# Somabha MukherjeeNational University of Singapore | NUS · Department of Statistics and Applied Probability

Somabha Mukherjee

Doctor of Philosophy

## About

40

Publications

3,397

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137

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Introduction

I am currently an Assistant Professor in the Department of Statistics and Data Science, National University of Singapore. Prior to joining this department, I completed PhD from the Department of Statistics, the Wharton School, University of Pennsylvania.
I am interested in both Statistics and Probability, mainly inference in graphical models, high-dimensional CLT, shape-restricted regression, inference in Markov random fields, combinatorial probability, large deviations and random graphs.

**Skills and Expertise**

## Publications

Publications (40)

In this paper we characterize all distributional limits of the quadratic chaos Tn=∑1≤u<v≤nau,vXuXv\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_n =\sum _{1\le u< v\...

In this paper, we derive distributional convergence rates for the magnetization vector in the tensor Curie-Weiss Potts model. Limit theorems for the magnetization vector have been derived recently in Bhowal & Mukherjee (2023), where several phase transition phenomena in terms of the scaling of the (centered) magnetization and its asymptotic distrib...

We develop a new approach for the estimation of a multivariate function based on the economic axioms of quasiconvexity (and monotonicity). On the computational side, we prove the existence of the quasiconvex constrained least squares estimator (LSE) and provide a characterisation of the function space to compute the LSE via a mixed-integer quadrati...

The estimator of a causal directed acyclic graph (DAG) with the PC algorithm is known to be consistent based on independent and identically distributed samples. In this paper, we consider the scenario when the multivariate samples are identically distributed but not independent. A common example is a stationary multivariate time series. We show tha...

In this paper, we derive results about the limiting distribution of the empirical magnetization vector and the maximum likelihood (ML) estimates of the natural parameters in the tensor Curie-Weiss Potts model. Our results reveal surprisingly new phase transition phenomena including the existence of a smooth curve in the interior of the parameter pl...

With advances in single cell RNA sequencing (scRNAseq), accurate detection of perturbed pathways between conditions or cell types of interest becomes a critical analytical step. The ability to model alterations in a set of genes corresponding of a biological function is particularly useful when comparing cells between healthy and tumor tissues. Yet...

The $k$-tensor Ising model is an exponential family on a $p$-dimensional binary hypercube for modeling dependent binary data, where the sufficient statistic consists of all $k$-fold products of the observations, and the parameter is an unknown $k$-fold tensor, designed to capture higher-order interactions between the binary variables. In this paper...

Pathway analysis is a key analytical stage in the interpretation of omics data, providing a powerful method for detecting alterations in cellular processes. We recently developed a sensitive and distribution-free statistical framework for multisample distribution testing, which we implement here in the open-source R package single-cell pathway anal...

The estimator of a causal directed acyclic graph (DAG) with the PC algorithm is known to be consistent based on independent and identically distributed samples. In this paper, we consider the scenario when the multivariate samples are identically distributed but not independent. A common example is a stationary multivariate time series. We show tha...

The $p$-tensor Ising model is a one-parameter discrete exponential family for modeling dependent binary data, where the sufficient statistic is a multi-linear form of degree $p \geqslant 2$. This is a natural generalization of the matrix Ising model that provides a convenient mathematical framework for capturing, not just pairwise, but higher-order...

In this paper we characterize all distributional limits of the random quadratic form $T_n =\sum_{1\le u< v\le n} a_{u, v} X_u X_v$, where $((a_{u, v}))_{1\le u,v\le n}$ is a $\{0, 1\}$-valued symmetric matrix with zeros on the diagonal and $X_1, X_2, \ldots, X_n$ are i.i.d.~ mean $0$ variance $1$ random variables with common distribution function $...

Next generation sequencing technologies have revolutionized the study of T cell biology, capturing previously unrecognized diversity in cellular states and functions. Pathway analysis is a key analytical stage in the interpretation of such transcriptomic data, providing a powerful method for detecting alterations in important biological processes....

Logistic regression is one of the most fundamental methods for modeling the probability of a binary outcome based on a collection of covariates. However, the classical formulation of logistic regression relies on the independent sampling assumption, which is often violated when the outcomes interact through an underlying network structure. This nec...

In this paper we study the fluctuations of the magnetization in the p-spin Curie–Weiss model, for \(p \geqslant 3\). We provide a complete description of the asymptotic distribution of the magnetization in the p-spin Curie–Weiss model, complementing the well-known results in the 2-spin case. Our results unearth various new phase transitions, such a...

The tensor Ising model is a discrete exponential family used for modeling binary data on networks with not just pairwise, but higher-order dependencies. In this exponential family, the sufficient statistic is a multi-linear form of degree $p\ge 2$, designed to capture $p$-fold interactions between the binary variables sitting on the nodes of a netw...

High-dimensional feature selection is a central problem in a variety of application domains such as machine learning, image analysis, and genomics. In this paper, we propose graph-based tests as a useful basis for feature selection. We describe an algorithm for selecting informative features in high-dimensional data, where each observation comes fr...

The Ising model is a celebrated example of a Markov random field, introduced in statistical physics to model ferromagnetism. This is a discrete exponential family with binary outcomes, where the sufficient statistic involves a quadratic term designed to capture correlations arising from pairwise interactions. However, in many situations the depende...

The Ising model is a celebrated example of a Markov random field, which was introduced in statistical physics to model ferromagnetism. More recently, it has emerged as a useful model for understanding dependent binary data with an underlying network structure. This is a discrete exponential family with binary outcomes, where the sufficient statisti...

An integer partition is called graphical if it is the degree sequence of a simple graph. We prove that the probability that a uniformly chosen partition of size $n$ is graphical decreases to zero faster than $n^{-.003}$, answering a question of Pittel. A lower bound of $n^{-1/2}$ was proven by Erd̋s and Richmond, meaning our work demonstrates that...

The $p$-tensor Ising model is a one-parameter discrete exponential family for modeling dependent binary data, where the sufficient statistic is a multi-linear form of degree $p \geq 2$. This is a natural generalization of the matrix Ising model, that provides a convenient mathematical framework for capturing higher-order dependencies in complex rel...

Given a sequence of s-uniform hypergraphs {Hn}n≥1, denote by Tp(Hn) the number of edges in the random induced hypergraph obtained by including every vertex in Hn independently with probability p∈(0,1). Recent advances in the large deviations of low complexity non-linear functions of independent Bernoulli variables can be used to show that tail prob...

In this paper we propose a nonparametric graphical test based on optimal matching, for assessing the equality of multiple unknown multivariate probability distributions. Our procedure pools the data from the different classes to create a graph based on the minimum non-bipartite matching, and then utilizes the number of edges connecting data points...

The $p$-tensor Curie-Weiss model is a two-parameter discrete exponential family for modeling dependent binary data, where the sufficient statistic has a linear term and a term with degree $p \geq 2$. This is a special case of the tensor Ising model and the natural generalization of the matrix Curie-Weiss model, which provides a convenient mathemati...

An integer partition is called graphical if it is the degree sequence of a simple graph. We prove that the probability that a uniformly chosen partition of size $n$ is graphical decreases to zero faster than $n^{-.003}$, answering a question of Pittel. A lower bound of $n^{-1/2}$ was proven by Erd\H{o}s and Richmond, and so this demonstrates that t...

We develop a new approach for the estimation of a multivariate function based on the economic axioms of monotonicity and quasiconvexity. We prove the existence of the nonparametric least squares estimator (LSE) for a monotone and quasiconvex function and provide two characterizations for it. One of these characterizations is useful from the theoret...

Consider the random quadratic form $T_n=\sum_{1 \leq u < v \leq n} a_{uv} X_u X_v$, where $((a_{uv}))_{1 \leq u, v \leq n}$ is a $\{0, 1\}$-valued symmetric matrix with zeros on the diagonal, and $X_1,$ $X_2, \ldots, X_n$ are i.i.d. $\mathrm{Ber}(p_n)$. In this paper, we prove various characterization theorems about the limiting distribution of $T_...

In this paper we propose a nonparametric graphical test based on optimal matching, for assessing the equality of multiple unknown multivariate probability distributions. Our procedure pools the data from the different classes to create a graph based on the minimum non-bipartite matching, and then utilizes the number of edges connecting data points...

Given a sequence of $s$-uniform hypergraphs $\{H_n\}_{n \geq 1}$, denote by $T_p(H_n)$ the number of edges in the random induced hypergraph obtained by including every vertex in $H_n$ independently with probability $p \in (0, 1)$. Recent advances in the large deviations of low complexity non-linear functions of independent Bernoulli variables can b...

Central limit theorems (CLTs) for high-dimensional random vectors with dimension possibly growing with the sample size have received a lot of attention in the recent times. Chernozhukov et al., (2017) proved a Berry--Esseen type result for high-dimensional averages for the class of hyperrectangles and they proved that the rate of convergence can be...

What is the chance that among a group of $n$ friends, there are $s$ friends all of whom have the same birthday? This is the celebrated birthday problem which can be formulated as the existence of a monochromatic $s$-clique $K_s$ ($s$-matching birthdays) in the complete graph $K_n$, where every vertex of $K_n$ is uniformly colored with $365$ colors...

M-estimators offer simple robust alternatives to the maximum likelihood estimator. Much of the robustness literature, however, has focused on the problems of location, location-scale and regression estimation rather than on estimation of general parameters. The density power divergence (DPD) and the logarithmic density power divergence (LDPD) measu...

In this article, we use the strong law of large numbers to give a proof of the Herschel-Maxwell theorem, which characterizes the normal distribution as the distribution of the components of a spherically symmetric random vector, provided they are independent. We present shorter proofs under additional moment assumptions, and include a remark, which...