/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

We introduce a new algorithm named WGAN, an alternative to traditional GAN training. In this new model, we show that we can improve the stability of learning, get rid of problems like mode collapse, and provide meaningful learning curves useful for debugging and hyperparameter searches. Furthermore, we show that the corresponding optimization problem is sound, and provide extensive theoretical work highlighting the deep connections to different distances between distributions.

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Wasserstein Generative Adversarial Networks

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

We present the Word Mover's Distance (WMD), a novel distance function between text documents. Our work is based on recent results in word embeddings that learn semantically meaningful representations for words from local cooccurrences in sentences. The WMD distance measures the dissimilarity between two text documents as the minimum amount of distance that the embedded words of one document need to "travel" to reach the embedded words of another document. We show that this distance metric can be cast as an instance of the Earth Mover's Distance, a well studied transportation problem for which several highly efficient solvers have been developed. Our metric has no hyperparameters and is straight-forward to implement. Further, we demonstrate on eight real world document classification data sets, in comparison with seven state-of-the-art baselines, that the WMD metric leads to unprecedented low k-nearest neighbor document classification error rates.

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From Word Embeddings To Document Distances

Complex objects can often be conveniently represented by finite sets of simpler components, such as images by sets of patches or texts by bags of words. We study the class of positive definite (p.d.) kernels for two such objects that can be expressed as a function of the merger of their respective sets of components. We prove a general integral representation of such kernels and present two particular examples. One of them leads to a kernel for sets of points living in a space endowed itself with a positive definite kernel. We provide experimental results on a benchmark experiment of handwritten digits image classification which illustrate the validity of the approach.

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Semigroup Kernels on Finite Sets

We focus in this paper on high-dimensional regression problems where each regressor can be associated to a location in a physical space, or more generally a generic geometric space. Such problems often employ sparse priors, which promote models using a small subset of regressors. To increase statistical power, the so-called multi-task techniques were proposed, which consist in the simultaneous estimation of several related models. Combined with sparsity assumptions, it lead to models enforcing the active regressors to be shared across models, thanks to, for instance L1 / Lq norms. We argue in this paper that these techniques fail to leverage the spatial information associated to regressors. Indeed, while sparse priors enforce that only a small subset of variables is used, the assumption that these regressors overlap across all tasks is overly simplistic given the spatial variability observed in real data. In this paper, we propose a convex reg-ularizer for multi-task regression that encodes a more flexible geometry. Our regularizer is based on unbalanced optimal transport (OT) theory, and can take into account a prior geometric knowledge on the regressor variables, without necessarily requiring overlapping supports. We derive an efficient algorithm based on a regularized formulation of OT, which iterates through applications of Sinkhorn's algorithm along with coordinate descent iterations. The performance of our model is demonstrated on regular grids with both synthetic and real datasets as well as complex triangulated geometries of the cortex with an application in neuroimaging.

/pdf/wasserstein-regularization-for-sparse-multi-task-regression-1px0fmo5xu.pdf

Wasserstein regularization for sparse multi-task regression

Optimal transport (OT) theory describes general principles to define and select, among many possible choices, the most efficient way to map a probability measure onto another. That theory has been mostly used to estimate, given a pair of source and target probability measures $(\mu, \nu)$, a parameterized map $T_\theta$ that can efficiently map $\mu$ onto $\nu$. In many applications, such as predicting cell responses to treatments, pairs of input/output data measures $(\mu, \nu)$ that define optimal transport problems do not arise in isolation but are associated with a context $c$, as for instance a treatment when comparing populations of untreated and treated cells. To account for that context in OT estimation, we introduce CondOT, a multi-task approach to estimate a family of OT maps conditioned on a context variable, using several pairs of measures $\left(\mu_i, \nu_i\right)$ tagged with a context label $c_i$. CondOT learns a global map $\mathcal{T}_\theta$ conditioned on context that is not only expected to fit all labeled pairs in the dataset $\left\{\left(c_i,\left(\mu_i, \nu_i\right)\right)\right\}$, i.e., $\mathcal{T}_\theta\left(c_i\right) \sharp \mu_i \approx \nu_i$, but should also generalize to produce meaningful maps $\mathcal{T}_\theta\left(c_{\text {new }}\right)$ when conditioned on unseen contexts $c_{\text {new }}$. Our approach harnesses and provides a novel usage for partially input convex neural networks, for which we introduce a robust and efficient initialization strategy inspired by Gaussian approximations. We demonstrate the ability of CondOT to infer the effect of an arbitrary combination of genetic or therapeutic perturbations on single cells, using only observations of the effects of said perturbations separately.

Supervised Training of Conditional Monge Maps

We propose a new divergence on the manifold of probability distributions, building on the entropic regularization of optimal transportation problems. As Cuturi (2013) showed, regularizing the optimal transport problem with an entropic term is known to bring several computational benefits. However, because of that regularization, the resulting approximation of the optimal transport cost does not define a proper distance or divergence between probability distributions. We recently tried to introduce a family of divergences connecting the Wasserstein distance and the Kullback-Leibler divergence from an information geometry point of view (see Amari, Karakida, & Oizumi, 2018). However, that proposal was not able to retain key intuitive aspects of the Wasserstein geometry, such as translation invariances, which play a key role when used in the more general problem of computing optimal transport barycenters. The divergence we propose in this work is able to retain such properties and admits an intuitive interpre...

Information Geometry for Regularized Optimal Transport and Barycenters of Patterns

Nonparametric two sample or homogeneity testing is a decision theoretic problem that involves identifying differences between two random variables without making parametric assumptions about their underlying distributions. The literature is old and rich, with a wide variety of statistics having being intelligently designed and analyzed, both for the unidimensional and the multivariate setting. Our contribution is to tie together many of these tests, drawing connections between seemingly very different statistics. In this work, our central object is the Wasserstein distance, as we form a chain of connections from univariate methods like the Kolmogorov-Smirnov test, PP/QQ plots and ROC/ODC curves, to multivariate tests involving energy statistics and kernel based maximum mean discrepancy. Some connections proceed through the construction of a \textit{smoothed} Wasserstein distance, and others through the pursuit of a "distribution-free" Wasserstein test. Some observations in this chain are implicit in the literature, while others seem to have not been noticed thus far. Given nonparametric two sample testing's classical and continued importance, we aim to provide useful connections for theorists and practitioners familiar with one subset of methods but not others.

Marco Cuturi

Papers

Semigroup Kernels on Finite Sets

Wasserstein regularization for sparse multi-task regression

Supervised Training of Conditional Monge Maps

Information Geometry for Regularized Optimal Transport and Barycenters of Patterns

On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests