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Bharath K. Sriperumbudur

Publications and source records attributed to Bharath K. Sriperumbudur.

At least 37 records · Page 2Linked to original sources
arXiv2023

Gromov-Wasserstein Distances: Entropic Regularization, Duality, and Sample Complexity

The Gromov-Wasserstein (GW) distance, rooted in optimal transport (OT) theory, quantifies dissimilarity between metric measure spaces and provides a framework for aligning heterogeneous datasets. While computational aspects of the GW problem have been widely studied, a duality theory and fundamental statistical questions concerning empirical convergence rates remained obscure. This work closes these gaps for the quadratic GW distance over Euclidean spaces of different dimensions $d_x$ and $d_y$. We treat both the standard and the entropically regularized GW distance, and derive dual forms that represent them in terms of the well-understood OT and entropic OT (EOT) problems, respectively. This enables employing proof techniques from statistical OT based on regularity analysis of dual potentials and empirical process theory, using which we establish the first GW empirical convergence rates. The derived two-sample rates are $n^{-2/\max\{\min\{d_x,d_y\},4\}}$ (up to a log factor when $\min\{d_x,d_y\}=4$) for standard GW and $n^{-1/2}$ for EGW, which matches the corresponding rates for standard and entropic OT. The parametric rate for EGW is evidently optimal, while for standard GW we provide matching lower bounds, which establish sharpness of the derived rates. We also study stability of EGW in the entropic regularization parameter and prove approximation and continuity results for the cost and optimal couplings. Lastly, the duality is leveraged to shed new light on the open problem of the one-dimensional GW distance between uniform distributions on $n$ points, illuminating why the identity and anti-identity permutations may not be optimal. Our results serve as a first step towards a comprehensive statistical theory as well as computational advancements for GW distances, based on the discovered dual formulations.

math.ST↗
arXiv2022

Shrinkage Estimation of Higher Order Bochner Integrals

We consider shrinkage estimation of higher order Hilbert space valued Bochner integrals in a non-parametric setting. We propose estimators that shrink the $U$-statistic estimator of the Bochner integral towards a pre-specified target element in the Hilbert space. Depending on the degeneracy of the kernel of the $U$-statistic, we construct consistent shrinkage estimators with fast rates of convergence, and develop oracle inequalities comparing the risks of the the $U$-statistic estimator and its shrinkage version. Surprisingly, we show that the shrinkage estimator designed by assuming complete degeneracy of the kernel of the $U$-statistic is a consistent estimator even when the kernel is not complete degenerate. This work subsumes and improves upon Krikamol et al., 2016, JMLR and Zhou et al., 2019, JMVA, which only handle mean element and covariance operator estimation in a reproducing kernel Hilbert space. We also specialize our results to normal mean estimation and show that for $d\ge 3$, the proposed estimator strictly improves upon the sample mean in terms of the mean squared error.

math.ST↗
arXiv2022

Shrinkage Estimation for the Diagonal Multivariate Exponential Families

We study shrinkage estimation of the mean parameters of a class of multivariate distributions for which the diagonal entries of the corresponding covariance matrix are certain quadratic functions of the mean parameter. This class of distributions includes the diagonal multivariate natural exponential families. We propose two classes of semi-parametric shrinkage estimators for the mean and construct unbiased estimators of the corresponding risk. We establish the asymptotic consistency and convergence rates for these shrinkage estimators under squared error loss as both $n$, the sample size, and $p$, the dimension, tend to infinity. Next, we specialize these results to the diagonal multivariate natural exponential families, which have been classified as consisting of the normal, Poisson, gamma, multinomial, negative multinomial, and hybrid classes of distributions. We establish the consistency of our estimators in the normal, gamma, and negative multinomial cases subject to the condition that $p n^{-1/3} (\log{n})^{4/3} \to 0$, and in the Poisson and multinomial cases if $p n^{-1/2} \to 0$, as $n,p \to \infty$. Simulation studies are provided to evaluate the performance of our estimators and we illustrate that, in the gamma and Poisson cases, our estimators achieve lower risk than the maximum likelihood estimator, thereby demonstrating the superiority of our estimators over the maximum likelihood estimator.

math.ST↗
arXiv2021

Cycle Consistent Probability Divergences Across Different Spaces

Discrepancy measures between probability distributions are at the core of statistical inference and machine learning. In many applications, distributions of interest are supported on different spaces, and yet a meaningful correspondence between data points is desired. Motivated to explicitly encode consistent bidirectional maps into the discrepancy measure, this work proposes a novel unbalanced Monge optimal transport formulation for matching, up to isometries, distributions on different spaces. Our formulation arises as a principled relaxation of the Gromov-Haussdroff distance between metric spaces, and employs two cycle-consistent maps that push forward each distribution onto the other. We study structural properties of the proposed discrepancy and, in particular, show that it captures the popular cycle-consistent generative adversarial network (GAN) framework as a special case, thereby providing the theory to explain it. Motivated by computational efficiency, we then kernelize the discrepancy and restrict the mappings to parametric function classes. The resulting kernelized version is coined the generalized maximum mean discrepancy (GMMD). Convergence rates for empirical estimation of GMMD are studied and experiments to support our theory are provided.

cs.LG↗
arXiv2021

Adaptive Clustering Using Kernel Density Estimators

We derive and analyze a generic, recursive algorithm for estimating all splits in a finite cluster tree as well as the corresponding clusters. We further investigate statistical properties of this generic clustering algorithm when it receives level set estimates from a kernel density estimator. In particular, we derive finite sample guarantees, consistency, rates of convergence, and an adaptive data-driven strategy for choosing the kernel bandwidth. For these results we do not need continuity assumptions on the density such as Hölder continuity, but only require intuitive geometric assumptions of non-parametric nature.

stat.ML↗
arXiv2021

Local minimax rates for closeness testing of discrete distributions

We consider the closeness testing problem for discrete distributions. The goal is to distinguish whether two samples are drawn from the same unspecified distribution, or whether their respective distributions are separated in $L_1$-norm. In this paper, we focus on adapting the rate to the shape of the underlying distributions, i.e. we consider \textit{a local minimax setting}. We provide, to the best of our knowledge, the first local minimax rate for the separation distance up to logarithmic factors, together with a test that achieves it. In view of the rate, closeness testing turns out to be substantially harder than the related one-sample testing problem over a wide range of cases.

math.ST↗
arXiv2020

On Distance and Kernel Measures of Conditional Independence

Measuring conditional independence is one of the important tasks in statistical inference and is fundamental in causal discovery, feature selection, dimensionality reduction, Bayesian network learning, and others. In this work, we explore the connection between conditional independence measures induced by distances on a metric space and reproducing kernels associated with a reproducing kernel Hilbert space (RKHS). For certain distance and kernel pairs, we show the distance-based conditional independence measures to be equivalent to that of kernel-based measures. On the other hand, we also show that some popular---in machine learning---kernel conditional independence measures based on the Hilbert-Schmidt norm of a certain cross-conditional covariance operator, do not have a simple distance representation, except in some limiting cases. This paper, therefore, shows the distance and kernel measures of conditional independence to be not quite equivalent unlike in the case of joint independence as shown by Sejdinovic et al. (2013).

math.ST↗
arXiv2020

Gaussian Sketching yields a J-L Lemma in RKHS

The main contribution of the paper is to show that Gaussian sketching of a kernel-Gram matrix $\boldsymbol K$ yields an operator whose counterpart in an RKHS $\mathcal H$, is a \emph{random projection} operator---in the spirit of Johnson-Lindenstrauss (J-L) lemma. To be precise, given a random matrix $Z$ with i.i.d. Gaussian entries, we show that a sketch $Z\boldsymbol{K}$ corresponds to a particular random operator in (infinite-dimensional) Hilbert space $\mathcal H$ that maps functions $f \in \mathcal H$ to a low-dimensional space $\mathbb R^d$, while preserving a weighted RKHS inner-product of the form $\langle f, g \rangle_Σ \doteq \langle f, Σ^3 g \rangle_{\mathcal H}$, where $Σ$ is the \emph{covariance} operator induced by the data distribution. In particular, under similar assumptions as in kernel PCA (KPCA), or kernel $k$-means (K-$k$-means), well-separated subsets of feature-space $\{K(\cdot, x): x \in \cal X\}$ remain well-separated after such operation, which suggests similar benefits as in KPCA and/or K-$k$-means, albeit at the much cheaper cost of a random projection. In particular, our convergence rates suggest that, given a large dataset $\{X_i\}_{i=1}^N$ of size $N$, we can build the Gram matrix $\boldsymbol K$ on a much smaller subsample of size $n\ll N$, so that the sketch $Z\boldsymbol K$ is very cheap to obtain and subsequently apply as a projection operator on the original data $\{X_i\}_{i=1}^N$. We verify these insights empirically on synthetic data, and on real-world clustering applications.

stat.ML↗
arXiv2019

On Kernel Derivative Approximation with Random Fourier Features

Random Fourier features (RFF) represent one of the most popular and wide-spread techniques in machine learning to scale up kernel algorithms. Despite the numerous successful applications of RFFs, unfortunately, quite little is understood theoretically on their optimality and limitations of their performance. Only recently, precise statistical-computational trade-offs have been established for RFFs in the approximation of kernel values, kernel ridge regression, kernel PCA and SVM classification. Our goal is to spark the investigation of optimality of RFF-based approximations in tasks involving not only function values but derivatives, which naturally lead to optimization problems with kernel derivatives. Particularly, in this paper, we focus on the approximation quality of RFFs for kernel derivatives and prove that the existing finite-sample guarantees can be improved exponentially in terms of the domain where they hold, using recent tools from unbounded empirical process theory. Our result implies that the same approximation guarantee is attainable for kernel derivatives using RFF as achieved for kernel values.

stat.ML↗
arXiv2018

Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings

This paper presents a convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than a Sobolev RKHS based on which a quadrature rule is constructed. We provide convergence guarantees based on two different assumptions on a quadrature rule: one on quadrature weights, and the other on design points. More precisely, we show that convergence rates can be derived (i) if the sum of absolute weights remains constant (or does not increase quickly), or (ii) if the minimum distance between design points does not decrease very quickly. As a consequence of the latter result, we derive a rate of convergence for Bayesian quadrature in misspecified settings. We reveal a condition on design points to make Bayesian quadrature robust to misspecification, and show that, under this condition, it may adaptively achieve the optimal rate of convergence in the Sobolev space of a lesser order (i.e., of the unknown smoothness of a test integrand), under a slightly stronger regularity condition on the integrand.

math.NA↗
arXiv2018

Minimax Estimation of Quadratic Fourier Functionals

We study estimation of (semi-)inner products between two nonparametric probability distributions, given IID samples from each distribution. These products include relatively well-studied classical $\mathcal{L}^2$ and Sobolev inner products, as well as those induced by translation-invariant reproducing kernels, for which we believe our results are the first. We first propose estimators for these quantities, and the induced (semi)norms and (pseudo)metrics. We then prove non-asymptotic upper bounds on their mean squared error, in terms of weights both of the inner product and of the two distributions, in the Fourier basis. Finally, we prove minimax lower bounds that imply rate-optimality of the proposed estimators over Fourier ellipsoids.

math.ST↗
arXiv2018

Characteristic and Universal Tensor Product Kernels

Maximum mean discrepancy (MMD), also called energy distance or N-distance in statistics and Hilbert-Schmidt independence criterion (HSIC), specifically distance covariance in statistics, are among the most popular and successful approaches to quantify the difference and independence of random variables, respectively. Thanks to their kernel-based foundations, MMD and HSIC are applicable on a wide variety of domains. Despite their tremendous success, quite little is known about when HSIC characterizes independence and when MMD with tensor product kernel can discriminate probability distributions. In this paper, we answer these questions by studying various notions of characteristic property of the tensor product kernel.

stat.ML↗
arXiv2016

Convergence guarantees for kernel-based quadrature rules in misspecified settings

Kernel-based quadrature rules are becoming important in machine learning and statistics, as they achieve super-$\sqrt{n}$ convergence rates in numerical integration, and thus provide alternatives to Monte Carlo integration in challenging settings where integrands are expensive to evaluate or where integrands are high dimensional. These rules are based on the assumption that the integrand has a certain degree of smoothness, which is expressed as that the integrand belongs to a certain reproducing kernel Hilbert space (RKHS). However, this assumption can be violated in practice (e.g., when the integrand is a black box function), and no general theory has been established for the convergence of kernel quadratures in such misspecified settings. Our contribution is in proving that kernel quadratures can be consistent even when the integrand does not belong to the assumed RKHS, i.e., when the integrand is less smooth than assumed. Specifically, we derive convergence rates that depend on the (unknown) lesser smoothness of the integrand, where the degree of smoothness is expressed via powers of RKHSs or via Sobolev spaces.

stat.ML↗
arXiv2015

Optimal Rates for Random Fourier Features

Kernel methods represent one of the most powerful tools in machine learning to tackle problems expressed in terms of function values and derivatives due to their capability to represent and model complex relations. While these methods show good versatility, they are computationally intensive and have poor scalability to large data as they require operations on Gram matrices. In order to mitigate this serious computational limitation, recently randomized constructions have been proposed in the literature, which allow the application of fast linear algorithms. Random Fourier features (RFF) are among the most popular and widely applied constructions: they provide an easily computable, low-dimensional feature representation for shift-invariant kernels. Despite the popularity of RFFs, very little is understood theoretically about their approximation quality. In this paper, we provide a detailed finite-sample theoretical analysis about the approximation quality of RFFs by (i) establishing optimal (in terms of the RFF dimension, and growing set size) performance guarantees in uniform norm, and (ii) presenting guarantees in $L^r$ ($1\le r<\infty$) norms. We also propose an RFF approximation to derivatives of a kernel with a theoretical study on its approximation quality.

math.ST↗
arXiv2010

Universality, Characteristic Kernels and RKHS Embedding of Measures

A Hilbert space embedding for probability measures has recently been proposed, wherein any probability measure is represented as a mean element in a reproducing kernel Hilbert space (RKHS). Such an embedding has found applications in homogeneity testing, independence testing, dimensionality reduction, etc., with the requirement that the reproducing kernel is characteristic, i.e., the embedding is injective. In this paper, we generalize this embedding to finite signed Borel measures, wherein any finite signed Borel measure is represented as a mean element in an RKHS. We show that the proposed embedding is injective if and only if the kernel is universal. This therefore, provides a novel characterization of universal kernels, which are proposed in the context of achieving the Bayes risk by kernel-based classification/regression algorithms. By exploiting this relation between universality and the embedding of finite signed Borel measures into an RKHS, we establish the relation between universal and characteristic kernels.

stat.ML↗
arXiv2010

Hilbert space embeddings and metrics on probability measures

A Hilbert space embedding for probability measures has recently been proposed, with applications including dimensionality reduction, homogeneity testing, and independence testing. This embedding represents any probability measure as a mean element in a reproducing kernel Hilbert space (RKHS). A pseudometric on the space of probability measures can be defined as the distance between distribution embeddings: we denote this as $γ_k$, indexed by the kernel function $k$ that defines the inner product in the RKHS. We present three theoretical properties of $γ_k$. First, we consider the question of determining the conditions on the kernel $k$ for which $γ_k$ is a metric: such $k$ are denoted {\em characteristic kernels}. Unlike pseudometrics, a metric is zero only when two distributions coincide, thus ensuring the RKHS embedding maps all distributions uniquely (i.e., the embedding is injective). While previously published conditions may apply only in restricted circumstances (e.g. on compact domains), and are difficult to check, our conditions are straightforward and intuitive: bounded continuous strictly positive definite kernels are characteristic. Alternatively, if a bounded continuous kernel is translation-invariant on $\bb{R}^d$, then it is characteristic if and only if the support of its Fourier transform is the entire $\bb{R}^d$. Second, we show that there exist distinct distributions that are arbitrarily close in $γ_k$. Third, to understand the nature of the topology induced by $γ_k$, we relate $γ_k$ to other popular metrics on probability measures, and present conditions on the kernel $k$ under which $γ_k$ metrizes the weak topology.

stat.ML↗
arXiv2009

On integral probability metrics, ϕ-divergences and binary classification

A class of distance measures on probabilities -- the integral probability metrics (IPMs) -- is addressed: these include the Wasserstein distance, Dudley metric, and Maximum Mean Discrepancy. IPMs have thus far mostly been used in more abstract settings, for instance as theoretical tools in mass transportation problems, and in metrizing the weak topology on the set of all Borel probability measures defined on a metric space. Practical applications of IPMs are less common, with some exceptions in the kernel machines literature. The present work contributes a number of novel properties of IPMs, which should contribute to making IPMs more widely used in practice, for instance in areas where $ϕ$-divergences are currently popular. First, to understand the relation between IPMs and $ϕ$-divergences, the necessary and sufficient conditions under which these classes intersect are derived: the total variation distance is shown to be the only non-trivial $ϕ$-divergence that is also an IPM. This shows that IPMs are essentially different from $ϕ$-divergences. Second, empirical estimates of several IPMs from finite i.i.d. samples are obtained, and their consistency and convergence rates are analyzed. These estimators are shown to be easily computable, with better rates of convergence than estimators of $ϕ$-divergences. Third, a novel interpretation is provided for IPMs by relating them to binary classification, where it is shown that the IPM between class-conditional distributions is the negative of the optimal risk associated with a binary classifier. In addition, the smoothness of an appropriate binary classifier is proved to be inversely related to the distance between the class-conditional distributions, measured in terms of an IPM.

cs.IT↗