arXiv subjects

Nabarun Deb

Publications and source records attributed to Nabarun Deb.

At least 19 recordsLinked to original sources

arXiv2026

LDP for Tensor Forms

In this paper, we study the large deviation principle (LDP) for a tensor-weighted functional of i.i.d. random variables, when the sequence of tensors converges under a variant of the "bad" cut norm. Using the LDP, we analyze a Gibbs measure with a tensor-valued Hamiltonian, and characterize the optimizers of the limiting variational problem in terms of a functional fixed point equation. As applications, we focus on several concrete examples, which include monochromatic subgraph counts in sparse random graphs, Erd\H{o}s-R\'enyi hypergraphs, and a generalized Potts statistic of order $v\ge 2$. Studying the optimization problem, we give sufficient conditions for uniqueness of the optimizer, as well as for existence of constant optimizers (replica symmetry). Our results demonstrate universal weak laws for a large class of tensor Gibbs models with approximately regular tensors.

math.PR↗

arXiv2026

Normal approximations in nonparametric empirical Bayes

Empirical Bayes analyses routinely model noisy measurements of latent parameters as normal, justifying this by an informal appeal to the central limit theorem (CLT). This paper puts this heuristic appeal on firmer analytical grounds. We show that the denoising regret of the nonparametric maximum likelihood estimator (NPMLE) and related sieve methods is controlled by the rate attained under exact normality, plus a term reflecting the quality of the CLT approximation. The CLT need only hold marginally for each coordinate, and moreover only on average, without needing high-dimensional normal approximations. We identify two asymptotic regimes in which the normal approximation is adequate and the empirical Bayesian prior remains informative, and we show that our guarantees are robust to dependence and to variance estimation.

math.ST↗

arXiv2026

Parametric Mean-Field empirical Bayes in high-dimensional linear regression

In this paper, we consider the problem of parametric empirical Bayes estimation of an i.i.d. prior in high-dimensional Bayesian linear regression, with random design. We obtain the asymptotic distribution of the variational Empirical Bayes (vEB) estimator, which approximately maximizes a variational lower bound of the intractable marginal likelihood. We characterize a sharp phase transition behavior for the vEB estimator -- namely that it is information theoretically optimal (in terms of limiting variance) up to $p=o(n^{2/3})$ while it suffers from a sub-optimal convergence rate in higher dimensions. In the first regime, i.e., when $p=o(n^{2/3})$, we show how the estimated prior can be calibrated to enable valid coordinate-wise and delocalized inference, both under the \emph{empirical Bayes posterior} and the oracle posterior. In the second regime, we propose a debiasing technique as a way to improve the performance of the vEB estimator beyond $p=o(n^{2/3})$. Extensive numerical experiments corroborate our theoretical findings.

math.ST↗

arXiv2025

Pivotal CLTs for Pseudolikelihood via Conditional Centering in Dependent Random Fields

In this paper, we study fluctuations of conditionally centered statistics of the form $$N^{-1/2}\sum_{i=1}^N c_i(g(\sigma_i)-\mathbb{E}_N[g(\sigma_i)|\sigma_j,j\neq i])$$ where $(\sigma_1,\ldots ,\sigma_N)$ are sampled from a dependent random field, and $g$ is some bounded function. Our first main result shows that under weak smoothness assumptions on the conditional means (which cover both sparse and dense interactions), the above statistic converges to a Gaussian \emph{scale mixture} with a random scale determined by a \emph{quadratic variance} and an \emph{interaction component}. We also show that under appropriate studentization, the limit becomes a pivotal Gaussian. We leverage this theory to develop a general asymptotic framework for maximum pseudolikelihood (MPLE) inference in dependent random fields. We apply our results to Ising models with pairwise as well as higher-order interactions and exponential random graph models (ERGMs). In particular, we obtain a joint central limit theorem for the inverse temperature and magnetization parameters via the joint MPLE (to our knowledge, the first such result in dense, irregular regimes), and we derive conditionally centered edge CLTs and marginal MPLE CLTs for ERGMs without restricting to the ``sub-critical" region. Our proof is based on a method of moments approach via combinatorial decision-tree pruning, which may be of independent interest.

math.ST↗

arXiv2025

CLT in high-dimensional Bayesian linear regression with low SNR

We study central limit theorems for linear statistics in high-dimensional Bayesian linear regression with product priors. Unlike the existing literature where the focus is on posterior contraction, we work under a non-contracting regime where neither the likelihood nor the prior dominates the other. This is motivated by modern high-dimensional datasets characterized by a bounded signal-to-noise ratio. This work takes a first step towards understanding limit distributions for one-dimensional projections of the posterior, as well as the posterior mean, in such regimes. Analogous to contractive settings, the resulting limiting distributions are Gaussian, but they heavily depend on the chosen prior and center around the Mean-Field approximation of the posterior. We study two concrete models of interest to illustrate this phenomenon -- the white noise design, and the (misspecified) Bayesian model. As an application, we construct credible intervals and compute their coverage probability under any misspecified prior. Our proofs rely on a combination of recent developments in Berry-Esseen type bounds for Random Field Ising models and both first and second order Poincar\'{e} inequalities. Notably, our results do not require any sparsity assumptions on the prior.

math.ST↗

arXiv2025

Phase Transition in Nonparametric Minimax Rates for Covariate Shifts on Approximate Manifolds

We study nonparametric regression under covariate shift with structured data, where a small amount of labeled target data is supplemented by a large labeled source dataset. In many real-world settings, the covariates in the target domain lie near a low-dimensional manifold within the support of the source, e.g., personalized handwritten digits (target) within a large, high-dimensional image repository (source). Since density ratios may not exist in these settings, standard transfer learning techniques often fail to leverage such structure. This necessitates the development of methods that exploit both the size of the source dataset and the structured nature of the target. Motivated by this, we establish new minimax rates under covariate shift for estimating a regression function in a general H\"older class, assuming the target distribution lies near -- but not exactly on -- a smooth submanifold of the source. General smoothness helps reduce the curse of dimensionality when the target function is highly regular, while approximate manifolds capture realistic, noisy data. We identify a phase transition in the minimax rate of estimation governed by the distance to the manifold, source and target sample sizes, function smoothness, and intrinsic versus ambient dimensions. We propose a local polynomial regression estimator that achieves optimal rates on either side of the phase transition boundary. Additionally, we construct a fully adaptive procedure that adjusts to unknown smoothness and intrinsic dimension, and attains nearly optimal rates. Our results unify and extend key threads in covariate shift, manifold learning, and adaptive nonparametric inference.

math.ST↗

arXiv2025

No-Regret Generative Modeling via Parabolic Monge-Amp\`ere PDE

We introduce a novel generative modeling framework based on a discretized parabolic Monge-Amp\`{e}re PDE, which emerges as a continuous limit of the Sinkhorn algorithm commonly used in optimal transport. Our method performs iterative refinement in the space of Brenier maps using a mirror gradient descent step. We establish theoretical guarantees for generative modeling through the lens of no-regret analysis, demonstrating that the iterates converge to the optimal Brenier map under a variety of step-size schedules. As a technical contribution, we derive a new Evolution Variational Inequality tailored to the parabolic Monge-Amp\`{e}re PDE, connecting geometry, transportation cost, and regret. Our framework accommodates non-log-concave target distributions, constructs an optimal sampling process via the Brenier map, and integrates favorable learning techniques from generative adversarial networks and score-based diffusion models. As direct applications, we illustrate how our theory paves new pathways for generative modeling and variational inference.

stat.ML↗

arXiv2025

Fluctuations in random field Ising models

This paper establishes a CLT for linear statistics of the form $\langle \mathbf{q},\boldsymbol{\sigma} \rangle$ with quantitative Berry-Esseen bounds, where $\boldsymbol{\sigma}$ is an observation from an exponential family with a quadratic form as its sufficient statistic, in the \enquote{high-temperature} regime. We apply our general result to random field Ising models with both discrete and continuous spins. To demonstrate the generality of our techniques, we apply our results to derive both quenched and annealed CLTs in various examples, which include Ising models on some graph ensembles of common interest (Erd\H{o}s-R\'{e}nyi, regular, dense bipartite), and the Hopfield spin glass model. Our proofs rely on a combination of Stein's method of exchangeable pairs and Chevet type concentration inequalities.

math.PR↗

arXiv2024

Distribution-free Measures of Association based on Optimal Transport

In this paper we propose and study a class of nonparametric, yet interpretable measures of association between two random vectors $X$ and $Y$ taking values in $\mathbb{R}^{d_1}$ and $\mathbb{R}^{d_2}$ respectively ($d_1, d_2\ge 1$). These nonparametric measures -- defined using the theory of reproducing kernel Hilbert spaces coupled with optimal transport -- 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 geometric graphs which include $k$-nearest neighbor graphs and minimum spanning trees. Additionally, these measures can also be readily used to construct an exact finite sample distribution-free test 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. The correlation coefficient proposed in Dette et al. (2013), Chatterjee (2021), Azadkia and Chatterjee (2021), at the population level, can be seen as a special case of this general class of measures.

math.ST↗

arXiv2024

Trade-off Between Dependence and Complexity for Nonparametric Learning -- an Empirical Process Approach

Empirical process theory for i.i.d. observations has emerged as a ubiquitous tool for understanding the generalization properties of various statistical problems. However, in many applications where the data exhibit temporal dependencies (e.g., in finance, medical imaging, weather forecasting etc.), the corresponding empirical processes are much less understood. Motivated by this observation, we present a general bound on the expected supremum of empirical processes under standard $\beta/\rho$-mixing assumptions. Unlike most prior work, our results cover both the long and the short-range regimes of dependence. Our main result shows that a non-trivial trade-off between the complexity of the underlying function class and the dependence among the observations characterizes the learning rate in a large class of nonparametric problems. This trade-off reveals a new phenomenon, namely that even under long-range dependence, it is possible to attain the same rates as in the i.i.d. setting, provided the underlying function class is complex enough. We demonstrate the practical implications of our findings by analyzing various statistical estimators in both fixed and growing dimensions. Our main examples include a comprehensive case study of generalization error bounds in nonparametric regression over smoothness classes in fixed as well as growing dimension using neural nets, shape-restricted multivariate convex regression, estimating the optimal transport (Wasserstein) distance between two probability distributions, and classification under the Mammen-Tsybakov margin condition -- all under appropriate mixing assumptions. In the process, we also develop bounds on $L_r$ ($1\le r\le 2$)-localized empirical processes with dependent observations, which we then leverage to get faster rates for (a) tuning-free adaptation, and (b) set-structured learning problems.

math.ST↗

arXiv2023

Wasserstein Mirror Gradient Flow as the limit of the Sinkhorn Algorithm

We prove that the sequence of marginals obtained from the iterations of the Sinkhorn algorithm or the iterative proportional fitting procedure (IPFP) on joint densities, converges to an absolutely continuous curve on the $2$-Wasserstein space, as the regularization parameter $\varepsilon$ goes to zero and the number of iterations is scaled as $1/\varepsilon$ (and other technical assumptions). This limit, which we call the Sinkhorn flow, is an example of a Wasserstein mirror gradient flow, a concept we introduce here inspired by the well-known Euclidean mirror gradient flows. In the case of Sinkhorn, the gradient is that of the relative entropy functional with respect to one of the marginals and the mirror is half of the squared Wasserstein distance functional from the other marginal. Interestingly, the norm of the velocity field of this flow can be interpreted as the metric derivative with respect to the linearized optimal transport (LOT) distance. An equivalent description of this flow is provided by the parabolic Monge-Amp\`{e}re PDE whose connection to the Sinkhorn algorithm was noticed by Berman (2020). We derive conditions for exponential convergence for this limiting flow. We also construct a Mckean-Vlasov diffusion whose marginal distributions follow the Sinkhorn flow.

math.PR↗

arXiv2023

Gibbs Measures with Multilinear Forms

In this paper, we study a class of multilinear Gibbs measures with Hamiltonian given by a generalized $\mathrm{U}$-statistic and with a general base measure. Expressing the asymptotic free energy as an optimization problem over a space of functions, we obtain sufficient conditions for replica-symmetry, and provide examples to show why these conditions are also necessary. Utilizing this, we obtain weak limits for a large class of statistics of interest, which includes the \enquote{local fields/magnetization}, the Hamiltonian, the global magnetization, etc. An interesting consequence is a universal weak law for contrasts under replica symmetry, namely, $n^{-1}\sum_{i=1}^n c_i X_i\to 0$ weakly, if $\sum_{i=1}^n c_i=o(n)$. Our results yield a probabilistic interpretation for the optimizers arising out of the limiting free energy. We also prove the existence of a sharp phase transition point in terms of the temperature parameter, thereby generalizing existing results that were only known for quadratic Hamiltonians. As a by-product of our proof technique, we obtain exponential concentration bounds on local and global magnetizations, which are of independent interest.

math.PR↗

arXiv2022

LDP for Inhomogeneous U-Statistics

In this paper we derive a Large Deviation Principle (LDP) for inhomogeneous U/V-statistics of a general order. Using this, we derive a LDP for two types of statistics: random multilinear forms, and number of monochromatic copies of a subgraph. We show that the corresponding rate functions in these cases can be expressed as a variational problem over a suitable space of functions. We use the tools developed to study Gibbs measures with the corresponding Hamiltonians, which include tensor generalizations of both Ising (with non-compact base measure) and Potts models. For these Gibbs measures, we establish scaling limits of log normalizing constants, and weak laws in terms of weak* topology, which are of possible independent interest.

math.PR↗

arXiv2021

Robustness and Efficiency of Rosenbaum's Rank-based Estimator in Randomized Trials: A Design-based Perspective

Mean-based estimators of causal effects in randomized experiments may behave poorly if the potential outcomes have a heavy tail or contain outliers. An alternative estimator proposed by Rosenbaum (1993) estimates a constant additive treatment effect by inverting a randomization test using ranks. We develop a design-based asymptotic theory for this rank-based estimator and study its robustness and efficiency properties. We show that Rosenbaum's estimator is robust against outliers with a breakdown point that uniformly dominates that of any weighted quantile estimator. When pretreatment covariates are available, a regression-adjusted version of Rosenbaum's estimator uses an agnostic linear regression on the covariates and bases inference on the ranks of residuals. Under mild integrability conditions, we show that this estimator is at most 13.6% less efficient, in the worst case, than the commonly used mean-based regression adjustment method proposed by Lin (2013); often outperforming it when the residuals have heavy tails. Moreover, under suitable assumptions, Rosenbaum's regression-adjusted estimator is at least as efficient as the unadjusted one. Finally, we initiate the study of Rosenbaum's estimator when the constant treatment effect assumption may be violated. To analyze the regression-adjusted estimator, we develop local asymptotics of rank statistics under the design-based framework, which may be of independent interest.

stat.ME↗

arXiv2021

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

Optimal transport maps between two probability distributions $\mu$ and $\nu$ 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 $\mu$ and $\nu$. 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 $W_2(\mu,\nu)$, the Wasserstein distance between $\mu$ and $\nu$, 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

Exact Detection Thresholds and Minimax Optimality of Chatterjee's Correlation Coefficient

Recently, Chatterjee (2021) introduced a new rank-based correlation coefficient which can be used to measure the strength of dependence between two random variables. This coefficient has already attracted much attention as it converges to the Dette-Siburg-Stoimenov measure (see Dette et al. (2013)), which equals $0$ if and only if the variables are independent and $1$ if and only if one variable is a function of the other. Further, Chatterjee's coefficient is computable in (near) linear time, which makes it appropriate for large-scale applications. In this paper, we expand the theoretical understanding of Chatterjee's coefficient in two directions: (a) First we consider the problem of testing for independence using Chatterjee's correlation. We obtain its asymptotic distribution under any changing sequence of alternatives converging to the null hypothesis (of independence). We further obtain a general result that gives exact detection thresholds and limiting power for Chatterjee's test of independence under natural nonparametric alternatives converging to the null. As applications of this general result, we prove a $n^{-1/4}$ detection boundary for this test and compute explicitly the limiting local power on the detection boundary for popularly studied alternatives in the literature. (b) We then construct a test for non-trivial levels of dependence using Chatterjee's coefficient. In contrast to testing for independence, we prove that, in this case, Chatterjee's coefficient indeed yields a minimax optimal procedure with a $n^{-1/2}$ detection boundary. Our proof techniques rely on Stein's method of exchangeable pairs, a non-asymptotic projection result, and information theoretic lower bounds.

math.ST↗

arXiv2021

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.

math.ST↗

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.

stat.ME↗