arXiv subjects

Bodhisattva Sen

Publications and source records attributed to Bodhisattva Sen.

At least 37 records · Page 2Linked to original sources

arXiv2023

Multivariate Symmetry: Distribution-Free Testing via Optimal Transport

The sign test (Arbuthnott, 1710) and the Wilcoxon signed-rank test (Wilcoxon, 1945) are among the first examples of a nonparametric test. These procedures -- based on signs, (absolute) ranks and signed-ranks -- yield distribution-free tests for symmetry in one-dimension. In this paper we propose a novel and unified framework for distribution-free testing of multivariate symmetry (that includes central symmetry, sign symmetry, spherical symmetry, etc.) based on the theory of optimal transport. Our approach leads to notions of distribution-free generalized multivariate signs, ranks and signed-ranks. As a consequence, we develop analogues of the sign and Wilcoxon signed-rank tests that share many of the appealing properties of their one-dimensional counterparts. In particular, the proposed tests are exactly distribution-free in finite samples with an asymptotic normal limit, and adapt to various notions of multivariate symmetry. We study the consistency of the proposed tests and their behavior under local alternatives, and show that the proposed generalized Wilcoxon signed-rank (GWSR) test is particularly powerful against location shift alternatives. We show that in a large class of such models, our GWSR test suffers from no loss in (asymptotic) efficiency, when compared to Hotelling's $T^2$ test, despite being nonparametric and exactly distribution-free. An appropriately score transformed version of the GWSR statistic leads to a locally asymptotically optimal test. Further, our method can be readily used to construct distribution-free confidence sets for the center of symmetry.

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arXiv2023

Pitman Efficiency Lower Bounds for Multivariate Distribution-Free Tests Based on Optimal Transport

Distribution-free tests such as the Wilcoxon rank sum test are popular for testing the equality of two univariate distributions. Among the important reasons for their popularity are the striking results of Hodges-Lehmann (1956) and Chernoff-Savage (1958), where the authors show that the asymptotic (Pitman) relative efficiency of Wilcoxon's test with respect to Student's $t$-test, under location-shift alternatives, never falls below $0.864$ (with the identity score) and $1$ (with the Gaussian score) respectively, despite the former being exactly distribution-free for all sample sizes. Motivated by these results, we propose and study a large family of exactly distribution-free multivariate rank-based two-sample tests by leveraging the theory of optimal transport. First, we propose distribution-free analogs of the Hotelling $T^2$ test (the natural multidimensional counterpart of Student's $t$-test) and show that they satisfy Hodges-Lehmann and Chernoff-Savage-type efficiency lower bounds over natural sub-families of multivariate distributions, despite being entirely agnostic to the underlying data generating mechanism -- making them the first multivariate, nonparametric, exactly distribution-free tests that provably achieve such efficiency lower bounds. As these tests are derived from Hotelling $T^2$, naturally they are not universally consistent (same as Wilcoxon's test). To overcome this, we propose exactly distribution-free versions of the celebrated kernel maximum mean discrepancy test and the energy test. These tests are indeed universally consistent under no moment assumptions, exactly distribution-free for all sample sizes, and have non-trivial Pitman efficiency. We believe this trifecta of properties hasn't yet been proven for any existing test in the literature.

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arXiv2022

A Kernel Measure of Dissimilarity between $M$ Distributions

Given $M \geq 2$ distributions defined on a general measurable space, we introduce a nonparametric (kernel) measure of multi-sample dissimilarity (KMD) -- a parameter that quantifies the difference between the $M$ distributions. The population KMD, which takes values between 0 and 1, is 0 if and only if all the $M$ distributions are the same, and 1 if and only if all the distributions are mutually singular. Moreover, KMD possesses many properties commonly associated with $f$-divergences such as the data processing inequality and invariance under bijective transformations. The sample estimate of KMD, based on independent observations from the $M$ distributions, can be computed in near linear time (up to logarithmic factors) using $k$-nearest neighbor graphs (for $k \ge 1$ fixed). We develop an easily implementable test for the equality of $M$ distributions based on the sample KMD that is consistent against all alternatives where at least two distributions are not equal. We prove central limit theorems for the sample KMD, and provide a complete characterization of the asymptotic power of the test, as well as its detection threshold. The usefulness of our measure is demonstrated via real and synthetic data examples; our method is also implemented in an R package.

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arXiv2022

On Efficient and Scalable Computation of the Nonparametric Maximum Likelihood Estimator in Mixture Models

In this paper we study the computation of the nonparametric maximum likelihood estimator (NPMLE) in multivariate mixture models. Our first approach discretizes this infinite dimensional convex optimization problem by fixing the support points of the NPMLE and optimizing over the mixture proportions. In this context we propose, leveraging the sparsity of the solution, an efficient and scalable semismooth Newton based augmented Lagrangian method (ALM). Our algorithm beats the state-of-the-art methods~\cite{koenker2017rebayes, kim2020fast} and can handle $n \approx 10^6$ data points with $m \approx 10^4$ support points. Our second procedure, which combines the expectation-maximization (EM) algorithm with the ALM approach above, allows for joint optimization of both the support points and the probability weights. For both our algorithms we provide formal results on their (superlinear) convergence properties. The computed NPMLE can be immediately used for denoising the observations in the framework of empirical Bayes. We propose new denoising estimands in this context along with their consistent estimates. Extensive numerical experiments are conducted to illustrate the effectiveness of our methods. In particular, we employ our procedures to analyze two astronomy data sets: (i) Gaia-TGAS Catalog~\cite{anderson2018improving} containing $n \approx 1.4 \times 10^6$ data points in two dimensions, and (ii) the $d=19$ dimensional data set from the APOGEE survey~\cite{majewski2017apache} with $n \approx 2.7 \times 10^4$.

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arXiv2022

Permuted and Unlinked Monotone Regression in $\mathbb{R}^d$: an approach based on mixture modeling and optimal transport

Suppose that we have a regression problem with response variable Y in $\mathbb{R}^d$ and predictor X in $\mathbb{R}^d$, for $d \geq 1$. In permuted or unlinked regression we have access to separate unordered data on X and Y, as opposed to data on (X,Y)-pairs in usual regression. So far in the literature the case $d=1$ has received attention, see e.g., the recent papers by Rigollet and Weed [Information & Inference, 8, 619--717] and Balabdaoui et al. [J. Mach. Learn. Res., 22(172), 1--60]. In this paper, we consider the general multivariate setting with $d \geq 1$. We show that the notion of cyclical monotonicity of the regression function is sufficient for identification and estimation in the permuted/unlinked regression model. We study permutation recovery in the permuted regression setting and develop a computationally efficient and easy-to-use algorithm for denoising based on the Kiefer-Wolfowitz [Ann. Math. Statist., 27, 887--906] nonparametric maximum likelihood estimator and techniques from the theory of optimal transport. We provide explicit upper bounds on the associated mean squared denoising error for Gaussian noise. As in previous work on the case $d = 1$, the permuted/unlinked setting involves slow (logarithmic) rates of convergence rooting in the underlying deconvolution problem. Numerical studies corroborate our theoretical analysis and show that the proposed approach performs at least on par with the methods in the aforementioned prior work in the case $d = 1$ while achieving substantial reductions in terms of computational complexity.

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arXiv2021

Tracing birth properties of stars with abundance clustering

To understand the formation and evolution of the Milky Way disk, we must connect its current properties to its past. We explore hydrodynamical cosmological simulations to investigate how the chemical abundances of stars might be linked to their origins. Using hierarchical clustering of abundance measurements in two Milky Way-like simulations with distributed and steady star formation histories, we find that abundance clusters of stars comprise different groups in birth place ($R_\text{birth}$) and time (age). Simulating observational abundance errors (0.05 dex), we find that to trace discrete groups of ($R_\text{birth}$, age) requires a large vector of abundances. Using 15-element abundances (Fe, O, Mg, S, Si, C, P, Mn, Ne, Al, N, V, Ba, Cr, Co), up to $\approx$ 10 clusters can be defined with $\approx$ 25% overlap in ($R_\text{birth}$, age). We build a simple model to show that it is possible to infer a star's age and $R_\text{birth}$ from abundances with precisions of $\pm$0.06 Gyr and $\pm$1.17 kpc respectively. We find that abundance clustering is ineffective for a third simulation, where low-$α$ stars form distributed in the disc and early high-$α$ stars form more rapidly in clumps that sink towards the galactic center as their constituent stars evolve to enrich the interstellar medium. However, this formation path leads to large age-dispersions across the [$α$/Fe]-[Fe/H] plane, which is inconsistent with the Milky Way's observed properties. We conclude that abundance clustering is a promising approach toward charting the history of our Galaxy.

astro-ph.GA↗

arXiv2021

Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections

Optimal transport maps between two probability distributions $μ$ and $ν$ on $\mathbb{R}^d$ have found extensive applications in both machine learning and statistics. In practice, these maps need to be estimated from data sampled according to $μ$ and $ν$. Plug-in estimators are perhaps most popular in estimating transport maps in the field of computational optimal transport. In this paper, we provide a comprehensive analysis of the rates of convergences for general plug-in estimators defined via barycentric projections. Our main contribution is a new stability estimate for barycentric projections which proceeds under minimal smoothness assumptions and can be used to analyze general plug-in estimators. We illustrate the usefulness of this stability estimate by first providing rates of convergence for the natural discrete-discrete and semi-discrete estimators of optimal transport maps. We then use the same stability estimate to show that, under additional smoothness assumptions of Besov type or Sobolev type, wavelet based or kernel smoothed plug-in estimators respectively speed up the rates of convergence and significantly mitigate the curse of dimensionality suffered by the natural discrete-discrete/semi-discrete estimators. As a by-product of our analysis, we also obtain faster rates of convergence for plug-in estimators of $W_2(μ,ν)$, the Wasserstein distance between $μ$ and $ν$, under the aforementioned smoothness assumptions, thereby complementing recent results in Chizat et al. (2020). Finally, we illustrate the applicability of our results in obtaining rates of convergence for Wasserstein barycenters between two probability distributions and obtaining asymptotic detection thresholds for some recent optimal-transport based tests of independence.

math.ST↗

arXiv2021

High dimensional asymptotics of likelihood ratio tests in the Gaussian sequence model under convex constraints

In the Gaussian sequence model $Y=μ+ξ$, we study the likelihood ratio test (LRT) for testing $H_0: μ=μ_0$ versus $H_1: μ\in K$, where $μ_0 \in K$, and $K$ is a closed convex set in $\mathbb{R}^n$. In particular, we show that under the null hypothesis, normal approximation holds for the log-likelihood ratio statistic for a general pair $(μ_0,K)$, in the high dimensional regime where the estimation error of the associated least squares estimator diverges in an appropriate sense. The normal approximation further leads to a precise characterization of the power behavior of the LRT in the high dimensional regime. These characterizations show that the power behavior of the LRT is in general non-uniform with respect to the Euclidean metric, and illustrate the conservative nature of existing minimax optimality and sub-optimality results for the LRT. A variety of examples, including testing in the orthant/circular cone, isotonic regression, Lasso, and testing parametric assumptions versus shape-constrained alternatives, are worked out to demonstrate the versatility of the developed theory.

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arXiv2021

Multivariate Ranks and Quantiles using Optimal Transport: Consistency, Rates, and Nonparametric Testing

In this paper we study multivariate ranks and quantiles, defined using the theory of optimal transport, and build on the work of Chernozhukov et al.(2017) and Hallin et al.(2021). We study the characterization, computation and properties of the multivariate rank and quantile functions and their empirical counterparts. We derive the uniform consistency of these empirical estimates to their population versions, under certain assumptions. In fact, we prove a Glivenko-Cantelli type theorem that shows the asymptotic stability of the empirical rank map in any direction. Under mild structural assumptions, we provide global and local rates of convergence of the empirical quantile and rank maps. We also provide a sub-Gaussian tail bound for the global L_2-loss of the empirical quantile function. Further, we propose tuning parameter-free multivariate nonparametric tests -- a two-sample test and a test for mutual independence -- based on our notion of multivariate quantiles/ranks. Asymptotic consistency of these tests are shown and the rates of convergence of the associated test statistics are derived, both under the null and alternative hypotheses.

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arXiv2021

Semiparametric Efficiency in Convexity Constrained Single Index Model

We consider estimation and inference in a single index regression model with an unknown convex link function. We introduce a convex and Lipschitz constrained least squares estimator (CLSE) for both the parametric and the nonparametric components given independent and identically distributed observations. We prove the consistency and find the rates of convergence of the CLSE when the errors are assumed to have only $q \ge 2$ moments and are allowed to depend on the covariates. When $q\ge 5$, we establish $n^{-1/2}$-rate of convergence and asymptotic normality of the estimator of the parametric component. Moreover, the CLSE is proved to be semiparametrically efficient if the errors happen to be homoscedastic. {We develop and implement a numerically stable and computationally fast algorithm to compute our proposed estimator in the R package~\texttt{simest}}. We illustrate our methodology through extensive simulations and data analysis. Finally, our proof of efficiency is geometric and provides a general framework that can be used to prove efficiency of estimators in a wide variety of semiparametric models even when they do not satisfy the efficient score equation directly.

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arXiv2020

Kernel Partial Correlation Coefficient -- a Measure of Conditional Dependence

In this paper we propose and study a class of simple, nonparametric, yet interpretable measures of conditional dependence between two random variables $Y$ and $Z$ given a third variable $X$, all taking values in general topological spaces. The population version of any of these measures captures the strength of conditional dependence and it is 0 if and only if $Y$ and $Z$ are conditionally independent given $X$, and 1 if and only if $Y$ is a measurable function of $Z$ and $X$. Thus, our measure -- which we call kernel partial correlation (KPC) coefficient -- can be thought of as a nonparametric generalization of the partial correlation coefficient that possesses the above properties when $(X,Y,Z)$ is jointly normal. We describe two consistent methods of estimating KPC. Our first method utilizes the general framework of geometric graphs, including $K$-nearest neighbor graphs and minimum spanning trees. A sub-class of these estimators can be computed in near linear time and converges at a rate that automatically adapts to the intrinsic dimension(s) of the underlying distribution(s). Our second strategy involves direct estimation of conditional mean embeddings using cross-covariance operators in the reproducing kernel Hilbert spaces. Using these empirical measures we develop forward stepwise (high-dimensional) nonlinear variable selection algorithms. We show that our algorithm, using the graph-based estimator, yields a provably consistent model-free variable selection procedure, even in the high-dimensional regime when the number of covariates grows exponentially with the sample size, under suitable sparsity assumptions. Extensive simulation and real-data examples illustrate the superior performance of our methods compared to existing procedures. The recent conditional dependence measure proposed by Azadkia and Chatterjee (2019) can be viewed as a special case of our general framework.

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arXiv2020

Measuring Association on Topological Spaces Using Kernels and Geometric Graphs

In this paper we propose and study a class of simple, nonparametric, yet interpretable measures of association between two random variables $X$ and $Y$ taking values in general topological spaces. These nonparametric measures -- defined using the theory of reproducing kernel Hilbert spaces -- capture the strength of dependence between $X$ and $Y$ and have the property that they are 0 if and only if the variables are independent and 1 if and only if one variable is a measurable function of the other. Further, these population measures can be consistently estimated using the general framework of graph functionals which include $k$-nearest neighbor graphs and minimum spanning trees. Moreover, a sub-class of these estimators are also shown to adapt to the intrinsic dimensionality of the underlying distribution. Some of these empirical measures can also be computed in near linear time. Under the hypothesis of independence between $X$ and $Y$, these empirical measures (properly normalized) have a standard normal limiting distribution. Thus, these measures can also be readily used to test the hypothesis of mutual independence between $X$ and $Y$. In fact, as far as we are aware, these are the only procedures that possess all the above mentioned desirable properties. Furthermore, when restricting to Euclidean spaces, we can make these sample measures of association finite-sample distribution-free, under the hypothesis of independence, by using multivariate ranks defined via the theory of optimal transport. The recent correlation coefficient proposed in Dette et al. (2013), Chatterjee (2019), and Azadkia and Chatterjee (2019) can be seen as a special case of this general class of measures.

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arXiv2020

Inference for local parameters in convexity constrained models

We consider the problem of inference for local parameters of a convex regression function $f_0: [0,1] \to \mathbb{R}$ based on observations from a standard nonparametric regression model, using the convex least squares estimator (LSE) $\widehat{f}_n$. For $x_0 \in (0,1)$, the local parameters include the pointwise function value $f_0(x_0)$, the pointwise derivative $f_0'(x_0)$, and the anti-mode (i.e., the smallest minimizer) of $f_0$. The existing limiting distribution of the estimation error $(\widehat{f}_n(x_0) - f_0(x_0), \widehat{f}_n'(x_0) - f_0'(x_0) )$ depends on the unknown second derivative $f_0''(x_0)$, and is therefore not directly applicable for inference. To circumvent this impasse, we show that the following locally normalized errors (LNEs) enjoy pivotal limiting behavior: Let $[\widehat{u}(x_0), \widehat{v}(x_0)]$ be the maximal interval containing $x_0$ where $\widehat{f}_n$ is linear. Then, under standard conditions, $$\binom{ \sqrt{n(\widehat{v}(x_0)-\widehat{u}(x_0))}(\widehat{f}_n(x_0)-f_0(x_0)) }{ \sqrt{n(\widehat{v}(x_0)-\widehat{u}(x_0))^3}(\widehat{f}_n'(x_0)-f_0'(x_0))} \rightsquigarrow σ\cdot \binom{\mathbb{L}^{(0)}_2}{\mathbb{L}^{(1)}_2},$$ where $n$ is the sample size, $σ$ is the standard deviation of the errors, and $\mathbb{L}^{(0)}_2, \mathbb{L}^{(1)}_2$ are universal random variables. This asymptotically pivotal LNE theory instantly yields a simple tuning-free procedure for constructing CIs with asymptotically exact coverage and optimal length for $f_0(x_0)$ and $f_0'(x_0)$. We also construct an asymptotically pivotal LNE for the anti-mode of $f_0$, and its limiting distribution does not even depend on $σ$. These asymptotically pivotal LNE theories are further extended to other convexity/concavity constrained models (e.g., log-concave density estimation) for which a limit distribution theory is available for problem-specific estimators.

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arXiv2020

Multivariate extensions of isotonic regression and total variation denoising via entire monotonicity and Hardy-Krause variation

We consider the problem of nonparametric regression when the covariate is $d$-dimensional, where $d \geq 1$. In this paper we introduce and study two nonparametric least squares estimators (LSEs) in this setting---the entirely monotonic LSE and the constrained Hardy-Krause variation LSE. We show that these two LSEs are natural generalizations of univariate isotonic regression and univariate total variation denoising, respectively, to multiple dimensions. We discuss the characterization and computation of these two LSEs obtained from $n$ data points. We provide a detailed study of their risk properties under the squared error loss and fixed uniform lattice design. We show that the finite sample risk of these LSEs is always bounded from above by $n^{-2/3}$ modulo logarithmic factors depending on $d$; thus these nonparametric LSEs avoid the curse of dimensionality to some extent. We also prove nearly matching minimax lower bounds. Further, we illustrate that these LSEs are particularly useful in fitting rectangular piecewise constant functions. Specifically, we show that the risk of the entirely monotonic LSE is almost parametric (at most $1/n$ up to logarithmic factors) when the true function is well-approximable by a rectangular piecewise constant entirely monotone function with not too many constant pieces. A similar result is also shown to hold for the constrained Hardy-Krause variation LSE for a simple subclass of rectangular piecewise constant functions. We believe that the proposed LSEs yield a novel approach to estimating multivariate functions using convex optimization that avoid the curse of dimensionality to some extent.

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arXiv2020

Tracing the assembly of the Milky Way's disk through abundance clustering

A major goal in the field of galaxy formation is to understand the formation of the Milky Way's disk. The first step toward doing this is to empirically describe its present state. We use the new high-dimensional dataset of 19 abundances from 27,135 red clump APOGEE stars to examine the distribution of clusters defined using abundances. We explore different dimensionality reduction techniques and implement a non-parametric agglomerate hierarchical clustering method. We see that groups defined using abundances are spatially separated, as a function of age. Furthermore, the abundance groups represent different distributions in the [Fe/H]-age plane. Ordering our clusters by age reveals patterns suggestive of the sequence of chemical enrichment in the disk over time. Our results indicate that a promising avenue to trace the details of the disk's assembly is via a full interpretation of the empirical connections we report.

astro-ph.GA↗

arXiv2019

Multivariate Rank-based Distribution-free Nonparametric Testing using Measure Transportation

In this paper, we propose a general framework for distribution-free nonparametric testing in multi-dimensions, based on a notion of multivariate ranks defined using the theory of measure transportation. Unlike other existing proposals in the literature, these multivariate ranks share a number of useful properties with the usual one-dimensional ranks; most importantly, these ranks are distribution-free. This crucial observation allows us to design nonparametric tests that are exactly distribution-free under the null hypothesis. We demonstrate the applicability of this approach by constructing exact distribution-free tests for two classical nonparametric problems: (i) testing for mutual independence between random vectors, and (ii) testing for the equality of multivariate distributions. In particular, we propose (multivariate) rank versions of distance covariance (Székely et al., 2007) and energy statistic (Székely and Rizzo, 2013) for testing scenarios (i) and (ii) respectively. In both these problems, we derive the asymptotic null distribution of the proposed test statistics. We further show that our tests are consistent against all fixed alternatives. Moreover, the proposed tests are tuning-free, computationally feasible and are well-defined under minimal assumptions on the underlying distributions (e.g., they do not need any moment assumptions). We also demonstrate the efficacy of these procedures via extensive simulations. In the process of analyzing the theoretical properties of our procedures, we end up proving some new results in the theory of measure transportation and in the limit theory of permutation statistics using Stein's method for exchangeable pairs, which may be of independent interest.

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arXiv2019

On Efficiency of the Plug-in Principle for Estimating Smooth Integrated Functionals of a Nonincreasing Density

We consider the problem of estimating smooth integrated functionals of a monotone nonincreasing density $f$ on $[0,\infty)$ using the nonparametric maximum likelihood based plug-in estimator. We find the exact asymptotic distribution of this natural (tuning parameter-free) plug-in estimator, properly normalized. In particular, we show that the simple plug-in estimator is always $\sqrt{n}$-consistent, and is additionally asymptotically normal with zero mean and the semiparametric efficient variance for estimating a subclass of integrated functionals. Compared to the previous results on this topic (see e.g., Nickl (2007), Gine and Nickl (2008), Jankowski (2014), and Sohl (2015)) our results hold for a much larger class of functionals (which include linear and non-linear functionals) under less restrictive assumptions on the underlying $f$ --- we do not require $f$ to be (i) smooth, (ii) bounded away from $0$, or (iii) compactly supported. Further, when $f$ is the uniform distribution on a compact interval we explicitly characterize the asymptotic distribution of the plug-in estimator --- which now converges at a non-standard rate --- thereby extending the results in Groeneboom and Pyke (1983) for the case of the quadratic functional.

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arXiv2019

Two-component Mixture Model in the Presence of Covariates

In this paper, we study a generalization of the two-groups model in the presence of covariates --- a problem that has recently received much attention in the statistical literature due to its applicability in multiple hypotheses testing problems. The model we consider allows for infinite dimensional parameters and offers flexibility in modeling the dependence of the response on the covariates. We discuss the identifiability issues arising in this model and systematically study several estimation strategies. We propose a tuning parameter-free nonparametric maximum likelihood method, implementable via the EM algorithm, to estimate the unknown parameters. Further, we derive the rate of convergence of the proposed estimators --- in particular, we show that the finite sample Hellinger risk for every `approximate' nonparametric maximum likelihood estimator achieves a near-parametric rate (up to logarithmic multiplicative factors). In addition, we propose and theoretically study two `marginal' methods that are more scalable and easily implementable. We demonstrate the efficacy of our procedures through extensive simulation studies and relevant data analyses --- one arising from neuroscience and the other from astronomy. We also outline the application of our methods to multiple testing. The companion R package $\texttt{NPMLEmix}$ implements all the procedures proposed in this paper.

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