.e.

knowing the value of ω completely describes all probabilistic aspects of the model universe, and all random aspects are described by the probability measure P. Ω, A and P are never known explicitly, but rather constitute the modeling assumption that any explicitly known distribution P X is derived from one and the same probability measure P through some random variable X.


Instead, [18] showed that in the case of coverage functions, it is possible to efficiently maximize f by lifting the problem to the continuous domain and using stochastic gradient methods on a continuous relaxation to reach a solution that is within a factor (1 − 1 e) of the optimum.

our work provides a general recipe with 1 2 approximation guarantee for problem (1.2) in which f θ 's can be any monotone submodular function.


To encourage consistency while reusing general purpose solvers, a PN-consistency potential [9] is often incorporated into the model: Hereby c is a positive constant that is tuned to penalize for the violation of consistency.

as c increases, the following constraint holds: usage of PN-potentials raises concerns: (i) increasing the number of pairwise constraints decreases computational efficiency, (ii) enforcing consistency in a soft manner requires tuning of an additional parameter c, (iii) large values of c reduce convergence, and (iv) large values of c result in corresponding beliefs being assigned zero probability mass which is not desirable.


One important remark about the k-gram conditional swap regret is that it is a random quantity that depends on the particular sequence of actions played.

a natural deterministic alternative would be of the form: by taking the expectation of Reg �k ��� T ) with respect to a T −1 � a T2 � .


scenario and adversarial on-line learning setting.

the key component of our learning guarantees is the discrepancy term .


First, the flow of information between tasks should not be dictated by a rigid diagram that reflects the relationship between the tasks themselves, such as hierarchical or temporal dependencies.

information should be exchanged across tasks whenever useful.


Finally, we simulate profile face images in various poses with pre-defined yaw angles.

the performance of the simulator decreases dramatically under large poses (e.g., yaw angles ) due to artifacts and severe texture losses, misleading the network to overfit to fake information only presented in synthetic images and fail to generalize well on real data.


Lloyd's algorithm can be parallelized in the MapReduce framework (Zhao et al., 2009) or even replaced by fast stochastic optimization techniques such as online or mini-batch k-Means (Bottou & Bengio, 1994;Sculley, 2010).

the seeding step requires k inherently sequential passes through the data, making it impractical even for moderate k. This highlights the need for a fast and scalable seeding algorithm.


In cases where the expectation is intractable to compute exactly, it can be approximated by a Monte Carlo estimate: where {v n : 1 ≤ n ≤ N } are a set of samples of X.

obtaining a good estimate will require many samples if f (X D ) has high variance.


In the first paper [1], independence between blocks and delays is avoided.

they require a step size that diminishes at 1/k and that the sequence of iterate is bounded (which in general may not be true).


Although the frustration in these networks is relatively mild, REC-BP did not converge in any of these cases.

rEC-I compensations were relatively well behaved, and produced monotonically decreasing upper bounds on the MAP value; see Figure 1 (center).


Note BPM [31]'s runtime is significantly more expensive than other methods (empirically an order higher than ours using the public source code) as it simultaneously seeks multiple paths for the best score (though accuracy is similar to ours).

our method focus on one path no matter the path following or multiplicative strategy is used.


Performance is convincing also for stripes and dots, especially since these attributes have generic appearance, and hence must be recognized based only on geometry and layout.

colors enjoy a very distinctive, specific appearance.


The problem of finding the M MPCs has been successfully solved within the junction tree (JT) framework.

to the best of our knowledge, there has been no equivalent solution when building a junction tree is infeasible.


In * Equal contributions particular, the popular Mahalanobis metric weights each feature (and their interactions) additively when calculating distances.

similarity can arise from a complex aggregation of comparing data instances on multiple subsets of features, to which we refer as latent components.


But to the best of our knowledge, no previous deep learning approaches on general graphs preserve the order of edges.

we propose a novel way of graph embedding that can preserve the information of edge ordering, and demonstrate its effectiveness for premise selection.


( 1), it may be tempting to treat the environment s as a discrete latent variable and learn it through amortised variational inference.

we found that in the continual learning scenario this is not a viable strategy.


In essence, these methods consider explicit factorization of a coupled tensor T ∈ R n1×n2×n3 and a matrix , respectively, with a common rank R and shared components a i , i = 1, . . . , R. Many variations of factorization models for coupled completion models have been proposed based on CP decomposition with shared and unshared components (Acar et al., 2014), Tucker decomposition (Ermis et al., 2015), and non-negative factorization (Ermis et al., 2015).

due to factorization, these coupled completion models are non-convex that lead to local optimal solutions.


Feedback loops are prevalent in animal behaviour, where they are normally called a "reflex".

the reflex has the disadvantage of always being too late.


In [4], the authors propose to minimize the cut between samples and features, which is equivalent to conducting spectral clustering on the bipartite graph.

in this method, since the original graph doesn't display an explicit cluster structure, it still calls for the post-processing step like K-mean clustering to obtain the final clustering indicators, which may not be optimal.


@ , 3 © ¡ ¦¢ at the same initial belief states used in [12], but without using them to tailor the policy.

pBVI achieved expected returns of ¡ © ¡ ¨ § , 3 © § ¦ and 3 © ¨ in ¦ ¢ )¦ , ¡ ¨¢ ¨¢ ©¦ and ¢ ©3 ©¦ with policies of ¤£ 3 , ¢ ¢ and ¥ § linear segments tailored to those initial belief states.


This demonstrates the main limitation of our approach to randomly sample 'seed' patches: it does not scale to arbitrarily large amounts of unlabeled data.

we do not see this as a fundamental restriction and discuss possible solutions in Section 5 .  saturates at around 100 samples.


This is especially true for the move-to-front rules.

the cue orders resulting from all learning rules but the validity learning rule do not correlate or correlate negatively with the validity cue order, and even the correlations of the cue orders resulting from the validity learning rule after 100 decisions only reach an average r = .12.


This leads to a branching law with exponent of 5/2.

the presence of reflections from branch points and active conductances is likely to complicate the picture.


Note that p∈P f x (p) − q∈S w q f x (q) does not equal p∈P cost(p, x) − q∈S w q cost(q, x).

it equals the difference between p∈P cost(p, x) and a weighted cost of the sampled points and the centers in the approximation solution.


The notion of universal equivalence might appear quite restrictive because condition (10) must hold for any underlying probability measure P (X, Y ).

this is precisely what we need when P (X, Y is unknown.


Valid causal inference in observational studies often requires controlling for confounders.

in practice measurements of confounders may be noisy, and can lead to biased estimates of causal effects.


In fact, alternative optimization is popular in generative adversarial networks [8], in which a generator and discriminator get alternatively updated.

alternative optimization has the following shortcomings.


Concretely, for decomposing an m×n matrix, say with m ≤ n, the best specialized implementations (typically first-order methods) have a per-iteration complexity of O m 2 n , and require O(1/ ) number of iterations to achieve an error of .

the usual PCA, which carries out a rankr approximation of the input matrix, has O(rmn) complexity per iteration -drastically smaller when r is much smaller than m, n. Moreover, PCA requires exponentially fewer iterations for convergence: an accuracy is achieved with only O (log(1/ )) iterations (assuming constant gap in singular values).


When it does succeed, it produces results that are comparable with LELVM, although somewhat less accurate visually.

even then GPLVM's latent space consists of continuous chunks spread apart and offset from each other; GPLVM has no incentive to place nearby two xs mapping to the same y.


The famous Newton step corresponds to a change of variables D 1 2 = H 1 2 which makes the new Hessian perfectly conditioned.

a change of variables only exists 2 when the Hessian H is positive semi-definite.


Hence, the likelihood function takes the right shape around the training samples, but not necessarily everywhere.

the code vector in an RBM is binary and noisy, and one may wonder whether this does not have the effect of surreptitiously limiting the information content of the code, thereby further minimizing the log partition function as a bonus.


In order to do this, we make regions of high density larger, and we make regions of low density smaller.

the Jacobi metric does not completely override the old notion of distance and scale; the Jacobi metric provides a compromise between physical distance and density of the probability measure.


Nonnegative Matrix Factorization (NMF) is a promising relaxation technique for clustering analysis.

conventional NMF methods that directly approximate the pairwise similarities using the least square error often yield mediocre performance for data in curved manifolds because they can capture only the immediate similarities between data samples.


[39] introduce an end-to-end imitation learning framework that learns to drive entirely from visual information, and test their approach on real-world scenarios.

their method uses behavior cloning by performing supervised learning over the state-action pairs, which is well-known to generalize poorly to more sophisticated tasks, such as changing lanes or passing vehicles.


Below, we show that a global solution of , the global solution is unique and diagonal.

when C A C B is degenerate, the global solutions are not unique because arbitrary rotation in the degenerate subspace is possible without changing the free energy.


When predicting with smaller than the one used for learning the results are marginally worse than when predicting with the same .

when predicting with larger , the results get significantly worse, e.g., learning with = 0.01 and predicting with = 1 results in 10 errors, and only 2 when = 0.01.


We do not consider features in this paper.

since most of the online game matching data has features associated with each game, it is our future work to explore this area.


It is possible to recursively insert minimum variance unbiased baseline terms into these expectations in order to reduce the variance on the baseline estimates.

the number of baseline parameters being estimated increases rapidly in this recursive process.


The Kendall's τ distance is a natural discrepancy measure when permutations are interpreted as rankings and is thus the most widely used in the preference learning literature.

the Hamming distance is particularly used when permutations represent matching of bipartite graphs and is thus also very popular (see Fathony et al.


Furthermore, the PY model's usage of thresholded Gaussian processes leads to a complex likelihood function, for which inference is a significant challenge.

ddCRP inference is carried out through a straightforward sampling algorithm, 5 and thus may provide a simpler foundation for building rich models of visual scenes.


We do not consider these approaches in this work, in part due to the fact that the bounded-delay assumptions associated with most asynchronous schemes limit fault tolerance.

it would be interesting to further explore the differences and connections between asynchronous methods and approximation-based, synchronous methods like MOCHA in future work.


For instance, we know that the sample complexity can be much smaller than the radius of the support of X, if the average norm E[ X 2 ] is small.

e[ X 2 ] is also not a precise characterization of the sample complexity, for instance in low dimensions.


2 While Theorem 1 is positive for finding variable settings that satisfy sentences, unsatisfiable sentences remain problematic when we are unsure that there exists γ > 0 or if we have an incorrect setting of f . We are unaware of an efficient method to determine all y ij for visited nodes in proofs of unsatisfiable sentences.

we expect that similar substructures will exist in satisfiable and unsatisfiable sentences resulting from the same application.


Early classification should become more effective once MI-RNN identifies instances that contain true and tight class signatures as these signatures are unique to that class.

since the two methods are trained separately, a naïve combination ends up reducing the accuracy significantly in many cases (see Figure 4).


Therefore, each iteration takes O(h 2 |L|) time, and the total time to obtain an optimal k-leaf-sparse solution is O(h 2 k|L|).

a brute-force search will take |L| k time.


The issue, however, is that the large terms are extremely volatile and could dominate all other components in an undesired way.

tWF makes use of only gradient components of typical sizes, which slightly increases the bias but remarkably reduces the variance of the descent direction.


Recent interest in such models has surged due to their biological plausibility and accuracy for characterizing early sensory responses.

fitting poses a difficult computational challenge due to the expense of evaluating the log-likelihood and the ubiquity of local optima.


For GMM, it is still possible to use a matched linear averaging which matches the mixture components of the different local models by minimizing a symmetric KL divergence; the same idea can be used on our linear control variates method to make it applicable to GMM.

because the parameters of PPCA-based models are unidentifiable up to arbitrary orthonormal transforms, linear averaging and linear control variates can no longer be applied easily.


These works focus on a multi-task setting in which meta-learning takes place on a distribution of training tasks, to facilitate fast adaptation on an unseen test task.

our work emphasises the (arguably) more fundamental problem of meta-learning within a single task.


For the convergence of this instance of the algorithm, it is required that all the states and actions are visited infinitely many times, which makes the analysis slightly more complicated.

given a generative model, the algorithm may be also formulated in a synchronous fashion, in which we first generate a next state y ∼ P (•|x, a) for each state-action pair (x, a), and then update the action-values of all the stateaction pairs using these samples.


Nevertheless, the emphasis has been on asymptotic analysis, characterizing the rates of convergence of test statistics under null hypotheses, as the number of samples tends to infinity.

we wish to study the following problem in the small sample regime: ⇧(C, "): Given a family of distributions C, some " > 0, and sample access to an unknown distribution p over a discrete support, how many samples are required to distinguish between p 2 C versus d TV (p, C) > "?


Arguably, one solution is to reduce the dimensionality of such ultra high dimensional data while preserving the original data distribution.


The GMM seems to support their intuition that learning separate linear subspace models for flat vs motion boundary is a good idea.


Since the distribution is unknown, the true error rate is not observable.

we can observe the empirical error rate, B ' C& D E0 F2 54 G (& ) H9 A A I ¦ QP ¦ SR I & T9 A .


SVGD has been applied to solve challenging inference problems in various domains; examples include Bayesian inference Feng et al., 2017), uncertainty quantification (Zhu & Zabaras, 2018), reinforcement learning (Liu et al., 2017;Haarnoja et al., 2017), learning deep probabilistic models Pu et al., 2017) and Bayesian meta learning (Feng et al., 2017;Kim et al., 2018).

the theoretical properties of SVGD are still largely unexplored.


Asynchronous-parallel algorithms have the potential to vastly speed up algorithms by eliminating costly synchronization.

our understanding of these algorithms is limited because the current convergence theory of asynchronous block coordinate descent algorithms is based on somewhat unrealistic assumptions.


Note that all settings fix b j = 1 since this yields the best rate as will be shown in Section 3.

in practice a reasonably large mini-batch size b j might be favorable due to the acceleration that could be achieved by vectorization; see Section 4 for more discussions on this point.


The RIP treats all possible K-sparse supports equally.

if we incorporate a probabilistic model on our signal supports and consider only the signal supports with the highest likelihoods, then we can potentially do much better in terms of the required number of measurements required for stable recovery.


A coarser but available alternative is to calculate the global Lipschitz constant.

prior work could provide only magnitudes of smaller certifications compared to the usual discretization of images even for small networks [29,24].


Labeled points participate in the information regularization in the same way as unlabeled points.

their conditionals have additional constraints, which incorporate the label information.


However, unlike them, it provides a principled framework for full Bayesian inference and can be used to determine how to trade off goodness-of-fit across summary statistics.

to the best of our knowledge, this potential has not been realised yet, and ABC approaches are not used for linking mechanistic models of neural dynamics with experimental data (for an exception, see [17]).


Thus, the solution above is communication-efficient only when λ is relatively large.

the situation immediately improves if we can use a without-replacement version of SVRG, which can easily be simulated with randomly partitioned data: The stochastic batches can now be simply subsets of each machine's data, which are statistically identical to sampling {f 1 (•), .


The surfaces and walls determine the stochastic dynamics of the world.

the agent also observes numerous other features in the environment.


Our framework, similar to the BicycleGAN, can be utilized to generate multiple realistic images for a single input, while does not require any supervision.

cycleGAN and UNIT learn one-to-one mappings as they learn only one domain-invariant latent code between the two modalities.


!, AQM-gen1Q-depA achieves a slight performance improvement over AQM-countQ-depA at 2-q (49.79% → 51.07%) outperforming the original deep SL method (46.8% in 5-q).

at 5-q, AQM-gen1Q performs slightly worse than AQM-countQ-depA (72.89% → 70.74%).


In fact, we are confident pseudo-counts may be used to prove similar results in non-tabular settings.

it may be difficult to provide theoretical guarantees about existing bonus-based intrinsic motivation approaches.


Although EM with power method [4] shares the same computational complexity as ours, there is no convergence guarantee for EM to the best of our knowledge.

we provide local convergence guarantee for our method.


[3], based on multiple kernel learning framework, further demonstrated that an additional text modality can improve the accuracy of SVMs on various object recognition tasks.

all of these approaches are discriminative by nature and cannot make use of large amounts of unlabeled data or deal easily with missing input modalities.


This is because some redundant attributes dominated the selection process and the attributes selected by comparing approaches had very unbalanced discrimination capability for different classes.

the attributes selected by our method have strong and similar discrimination capability for each class.


As expected, no method wins across all experiments.

the results show that the method that wins the most (out of the 9 options) is either the combination of SeLU and VCL or that of ELU and VCL.


For small values of a, our algorithm performs worse than the baseline of directly using θ 0 , likely due to finite-sample effects.

our algorithm is far more robust as a increases, and tracks the performance of an oracle that was trained on the same distribution as the test examples.


At first sight this may appear to be restrictive.

as we show in the supplementary material, one can construct Lasso problems using a Gaussian basis W m which lead to penalty parameter bounds ratios that converge in distribution to those of the Lasso problem in Eq.


Loewenstein & Seung (2006) demonstrated that matching behavior is a steady state of learning in neural networks if the synaptic weights change proportionally to the covariance between reward and neural activities.

their proof did not take into account the change in entire synaptic distributions.


We show that the quality of the tensor sketch does not depend on sparseness, uniform entry distribution, or any other properties of the input tensor.

previous works assume specific settings such as sparse tensors [24,8,16], or tensors having entries with similar magnitude [27].


[10]) has similar goals, but is technically quite distinct: the canonical problem in preference learning is to learn a ranking on distinct elements.

the problem we consider here is to predict the outcome of a continuous optimization problem as a function of varying constraints.


However, these methods reason about the image only on the global level using a single, fixed-sized representation from the top layer of a Convolutional Neural Network as a description for the entire image.

our model explicitly reasons about objects that make up a complex scene.


Methods exist for learning such systems from data [18,19]; these methods are able to handle multivariate target variables and models that repeat in the sequence.

they are consequently more complex and computationally intensive than the much simpler changepoint detection method we use, and they have not been used in the context of skill acquisition.


Quite a few papers studied alternative models, where the actions are endowed with a richer structure.

in the large majority of such papers, the feedback structure is the same as in the standard multi-armed bandits.


The framework presented here has some similarities with the very interesting and more explicitly physiological model proposed by Buonomano and colleagues [5,18], in which time is implicitly encoded in deterministic 2 neural networks through slow neuronal time constants.

temporal information in the network model is lost when there are stimulus-independent fluctuations in the network activity, and the network can only be used as a reliable timer when it starts from a fixed resting state, and if the stimulus is identical on every trial.


The process is asymptotically stationary if lim l!1 (l) = 0.

the most important property of the discrepancy is that, as shown later in Section 4, it can be estimated from data under some additional mild assumptions.


It can be seen that the PRFs look very different to the usual center-surrond structure of retinal ganglion.

one should keep in mind that it is really the space spanned by the PRFs that is relevant, and thus be careful when interpreting the actual filter shapes [ 1 5] .


Intuitively, imagine that we grow a ball of radius r around each sample point.

the union of these balls roughly captures the hidden domain at scale r. the topological structure of the union of these balls is captured by the so-called Čech complex, which mathematically is the nerve of this union of balls.


Additive noising introduces a product-form penalty depending on both and A 00 .

the full potential of artificial feature noising only emerges with dropout, which allows the penalty terms due to and A 00 to interact in a non-trivial way through the design matrix X (except for linear regression, in which all the noising schemes we consider collapse to ridge regression).


We demonstrate that under favorable conditions, we can construct logarithmic depth trees that have leaves with low label entropy.

the objective function at the nodes is challenging to optimize computationally.


As we only wish to model the hidden harmonic state given the melody, rather than construct a full generative model of the data, Conditional Random Fields (CRFs) [14] provide a related but alternative framework.


In standard GP applications, one has access to a single realisation of data y, and performs kernel learning by optimizing the marginal likelihood of the data with respect to covariance function hyperparameters θ (supplement).

with only a single realisation of data we are highly constrained in our ability to learn an expressive kernel function -requiring us to make strong assumptions, such as RBF covariances, to extract useful information from the data.


Using the above analyses, any γ that successfully disentangles e 1 and e 2 should be sufficient.

α and β can be selected by starting with α ≫ β and gradually increasing β as long as the performance of the prediction task improves.


Here, each event defines a relationship, e.g., whether in the event two entities' group(s) behave the same way or not.

in our model a relation may also have multiple attributes.


In this decomposition approach, the features specific to a topic, not a label, are regarded as important features.

the approach may result in inefficient learning as we will explain in the following example.


(2013) assumes that the buyer is fully strategic whereas we only require the buyer to be ✏-strategic.

the authors assume that the distribution satisfies a Lipchitz condition which technically allows them to bound the number of lies in the same way as in Proposition 2.


Matrix factorization (MF) collaborative filtering is an effective and widely used method in recommendation systems.

the problem of finding an optimal trade-off between exploration and exploitation (otherwise known as the bandit problem), a crucial problem in collaborative filtering from cold-start, has not been previously addressed.


[13,21] proposed object-level saliency maps to explain RL policies in visual domains by measuring the impact on action choices when we replace an object in the input image by the surrounding background.

templates for each object must be hand-engineered.


The above-mentioned work all addresses the tensor completion problem in which the locations of the missing entries are known, and moreover, observation noise is assumed to be Gaussian.

in practice, a fraction of the tensorial entries can be arbitrarily corrupted by some large errors, and the number and the locations of the corrupted entries are unknown.


A single inclusion mistake is sufficient for the OGMB learner to learn this hypothesis space.

the teacher can supply the KWIK learner with an exponential number of positive examples, because the KWIK learner cannot ever know that the target does not include all possible instances; this implies that the number of abstentions is not polynomially bounded.


To map the complex state h into a real output o r , we use a linear combination of the real and imaginary components, similar to [1], with W o and b o as weights and offset: In [1], it was proven that a unitary 4 W would prevent vanishing and exploding gradients of the cost function C with respect to h t , since the gradient magnitude is bounded.

this proof hinges on the assumption that the derivative of f a is also unity.


Two conditions are met in this scenario: (1) Temporal features are strong enough for classification tasks.

fine-grained spatial appearances prove to be less significant; (2) There are no complex visual structures to be modeled in the expected outputs so that spatial representations can be highly abstracted.


This was overcome by the use of Gradient Difference Loss (GDL) [14], which showed significant improvement over the past approaches when compared using similarity and sharpness measures.

this approach, although producing satisfying results for the first few predicted frames, tends to generate blurry results for predictions far away (∼6) in the future.


They demonstrate gains when parallelizing the computation across multiple machines in a cluster.

their approach requires the employed processing units to run in synchrony.


In [45], it was mentioned that any continuously differentiable value function (VF) can be approximated by increasing the number of independent basis functions to infinity in CT scenarios, and a CT policy iteration was proposed.

without resorting to the theory of reproducing kernels [3], determining the number of basis functions and selecting the suitable basis function class cannot be performed systematically in general.


A similar informal formulation appears in the work [1] that is devoted to optimizing a generalization of the ICA objective.

the actual problem considered only concerns the case of tree-structured dependence, which allows for a solution based on pairwise measurements of mutual information.


Until now the same could not be said for automatic speech recognition systems.

we have recently introduced a system which in many conditions performs this task better than humans [1][2].


Generic submodular maximization admits efficient algorithms that can attain approximate optima with global guarantees; these algorithms are typically based on local search techniques [16,35].

although polynomial-time solvable, submodular function minimization (SFM) which seeks to solve poses substantial algorithmic difficulties.


This setting is analogous to the problem of joint source-channel coding.

just like before, each encoded bit must depend on at most ∆ bits.


it can be anticipated that no element in the estimated 𝚪 and 𝚿 will be exactly zero, resulting in a model which is not interpretable, i.e., poor identification of diseaserelated regions.


In general, when one considers the temporal dynamics of an episode, more than two goals, or more complicated reward structures, the relationship becomes more complicated.

information is useful in abstracting away that complexity, and preparing Alice generically for a plethora of possible task setups.


This problem is often tackled in the same manner as object recognition and localization in images.

extension to a temporal domain comes with many challenges.


With heterogeneous costs, the quality of an action k under a context j is roughly captured by its normalized expected reward, defined as η j,k = u j,k /c j,k .

the agent cannot only focus on the "best" action, i.e., k * j = arg max k∈A η j,k , for context j.


Let G * be the highest-scoring decomposable DAG from step 2.

the upper bounds obtained via BNSL can be at times can be quite weak when the network structures contain many immoralities.


Deep features learned by CNNs can disentangle explanatory factors of variations behind data distributions to boost knowledge transfer [19,17].

the latest literature findings reveal that deep features can reduce, but not remove, the cross-domain distribution discrepancy [3], which motivates the state of the art deep feature adaptation methods [5,6,7].


Recently, several works [19,14] have made efforts to extend ADMM to its online or stochastic versions.

they suffer relatively low convergence rates.


For these more difficult grids, however, STRIPES was the fastest of the algorithms, taking 0.5 -5 minutes.

the Nilsson and BMMF algorithms took 18 minutes and 2.5 -7 minutes, respectively.


To indicate the form of a representer theorem, suppose we solve for the optimal parameters Θ * = arg min Θ 1 n n i L(x i , y i , Θ) + g(||Θ||) for some nondecreasing g. We would then like our pre-activation predictions Φ(x t , Θ) to have the decomposition: Given such a representer theorem, α i k(x t , x i ) can be seen as the contribution of the training data x i on the testing prediction Φ(x t , Θ).

such representer theorems have only been developed for non-parametric predictors, specifically where Φ lies in a reproducing kernel Hilbert space.


Another approach is to pick from (m 1 , m 2 ) before reaching the factor f , so that the model becomes In this case, the message from x to f is not blurred, and the upward messages to (m 1 , m 2 ) are blurred, which is correct.

the downward messages from (m 1 , m 2 ) to f are blurred before reaching f , which is incorrect.


there is at least one pair of priors p i and p j with overlapping support), then one can do better by utilizing the "super prior" p = 1 k i p i within the original Theorem 3.2.

note that when the supports are disjoint, these two views (of multiple priors and a super prior) are equivalent.


This assumption is rather permissive and is satisfied by many first-order algorithms, e.g., SAG and SAGA [6].

the lower bound stated in the paper faces limitations in a few aspects.


It is worth pointing out that, similar to the MMTM, our PTM is not a generative model of networks per se because (a) empty and single-edge motifs are not modeled, and (b) one can generate a set of triangles that does not correspond to any network, because the generative process does not force overlapping triangles to have consistent edge values.

given a bag of triangular motifs E extracted from a network, the above procedure defines a valid probabilistic model p(E | α, λ) and we can legitimately use it for performing posterior inference p(s, θ, B | E, α, λ).


As work in [2] suggests, it is unlikely that a polynomial time algorithm can avoid such dependence.

once we are near the solution, as we show, this two-step procedure achieves the optimal error rate of s log p/n.


In some cases manual alignment to salient external events or behavioural time-course may be used to reduce temporal misalignment [8,9].

just as with variability in the trajectories themselves, temporal variations in purely internal states must ultimately be identified from neural data alone [10].


Overall both algorithms LINKAGE++ and Single-linkage perform considerably better when it comes to real-world data and LINKAGE++ and PCA+ dominate on our synthetic datasets.

in general there is no reason to believe that PCA+ would perform well in clustering truly hierarchical data: there are regimes of the HSBM for which applying only phase 1 of the algorithm might lead to a high missclassification error and high cost and for which we can prove that LINKAGE++ is an p1`εq-approximation.


Some families have an elliptical dependence structure, similar to the multivariate normal distribution.

it is also possible to use completely different dependence structures which are more appropriate for the data at hand.


Optimization problem 8 is a mixed-integer quadratic programming problem (MIQP), which is NPhard in general.

we can exploit the structure of the problem via a block-coordinate descent algorithm where W and {ρ i,m } are optimized iteratively: 1.


For example, a conversion from scales to frequencies can be estimated using the center frequency of the mother wavelet F c , F s = Fc s [10].

converting from scales to frequency is not useful unless prior knowledge about the signal is available or assumptions are made on the relevant frequency content of the signal.


Critically, this is where we diverge from previous analyses that assumed this distribution was factorised, or only trivially correlated due to reciprocal synapses being precisely (anti-)symmetric [1,2,4].

we explicitly study the emergence and effects of non-trivial correlations in the synaptic weight matrixdistribtion, because almost all synaptic plasticity rules induce statistical dependencies between the synaptic weights of each neuron (Fig.


Ideally, we would have that E g 2 2 ≤ α(a i x) for some value of α, as previous methods do.

instead we settle for a bound of the form E g 2 2 ≤ α(a i x)+β x 2 2 .


The conversion strategies in this family also begin by using A to generate the sequence of online hypotheses.

instead of relying on a single hypothesis from the sequence, they set h to be some combination of the entire sequence.


Kernel-or nearest-neighbor-based methods, including nearly all of the methods described in Section 3, tend to require storing previously observed data, resulting in O(n) space requirements.

orthogonal basis estimation requires storing only O(Z D n ) estimated Fourier coefficients.


For small dimensionalities eigenvalues are small and therefore there is no advantage for oc-shrinkage.

the higher the order of oc-shrinkage, the larger the error by projecting out spurious large eigenvalues which should have been subject to regularization.


A symmetric prior over Φ only makes a prior statement (determined by the concentration parameter β) about whether topics will have more sparse or more uniform distributions over words, so the topics are free to be as distinct and specialized as is necessary.

it is still necessary to account for power-law word usage.


Previous GP-based classifiers did not use f within a margin-based classifier as in (6), implemented here via p(u n ) = N (−λ n , γ −1 λ n ), where u n = 1−y n f n . It has been shown empirically that nonlinear SVMs and GP classifiers often perform similarly [8].

for the latter, inference can be challenging due to the non-conjugacy of multivariate normal distribution to the link function.


1(a)), 'MF' performs slightly better than our GenDeR when 300 ≤ k ≤ 400; and for the other values of k, the two methods mix with each other.


Intuitively this is because EM attempts to maximize the overall likelihood.

our algorithm has significantly superior performance with respect to the edit distance which is the error in estimating the tree structure in the two components, as seen in Fig 2 .


When the target distribution is not Gibbs, we demonstrate that the second approach need not produce the optimal Gibbs distribution (with respect to log loss) even in the limit of infinitely many samples.

we prove that it produces models that are almost as good as the best Gibbs distribution according to a certain Bregman divergence that depends on the selection bias.


Dual decomposition methods are typically employed to solve inference tasks over combinatorial structures (e.g., [12; 13]).


We have shown so far that our trained policies outperform appropriate baselines for the task when tested on novel environments.

we are still training and testing on the same settings, such as noise level, trajectory length and distribution of trajectories.


However, in cases where all of the input densities have little overlap with the product density, mixture IS performs very poorly (see Figure 4(c)).

multiscale samplers perform very well in such situations, because they can discard large numbers of low weight product density kernels.


Nguyen et al [23], Singh and Póczos [24], and Krishnamurthy et al [25] each proposed divergence estimators that achieve the parametric convergence rate (O 1 T ) under weaker conditions than those given in [1].

solving the convex problem of [23] can be more demanding for large sample sizes than the estimator given in [1] which depends only on simple density plug-in estimates and an offline convex optimization problem.


For an input image I and a given pair of kernels, we can measure the data log-likelihood by associating each window with the maximum likelihood kernel: We search for a blurring model p k0 such that, when combined with the model p 1 (derivatives of the unblurred image), will maximize the log-likelihood of the observed derivatives: One problem we need to address in defining the likelihoods is the fact that uniform areas, or areas with pure horizontal edges (the aperture problem) don't contain any information about the blur.

uniform areas receive the highest likelihoods from wide blur kernels (since the derivatives distribution for wide kernels is more concentrated around zero, as can be observed in figure 1(c)).


At first glance, incorporating supervision into the WMD appears computationally prohibitive, as each individual WMD computation scales cubically with respect to the (sparse) dimensionality of the documents.

we devise an efficient technique that exploits a relaxed version of the underlying optimal transport problem, called the Sinkhorn distance [6].


Subsequently, for any x * i ∈ [0, 1]: If x * i is the expected allocation of player i under the efficient allocation rule X * i (v) ≡ 1{v i = max j v j }, then taking expectation of Equation ( 9) over v i and adding across all players we get: The theorem then follows by invoking the fact that for any feasible allocation x: , using the fact that expected total agent utility plus total revenue at equilibrium is equal to expected welfare at equilibrium and setting µ = µ(D).

comparison with worst-case POA In the worst-case, µ(D) is upper bounded by 1, leading to the well-known worst-case price of anarchy ratio of the single-item first price auction of (1 − 1/e) −1 , irrespective of the bid distribution D. if we know the distribution D then we can explicitly estimate µ, which can lead to a much better ratio (see Figure 1).


Note that when the observations are either discrete or parametric, it is possible to estimate the distribution using O(1/" 2 ) samples to achieve " error in a suitable metric, say, using the maximum likelihood estimate.

the nonparametric setting is inherently more difficult and therefore the rate of convergence is slower.


Inference using deep neural networks is often outsourced to the cloud since it is a computationally demanding task.

this raises a fundamental issue of trust.


All synapses connecting the input and output layers are equally likely to be active during an anti-causal regime.

the increase in average contribution to the postsynaptic membrane potential for the correlated group of neurons renders this population slightly more likely to be active during the causal regime than any single member of the uncorrelated group.


In order to encourage the reinforcement learning agent to discover positive symptoms more quickly, a simple heuristic is to provide the agent with an auxiliary piece of reward when a positive symptom is queried, and a relatively smaller (or even negative) reward when a negative symptom is queried.

this heuristic suffers from the risk of changing the optimal policy of the original MDP.


The non-smoothness of f can be challenging to tackle.

in many cases of interest, the function f enjoys a favorable structure that allows to tackle it with smoothing techniques.


The Input-Output HMM [3] extends HMMs by conditioning both their dynamics and emission model on an input sequence.

the IOHMM is representationally limited by its simple discrete state in the same way as a HMM.


Note that when τ " 1 this recovers the setting in [8].

we empirically found that using a small τ would result in accumulated ambiguity when generating words in our experiment.


Note that degenerate cluster allocations are generally suboptimal under the objective (1), as they would lead to a reduction in the marginal entropy H(y).

it is intuitive that maximization of the mutual information I(x, y) favors hard assignments of cluster labels to equiprobable data regions, as this would result in the growth in H(y) and reduction in H(y|x).


Return MDP assessment (π τ , T τ ) and This learning process is not guaranteed to converge, so upon termination, it could return an optimal, δ-stable MDP assessment for some very large δ.

it has been shown to be successful experimentally in simultaneous auction games [24] and other large games of imperfect information [7].


Recall d β (x, x ′ ) from ( 1) and consider If the above value is finite, then using any value even slightly larger than this would declare any x ′ ̸ ∈ S(x) correctly as such, hence no false positives.

the above infimum may not exist for a general configuration.


The results in this paper are a generalization of the results of Zhang and Yu [24] to the online setting.

we emphasize that this generalization is nontrivial and requires di↵erent algorithmic ideas and proof techniques.


Several machine learning frameworks such as TensorFlow [1], MXNet [2], and Caffe2 [3], come with distributed implementations of popular training algorithms, such as mini-batch SGD.

the empirical speed-up gains offered by distributed training, often fall short of the optimal linear scaling one would hope for.


We thus advocate using it as a replacement for the elastic net.

we also show that the gap between the elastic net and the k-support norm is at most a factor of √ 2, corresponding to a factor of two difference in the sample complexity.


Projected on the important dimension, clusters will be concentrated into two distinct points.

when the Euclidean distance is adopted as in K-Means, it is difficult to recover true clusters because two "lines" are close to each other.


For instance, regression with a decomposable kernel boils down to solving a Sylvester equation (which can be done efficiently) [10] and vector-valued Support Vector Machine (SVM) without intercept can be learned with a coordinate descent algorithm [21].

these methods can not be used in our setting since the loss function is different and considering the intercept is necessary for the quantile property.


Secondly, only the healthy subjects were remunerated.

repeating the analyses presented using only the MDD subjects yields the same results (data not shown).


Nowadays, to translate programs between different programming languages, typically programmers would manually investigate the correspondence between the grammars of the two languages, then develop a rule-based translator.

this process can be inefficient and error-prone.


Ideally one should design convex relaxations for each domain of Θ.

m exhibits some nice properties for any Θ: M 0, M I, tr(M ) = tr((ΘΘ ) † (ΘΘ )) = rank(ΘΘ ) = rank(Θ).


we have the following proposition about the gradient of f (L).


From this perspective HGMDPs appear to be a significant restriction over general POMDPs.

our first result shows that despite this restriction the worst-case complexity is not reduced even for deterministic dynamics.


A large number of previous studies of learning similarities have focused on metric learning, like in the case of a positive semidefinite matrix that defines a Mahalanobis distance [19].

similarity learning algorithms are often evaluated in a context of ranking [16,5].


These tasks are challenging, as code can be modified such that it syntactically differs (for instance, via different or reordered operations, or written in a different language altogether), but remains semantically equivalent (i.e., produces the same result).

these tasks are also ideal for machine learning, since they can be represented as classic regression and classification problems.


A similar situation was also observed in the server with 86G when n = 10 × 2 10 .

the memory required by BADMM is O(n 2 )-even when n = 15 × 2 10 (more than 0.2 billion parameters), BADMM can still run on a single GPU with only 5G memory.


The results are presented in Table 3 in Appendix F. In general, we observe P 3 & P 2 for all cases we have studied, which supports our conjecture.

this procedure is generally more time-consuming than EM for Model 2 since k!


[12] showed that the metric of the NSG corresponds with the changes in the stationary state-action joint distribution.

the metric of the NPG takes into account only changes in the action distribution and ignores changes in the state distribution, which also depends on the policy in general.


If we denote the predictions of the teacher by p(y|x, D N ) and the parameters of the student network by w, our objective becomes Unfortunately, computing this integral is not analytically tractable.

we can approximate this by Monte Carlo: where Θ is a set of samples from p(θ|D N ).


Ikeda pointed out that the problem we need to consider is the semiparametric model [10].


As can be seen from Table 1, layer normalization (Ba et al., 2016) improves the performance of PAG significantly.

according to our results on En→De, layer norm affects the performance of rPAG only marginally.


The representations that the network computes at different layers are related to the inference in an implicit latent variable model but the designer of the model does not need to know about them.

it is actually tremendously valuable to understand what kind of inference is required by different types of probabilistic models in order to design an efficient network architecture.


We observe significant improvements in 5 datasets (acoustic, census, heart, ionosphere, letter), demonstrating the advantage of using second order information.

we found that Oja-SON was outperformed by ADA-GRAD on most datasets, mostly because the diagonal adaptation of ADAGRAD greatly reduces the condition number on these datasets.


Their approach, similar to ours, generates a set of ranked class-agnostic proposals.

our model generates segmentation proposals instead of the less informative bounding box proposals.


Character-level convolutional filters are used to shrink the size of the inputembedding matrix in [13].

it still suffers from the gigantic output-embedding matrix.


Therefore, the condition number depends only on the largest eigenvalue of This shows that for large regularization parameters, the condition number is close to one and convergence is fast.

for small regularization parameters, the condition number gets very large, even if X is well-conditioned.


This may be undesired if we were actually interested in predicting noisy labels.

our goal is to predict clean labels, and the proposed framework benefits from the regularization that is imposed on the variational distribution.


While in standard control settings the sensors are assumed fixed, biological systems often gain from the extra flexibility of optimizing the sensors themselves.

this sensory adaptation is geared towards control rather than perception, as is often assumed.


Performance on large, realistic datasets is inarguably a better metric of architecture quality than performance on smaller datasets such as PTB.

such metrics make comparison among models nearly impossible: performance on large datasets is non-standard because evaluation at this scale is infeasible in many research settings simply because of limited hardware access.


Computing the optimal twisting functions boils down to performing exact inference in the model, which is assumed to be intractable.

this is where the use of deterministic inference algorithms comes into play.


The online incremental gradient (or backpropagation) algorithm is widely considered to be the fastest method for solving large-scale neural-network (NN) learning problems.

we show that an appropriately implemented iterative batch-mode (or block-mode) learning method can be much faster.


As we increase the scale of W, performance of the vanilla-RNN improves, suggesting that the model is able to better utilize the input information.

mI-RNN is much more robust to different initializations, where the scaling has almost no effect on the final performance.


On the one hand, a successful learner uncovers substantial structure of the target distribution.

this objective is clearly impossible when the means and covariances are extremely close.


Solving this equation analytically is not always possible.

for a broad class of functions, we can obtain an analytic solution.


Formally, this can be seen by comparing the variation of the log-determinant and trace functions with respect to the eigenvalues of the PSD matrix K, The gradient of the log-determinant is largest in the direction of the smallest eigenvalue of the error covariance matrix.

the MSE gives equal weight to all directions of the space.


CNN has the advantage of parallel processing because there is no dependency between the input and output.

cNN usually requires a large amount of feature maps, while RNN only needs to store the current cell and output state vectors in each layer.


Figure 2 shows that OLS leads to accurate ATE estimation for Gaussian additive noise when the number of covariates is sufficiently large, which is consistent with Corollary 1.1.

for high dimensional data, matrix factorization preprocessing dominates all other feasible methods and its RMSE is very close to the oracle regression for sufficiently large number of covariates.


First, the two square-root terms of the bound depend on r in opposite ways: the first is monotonically increasing, while the second is monotonically decreasing.

one could expect to optimize the bound by minimizing over r. the bound also depends on r indirectly via other quantities (e.g.


For sufficiently smooth objectives, the same algorithm is also optimal even if prox access is allowed, since Theorem 2 implies a lower bound of: That is, for smooth objectives, having access to a prox oracle does not improve the optimal complexity over just using gradient access.

for non-smooth or insufficiently smooth objectives, there is a gap between ( 11) and ( 12).


Hence, if both players play what we call optimal strategies, then neither player can improve and they are at Nash equilibrium.

suppose player 1 selects a strategy σ 1 that does not guarantee him payoff at least v − c 2 (σ 2 ).


While simple, this has no obvious probabilistic interpretation, and other divergences perform better in the experiments below.

it also forms the basis of our projected gradient descent strategy for computing the projection in Eq.


[22]'s adversarial loss encourages the hidden-state dynamics of teacher-forced and soft-sampled sequences to be similar.

there remains a gap between the dynamics of these sequences and sequences hard-sampled at test time.


with zero phase-lag [13][14][15][16][17], or strictly sequential patterns as in synfire chains [18][19][20][21] (see figure 1b).

some experimental studies have suggested that cortical spiking activity may harbor motifs with more complex structure [5,22] (see figure 1c).


Other rigorous variational lower bounds on the softmax have been used before [4,5], however they are not easily scalable since they require optimizing data-specific variational parameters.

the bound we introduce in this paper does not contain any variational parameter, which greatly facilitates stochastic minibatch training.


Note that while the set of rank-L PSD matrices is non-convex, we can still project onto this set efficiently using the eigenvalue decomposition of where r = min( L, L + M ) and L + M is the number of positive eigenvalues of M . Λ M (1 : r) denotes the top-r eigenvalues of M and U M (1 : r) denotes the corresponding eigenvectors.

while the above update restricts the rank of all intermediate iterates M t to be at most L, computing rank-L eigenvalue decomposition can still be fairly expensive for large n. by using special structure in the update (6), one can significantly reduce eigenvalue decomposition's computation complexity as well.


Some of the techniques in maximizing multilinear extensions [13; 7; 8] have inspired this work.

we are the first to explore the rich properties and devise algorithms for the general constrained DR-submodular maximization problem over continuous domains.


The results here provide substantial insight into the nature of GAN optimization, perhaps even offering some clues as to why these methods have worked so well despite not being convex-concave.

we also emphasize that there are substantial limitations to the analysis, and directions for future work.


Normalizing the bin histograms can then give us probability estimates that can be used to calculate entropies.

a loss based on entropy calculated with hard counts cannot be used to regularize the network, since the indicator quantization function is non-differentiable.


Efficient implementations of tensor operators, such as matrix multiplication and high dimensional convolution, are key enablers of effective deep learning systems.

current systems rely on manually optimized libraries, e.g., cuDNN, that support only a narrow range of server class GPUs.


Specifically, the number of assets under management is usually much larger than the sample size of exploitable historical data.

extreme events are typical in financial asset prices, leading to heavy-tailed asset returns.


The normalization To convert an input matrix X 2 R p (p = mp 0 ) into a score vector t 2 R m , it seems that we need to learn a matrix W 2 R m⇥mp0 .

a natural permutation invariance requirement (if the documents associated are presented in a permuted fashion, the output scores should also get permuted in the same way) reduces the dimensionality of W to p 0 (see, e.g., [14] for more details).


It has been proved in (Nesterov and Polyak, 2006, Theorem 2) that all limit points of {x k } k generated by CR are second order stationary points.

the sequence is not guaranteed to be convergent and no convergence rate is established.


We formulate the minimization of α-divergence with α = ∞ within the fractional covering framework [24].

the standard iterative algorithm for solving fractional covering is not readily applicable to our problem due to its small stepsize.


Finite differencing, where the perturbation directions d are the n standard basis vectors, is a default approach for Jacobian estimation.

it requires n function evaluations which may be prohibitively expensive for large n. Another natural approach, when the number of measurements, say k, is smaller than n, is to estimate the Jacobian via linear regression, where an l 2 regularizer is added to handle the underdetermined setting and • F stands for the Frobenius norm.


For example, according to the LWF Markov property, in the chain graph model in Figure 1(a), x 1 ⊥ x 3 |x 2 as x 1 and x 3 are separated by x 2 in the moralized graph in Figure 1(b).

the corresponding AMP Markov property implies a different probabilistic independence relationship, x 1 ⊥ x 3 .


CRFs with sufficiently expressive feature representation are consistent estimators of the marginal probabilities of variables in cliques of the graph [9], but are oblivious to the evaluative loss metric during training.

sSVMs directly incorporate the evaluative loss metric in the training optimization, but lack consistency guarantees for multiclass settings [10,11].


as we will now see, CPTs for the slow model need not be estimated or guessed-they can be derived from the fast model.


The 3D co-ordinates of a surface imaged by a perspective camera are physically ambiguous along a ray passing through the camera center.

a natural image often contains multiple cues that can indicate aspects of the scene's underlying geometry.


In all these works, the Markov chain is required to be reversible, and all functions f i , i ∈ [M ], are assumed to be convex.

non-reversible chains can have substantially faster convergence and thus more numerically efficient.


This requires a stronger separation condition between α and γ.

such "strong consistency" is only possible when the average degree grows faster than (n log n) 1/3 .


The straightforward approach is the well-known on-line K-means clustering algorithm, in which only the nearest reference vector to the sample vector is adjusted; G § )H IQP R S H ( T % U FV C# W4 6 § "H X# 8 (2) where, P is the step size and S H ¡ is the Kronecker delta.

this simple clustering algorithm is often stuck in a local minimum.


Both models correctly recover the stimulus disparity at most locations with gradual disparity changes, but tend to make errors at the depth boundaries.

the proposed model generates more accurate estimates.


As in the case of nearest neighbors-based methods, there is no guarantee that the points in the same manifold form a single connected component of the similarity graph built by SMCE.

this has always been the case in our experiments, as we will show next.


Moreover, an efficient training algorithm (EM) is available, as well as an efficient decoding algorithm (Viterbi), which provides the optimal sequence of states (and the corresponding sequence of high level events) associated with a given sequence of low-level data.

multimodal information processing is currently a very challenging framework of applications including multimodal person authentication, multimodal speech recognition, multimodal event analyzers, etc.


However, they tend to struggle when data is scarce or when they need to adapt to new tasks within a few numbers of steps.

humans are able to learn new concepts quickly, given just a few examples.


When the pattern that was actually presented is older than this estimate, the resulting memory signal is weaker than expected, suggesting that the initial pattern was very sparse (since a pair of inactive elements does not induce any synaptic changes according to our learning rule).

less reasonable is the fact that averaging over the prior distribution of recall times t (Eq.


As can be verified, mean map computes the means of x and y independently.

mean map computes the means of each of the C class-conditional distributions P (x|y).


[8] that the lower bounds of sample complexity of TOPK and MB problems are ⌦ H log(1/ ) , for the cases of Gaussian reward distributions.

for general decision classes with non-constant widths, we see that there is a gap of ⇥(width(M) 2 ) between the upper bound Eq.


From a qualitative standpoint, SDA-DP discovers sensible clusters in the data, as demonstrated in Figures 4a-4c.

an important quantitative result is highlighted by Table 4d: the larger a dataset is, the more the benefits of parallelism provided by SDA-DP become apparent.


If f is invertible, then the solution is trivial.

it is often the case that f is not trivially invertible, such as the CMB model mentioned in §1.


By standard Bayesian arguments, one can show that the optimal estimator of z to minimize the bit-wise error rate is given by ẑi = arg max Note that the EM algorithm (1), which maximizes rather than marginalizes q j , is not equivalent to the Bayesian estimator (3), and hence is expected to be suboptimal in terms of error rate.

calculating the marginal p(z i |L, θ) in (3) requires integrating all q and summing over all the other z i , a challenging computational task.


Noting that x * ∈ K, g(x * ) ≤ 0, we have F (x * ) ≤ f (x * ) + γ ln 2.

f ( x T ) = f ( x T ) + γ ln 1 + exp λ 0 g( x T ) γ ≥ f ( x T ) + max (0, λ 0 g( x T )) .


It is simple to specify a flat Dirichlet distribution by setting all ↵c = 1.

directly setting the in-distribution target ↵c is not convenient.


Note that it might be tempting to make the entire model non-linear and non-stationary.

the sample complexity of learning very general non-stationary, non-linear models is likely to be so high that they will not be useful in mHealth where data is often noisy, missing, or collected only over a few hundred decision points.


One popular approach for addressing this challenge is the options framework [29].

only recently in [1] was a policy gradient theorem derived for online learning of general purpose options in an end to end fashion.


User crash reports are used to direct debugging efforts toward those bugs which seem to affect the most people.

we can do much more with the information users may provide.


This occurs if one uses only appearance and relative geometric features.

our results suggest that if one uses absolute configuration features as well as different appearance features for left and right legs (implicit in the structure learning procedure), the left and right legs are identified correctly.


As shown in the plots, sample-aggregation algorithms have poor utility unless the privacy parameter ε is truly large (which means very little privacy protection).

both Gibbs sampling and SuLQ subspace clustering give reasonably good performance.


Indeed, we were not able to directly generalize the analysis in [24] by simply adapting the techniques used in recent online boosting work [4,2], but we made use of the classical Frank-Wolfe algorithm [8].

while an important part of the convergence analysis for the batch setting is to show statistical consistency of the algorithms [24,1,22], in the online setting we only need to study the empirical convergence (that is, the regret), which makes our analysis much more concise.


NTPs are constructed from Prolog's backward chaining and are thus related to Unification Neural Networks [55,56].

nTPs operate on vector representations of symbols instead of scalar values, which are more expressive.


The reader can verify that all of these examples also satisfy the remaining global optimization pre-conditions of Corollary 4.2 and Theorem 3.2.

these linear-growth examples do not satisfy dissipativity (Condition 2) when paired with the Gibbs measure Langevin diffusion.


The conventional role of analog circuits in mixed-signal VLSI is providing the I/O interface to the digital core of the chip -which realizes all the signal processing.

this approach may not be optimum for the processing of multi-dimensional sensory signals, such as those found in vision applications.


Finally, classifiers such as SVMs use both positive and negative examples and provide state-of-theart performance when used with appropriate kernels [5,6,7].

these classifiers still require an auxiliary method (such as PSI-BLAST) to handle unlabeled data: one generally adds predicted homologs of the positive training examples to the training set before training the classifier.


However, the rest of the network remains a conventional convolution network.

sphereNet not only generalizes the hyperspherical constraint to every layer, but also to different nonlinearity functions of input angles.


Second, the principal components are uncorrelated, which can aid with interpretation or subsequent statistical analysis.

pCA has a number of well-documented disadvantages as well.


Thus, our vq(n) is closely related to vpdn.

vpdn has an over-counting problem, which we avoid due to the way we dynamically update vq(n).


The independent work Neon by Xu and Yang [31] is actually the first recorded theoretical result that proposed this approach.

neon needs O(1/δ 3 ) stochastic gradients, because it uses full gradient descent to find NC (on a sub-sampled objective) inspired by power method and [16]; instead, Neon2 online uses stochastic gradients and is based on the recent result of Oja's algorithm [7].


Unfortunately, computing p(v i = w|h) to sample the words during Gibbs sampling is linear in V and H, where V tends to be quite large.

fortunately, given h, it needs to be computed only once before sampling all D words in v. when h is re-sampled, p(v i = w|h) must be recomputed.


Here we only use the observation that when the network is initialized the weights are usually small and therefore these nonlinearities will be in the linear region, and the linear analysis applies.

as the weights grow bigger during training their effect becomes hard to analyze.


These settings were theoretically analyzed in a series of works [18,19,23,21] that have demonstrated that information transfer can lead to provable sample complexity reductions compared to solving each task independently.

all these results rely on Baxter's model of lifelong learning and therefore assume access to the training data for all (observed) tasks and provide guarantees only on average performance over all tasks.


Since invariance is a special case of independent changes, based on PIC, change in one causal module does not force any changes in another causal module, i.e., a change in, say, 2 C|; , will not enforce any changes on E|C or 2 E|C .

in the reverse direction, as it can be seen from equations in (1) and (2), if any of the variables a, 2 C , and 2 E varies across two domains, by Assumption 1, all three variables 2 E|; , C|E , and 2 C|E will change.


Note that traditional model selection criteria, such as BIC/MDL, aim at fitting the data well and minimizing the representation complexity of the learned model (i.e., the total number of parameters).

as demonstrated in [2], such criteria are unable to capture the inference complexity: two models that have similar representation complexity and fit data equally well may have quite different graph structures, making one model exponentially slower for inference than the other.


Unlike previous methods that capture only a sparse set of correspondences, this geometric estimation CNN captures interactions between a full set of dense correspondences.

these methods currently only estimate a low complexity parametric transformation, and therefore their application is limited to only coarse image alignment tasks.


Approximate inference in these models via variational techniques [1,3] and sampling based methods [4,5] has been investigated.

many real-world problems have neither purely log-submodular nor log-supermodular formulations, but can be naturally expressed in the form P (A) ∝ exp(F (A) − G(A)), where both F (A) and G(A) are submodular functions -we call these types of models mixed PSMs.


The MC method overcomes the above limitations since it has an unbiased estimator of the probability mass.

the high stochasticity in point processes leads to a large value of VAR[f (N (t))], which requires a large number of samples to achieve a small error.


We can think of actor-critic architectures, generalizing to the option-critic architecture as well, as pairing a critic with each actor network so that the critic has additional information about the value of the actor's actions that can be used to improve the actor's learning.

this is derived by taking gradients with respect to the parameters of the policy while optimizing for the expected discounted return.


This may be related to the observation that sampling without replacement can often speed up randomized algorithms.

on the other hand, it is well known that full gradient methods fail to achieve a good convergence rate for other objectives with the same parameters µ, L, n (e.g.


We note that our algorithm is similar to the algorithm of [19] for the problem of offline matrix completion without noise.

our setting, with the presence of noise (which might conceivably propagate through the course of the algorithm), makes our analysis significantly more subtle.


However, when λ/λ max < 0.5, strong rule can not identify any inactive features and thus the running time is almost the same as that of the solver without screening.

slores is still able to identify more than 80% of the inactive features for the data sets cultured from the Yahoo web pages data sets and thus the efficiency is improved by roughly 5 times.


The problem of maximizing objective (1) with constraint ( 2) is a well-known quadratic optimization problem with a norm constraint, the solution of which is given by an eigenvalue-eigenvector problem (see below).

in order to handle the case where C is not invertible -which will be the case below in our experiments -and to attenuate the noise in the data, we first use a technique called dimensionality reduction (see, e.g., [15]).


In our experiments we showed the benefits of our formulation over direct methods -we are able to generalize to new object categories and produce compelling reconstructions with fewer images than classical  systems.


By applying that movie was cool the same computation recursively to compute the contribution of children to their parents, and the same computation to produce an output response, we are, in fact, representing every internal node (phrase) in the same space [6,8].

in the more conventional stacked deep learners (e.g.


This is coherent with our theoretical analysis, since GFM needs to estimate more parameters.

in the case of dependent labels, MEUF performs poorly, even for a larger sample size, since the underlying assumption is not satisfied.


The 7 While smoothing allows one to use gradient-based algorithms with guaranteed convergence, the obtained solution is a perturbed version of the targeted one and thus a rounding step (such as solving a linear program [18]) is required.

as illustrated in Figure 3, solving one linear program is more expensive than the PSGM for (2) when the data set is relatively large, thus indicating that using a smooth surrogate is not always beneficial.


Since the angle is a continuous variable the first approach that comes to mind is using a linear or non-linear regressors.

we found that regression algorithms such as Widrow-Hoff [10] yielded inferior results.


Thus far we have considered only dual variables.

it is often important to recover the primal variables.


Here only one of the responsibilities of each datapoint is set to 1: Because the number of paths is the product of the number of components per layer ( Q k j N j ), computing the responsibilities can become intractable for big Deep GMM networks.


However, when the exact parametric form of data is known and when it is utilized to learn the parameters, MLE with this full knowledge performs better.


In contrast to the previous case, since the label is costly here, if the algorithm decides to see the label it does not even have to reason about which features to observe, as querying the label requires paying a cost that is a constant over the cost of the best predictor in hindsight, already resulting in the ⇥(T 2/3 ) regret rate.

in practice (for shorter horizons) it still makes sense to select the features that provide the best balance between the feature-cost and the prediction loss.


By choosing linked anchor words, the algorithm can align topics cross-lingually so that the aligned topics form one multilingual topic.

randomly choosing translation pairs as anchor links will not produce coherent multilingual topics.


Given a feasible set S of experiments (i.e., ∑ i∈S x i x ⊤ i is invertible), the Gauss-Markov theorem shows that the lowest variance for an unbiased estimate θ satisfies Var[ θ] = ( ∑ i∈S x i x ⊤ i ) −1 .

var[ θ] is a matrix, and matrices do not admit a total order, making it difficult to compare different designs.


As we discussed previously in Section 2.2, minimizing the KL divergence will ensure that statistical properties of the protocol are going to be similar to those of natural language.

words are not likely to preserve their original meaning (e.g.


Research so far has focused on single neuroimaging modality.

as different modalities provide complementary measures for the same disease pathology, fusion of multi-modality data may increase the statistical power in identification of disease-related brain regions.


In this way, simple priors on latent variables give rise to complex conditional distributions over data.

learning the neural network weights is far from trivial because the marginal log probability of the data, or evidence, is intractable.


LinUCB [16], Epoch-Greedy [15] for the stochastic setting and EXP4, EXP4.P [2] for the adversarial setting.

most of these algorithms only allow the decision makers to pull one single arm in each round.


The optimal solution is F ⇤ = (1 ↵)(I ↵S)     method); another is to solve F ⇤ as a system of linear equations by using an iterative solver like conjugate gradient (CG) [10].

both methods suffer from slow convergence, especially when the number of labels, i.e., columns of Y , grows dramatically.


Large regions of visual and auditory cortex stimulated by a movie viewing do indeed show consistent inter-subject synchrony [6].

other areas in the intrinsic [7] or default [8] system fail to exhibit significant correlations across repeated stimulus trials.


They are able to scale to significantly larger problems than off-the-shelf QP solvers.

they are limited in scope in that they rely on dynamic programming to compute essential quantities such as gradients.


Through some theoretical analysis, we can prove that UC + AccProxSVRG also has the similar rate and mini-batch efficiency to our proposed method, although these results are not obtained in any literature.

our proposed method is superior to this algorithm both theoretically and practically, because the algorithm has several drawbacks due to the use of UC as follows.


As an extension of ART model, RART clusters together entities with similar roles.

the GT model presented here clusters entities into groups based on their relations to other entities.


Under a mild technical assumption, Proposition 9 suggests that losing connections monotonically increases the MSD, and individuals tend to maintain their friendships to obtain a lower MSD as a global objective.

this does not elaborate on the existence of individuals with whom losing or making connections could have an immense impact on learning.


Only slow rates of convergence of order 1/ T are derived for the variant of Frank-Wolfe, while we aim at fast rates, which are optimal.

in the adversarial case [16,32], there are instances of the problem where the average regret cannot decrease to zero [26].


Since conventional CRFs need not maximize over possible segment lengths d, inference for semi-CRFs is more expensive.

equation 4shows that the additional cost is only linear in L. For NER, a reasonable value of L might be four or five.


The hitting time, commute time, and resistance distance are popular measures of distance based upon the random walk which are believed to be robust and capture the cluster structure of the network.

it was shown in a surprising result in [19] that on undirected geometric graphs the scaled expected hitting time from x i to x j converges to inverse of the degree of x j .


If (G, s) is weakly U-type, then its signed minors are weakly U-type; this is obvious from Lemma 1.

direct computation of the zeta of (B 2 , s + ) shows that this signed graph is not weakly U-type.


Most recently, CNNs have shown promise on facial AU recognition.

the learned CNNs are often overfitted and do not generalize well to unseen subjects due to limited AU-coded training images.


CRH is closely related to a recent multimodal metric learning method called Multiview Neighborhood Preserving Projections (Multi-NPP) [23], because CRH uses a loss function for inter-modality point pairs which is similar to Multi-NPP.

cRH is a general framework and other loss functions for inter-modality point pairs can also be adopted.


our guarantee improves on the distribution-independent regret bounds of UCB that are of order √ KT log T . This is shown in the following corollary.


Although KRONDPPS have tractable learning algorithms, the memory requirements remain high for non-stochastic updates, as the matrix Yi U ⊤ i needs to be stored, requiring O(N 2 ) memory.

if the training set can be subdivided such that Θ can be decomposed as Due to the bound in Eq.


We discuss in the experiments differences and properties of the individual balancing terms.

it is out of the scope of this paper to answer the question which balancing term is the "best".


Thus, the detection is performed in the sliding window fashion followed by non maxima suppression.

we do not use the root filter, which is an appearance classifier of the whole detection window.


Minimizing it will reduce the similarity of the parameter vectors and make them "orthogonal" to each other.

orthogonality does not necessarily imply disjoint features.


Even high-dimensional distributions can be fitted, because all inference operations have quadratic complexity.

entropy and MI estimation can be problematic, because MC integration can become unfeasible for very high-dimensional problems.


Our version of the model that does not use any aligned data falls short of the supervised models.

in the semi-supervised setting we observe an improvement of more than 2 BLEU points over the purely supervised baselines.


has made it easy to collect low-cost labels from the crowd to generate training datasets for machine learning applications.

these applications remain vulnerable to noisy labels introduced either unintentionally by unreliable workers or intentionally by spammers and malicious workers [10,11].


It is known that estimators such as LASSO offer certain optimal min-max performance over a large class of sparse distributions [26].

for many particular distributions, there is a potentially large performance gap between LASSO and MMSE estimator with the correct prior.


Note that since (8) and ( 10) are expressed solely in terms of equivalence relations, and do not depend on the specific values of hidden variables in any way, this formulation is not subject to the triviality result of Lemma 1.

there are still some details left to consider.


Since larger singular values are multiplied to several leftmost columns, it is reasonable to assume that the most important portion of the output vector is already computed with the preview window.

we find that the preview outputs do not suffice to obtain accurate results.


[2017] also provide an optimality gap for ASOSPs of (PhaseCut) without perturbation.

their result is more general in the sense that it holds for all possible values of k. when k is large, it does not accurately capture the fact that SOSPs are optimal, thus incurring a larger bound on the optimality gap of ASOSPs.


And the differential graph is defined as the difference between the two latent precision matrices A simple procedure is estimating ⇥ ⇤ X and ⇥ ⇤ Y separately, followed by calculating their difference.

it requires estimating 2d 2 parameters (i.e., ⇥ ⇤ X and ⇥ ⇤ Y ), while our ultimate goal is only estimating d 2 parameters (i.e., ⇤ ).


Agents secure individual rewards when they appropriate resource units from a CPR.

the cost of such appropriation, felt via its impact on the CPR stock, affects all agents in the community equally.


One approach to solve this is imposing L1 regularization on the object masks to encourage sparsity.

if we only regularize the object masks, there is still a degenerate solution consisting of object masks with very small values coupled with large object translations.


A central task in statistical inference is learning properties of a high-dimensional distribution over some variables of interest given observational data.

probability distributions only capture the association between variables of interest and may not suffice to predict what the consequences would be of setting some of the variables to particular values.


Details of this model can be found in [18,35].

in practice, most given features X and Y will not be perfect.


We partially circumvent this issue by proposing an admittedly artificial learning-rate schedule that actually guarantees near-optimal performance.

a serious limitation of this schedule is that it relies on prior knowledge of problem parameters ∆ and T that are typically unknown at the beginning of the learning procedure.


A simple method would be to re-train our detector with both datasets using Equation 1.

we typically aren't certain that the labels h are correct, especially in the first iteration when our detector is trained solely from the image examples.


Recently, edge-exchangeable models have been proposed and shown to exhibit sparsity [8,9,16].

these models assume that networks are fully observed.


For instance, when M turns out to be a random bipartite graph 9 the expected number of matches that any algorithm can achieve is always upper bounded by O T n 2 M (this is how Theorem 1 is proven).

in order to have M * T = Θ(M ) as n grows large, it is sufficient that deg M (u) ≤ T n holds for all users u ∈ B ∪ G, even with such a random M. In order to avoid the pitfalls of M being a random bipartite graph (and hence the negative result of Theorem 1), we need to slightly depart from our general model of Section 2, and make structural assumptions on the way matches can be generated.


A quasi newton second order method was used in [10] to directly solve the symmetric NMF optimization problem (2).

both the element-wise updating approach and the second order method are observed to be computationally expensive in large scale applications.


Then to find y f that maximizes (26), we compare the maximizers in 4 cases: .

in practice we observe an O(1) cost for case (i) and the bottleneck is actually case (iv), which requires O(|A i ||A j |).


It is well known that the Maximum Likelihood Estimate (MLE) can be computed as P M LE (e i = a) = ca m where c a denotes the number of times the event a appears in the observed sequence and m = j c j denotes the total number of observed events.

mLE of the multinomial distribution is spiky since it assigns zero probability to the events that did not occur in the observed sequence.


A generative model, referred as the Generator G, is trained to replicate the data from noise, while an adversarial discriminative model, referred as the Discriminator D, is trained to identify whether a sample comes from the true data or from G. The GAN training objective is based on a minimax game between the two networks and approximately optimizes a Jensen-Shannon divergence.


STMVL performs well on the air quality data.

it is specifically designed for geographical data, and cannot be applied in the other datasets.


As in standard approaches such as Sparse Coding or Independent Component Analysis, the model assumes a sparse prior with independent hidden variables.

in the place where standard approaches use the sum to combine basis functions we use the maximum.


Automated reasoning and learning theory have been the subject of intensive investigation since the early developments in computer science [14].

while (machine) learning has focused mainly on quantitative and connectionist approaches [16], the reasoning component of intelligent systems has been developed mainly by formalisms of classical and non-classical logics [7,9].


When W is the identity, ( 14) becomes a standard convex relaxation of (13) and its globally optimal solution can be reached efficiently [27].

the 0 norm can also be approximated by reweighting 1 , as proposed by Candes et al.


This is exponentially smaller than the usual Ω(d/ǫ) sample complexity of learning linear classifiers in a supervised setting.

generalizing this result is non-trivial.


Conditional on all parameters, H C and H T are independent.

focus for a moment on H C . Since we wish to view it as a sample from a Wright-Fisher recombining population, its likelihood corresponds to the probability, under the coalescent-with-recombination distribution, of the set of all ARGs that could have generated H C . using the nucleotide mutation model described below, even Monte Carlo approximation of this probability is computationally intractable [12].


Each matrix in each of the nets is determined before the sampling begins, so it can be used in Theorem 3.4.

the construction of the nets guarantees that the N th net contains a matrix Q which is sufficiently close to QN to be used as a substitute for QN in the above process.


In particular, CCA is equivalent to first seeking projection directions W z of Z, and then performing a multi-output linear regression from x n to W z z n , under the constraints W x X T XW T x = I, to obtain W x .

it has not been seriously studied how to use CCA for LSDR because W z does not contain orthogonal rows.


However, the features for the affinity term are mostly simple ones such as color and location.

we have designed a unique planar surface affinity term and a novel compatibility term to enable 3D planar regularization over geometry estimation.


Models developed for these settings have proven useful in practice.

they are chiefly designed for supervised learning and do not scale to the size of the problem we consider in this work.


We also showed that this algorithm performed well on sound data, even though the topology might be less obvious in that case.

in this paper, we only considered the simple case where we knew in advance the dimensionnality of the coordinates.


A small β means epiphany is less likely per query, thus we expect the learner to spend more queries trying to induce epiphany in the case of U = [0.4, 0.6].

β may not matter much in the case of U = [0.7, 0.9] since epiphany may not be required.


Table 3 shows that memory-less methods (like "single" and "multimodal") exhibit higher negative BWT, leading to lower ACC.

memory-based methods such as EWC and GEM lead to higher ACC as the number of passes through the data increases.


All convergence proofs crucially use the Lipschitz condition on the gradient to bound the noise terms derived from asynchrony.

in the composite case, the gradient mapping term (Beck & Teboulle, 2009), which replaces the gradient in proximal-gradient methods, does not have a bounded Lipschitz constant.


Different learning paradigms have been explored, including the maximum conditional likelihood [7] and max-margin learning [2,12,13], with remarkable success.

the problem of structured input/output learning can be intriguing and significantly more difficult when there exist hidden substructures in the data, which is not uncommon in realistic problems.


A tight bound requires the Rademacher complexity to be close to zero.

if the Rademacher complexity is large, or n is too small, or the requested confidence 1 − δ is overly stringent, the bound can be loose.