arXiv subjects

Minerva Mukhopadhyay

Publications and source records attributed to Minerva Mukhopadhyay.

9 recordsLinked to original sources

arXiv2025

Ultrahigh-dimensional Quadratic Discriminant Analysis Using Random Projections

This paper investigates the effectiveness of using the Random Projection Ensemble (RPE) approach in Quadratic Discriminant Analysis (QDA) for ultrahigh-dimensional classification problems. Classical methods such as Linear Discriminant Analysis (LDA) and QDA are used widely, but face significant challenges in their implementation when the data dimension (say, $p$) exceeds the sample size (say, $n$). In particular, both LDA (using the Moore-Penrose inverse for covariance matrices) and QDA (even with known covariance matrices) may perform as poorly as random guessing when $p/n \to \infty$ as $n \to \infty$. The RPE method, known for addressing the curse of dimensionality, offers a fast and effective solution without relying on selective summary measures of the competing distributions. This paper demonstrates the practical advantages of employing RPE on QDA in terms of classification performance as well as computational efficiency. We establish results for limiting perfect classification in both the population and sample versions of the proposed RPE-QDA classifier, under fairly general assumptions that allow for sub-exponential growth of $p$ relative to $n$. Several simulated and gene expression data sets are analyzed to evaluate the performance of the proposed classifier in ultrahigh-dimensional~scenarios.

stat.ME↗

arXiv2024

Bayesian Semiparametric Longitudinal Inverse-Probit Mixed Models for Category Learning

Understanding how the adult human brain learns novel categories is an important problem in neuroscience. Drift-diffusion models are popular in such contexts for their ability to mimic the underlying neural mechanisms. One such model for gradual longitudinal learning was recently developed by Paulon et al. (2021). Fitting conventional drift-diffusion models, however, requires data on both category responses and associated response times. In practice, category response accuracies are often the only reliable measure recorded by behavioral scientists to describe human learning. However, To our knowledge, drift-diffusion models for such scenarios have never been considered in the literature. To address this gap, in this article, we build carefully on Paulon et al. (2021), but now with latent response times integrated out, to derive a novel biologically interpretable class of `inverse-probit' categorical probability models for observed categories alone. However, this new marginal model presents significant identifiability and inferential challenges not encountered originally for the joint model by Paulon et al. (2021). We address these new challenges using a novel projection-based approach with a symmetry-preserving identifiability constraint that allows us to work with conjugate priors in an unconstrained space. We adapt the model for group and individual-level inference in longitudinal settings. Building again on the model's latent variable representation, we design an efficient Markov chain Monte Carlo algorithm for posterior computation. We evaluate the empirical performance of the method through simulation experiments. The practical efficacy of the method is illustrated in applications to longitudinal tone learning studies.

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arXiv2023

Bayesian Variable Selection Under High-dimensional Settings With Grouped Covariates

Consider the normal linear regression setup when the number of covariates p is much larger than the sample size n, and the covariates form correlated groups. The response variable y is not related to an entire group of covariates in all or none basis, rather the sparsity assumption persists within and between groups. We extend the traditional g-prior setup to this framework. Variable selection consistency of the proposed method is shown under fairly general conditions, assuming the covariates to be random and allowing the true model to grow with both n and p. For the purpose of implementation of the proposed g-prior method to high-dimensional setup, we propose two procedures. First, a group screening procedure, termed as group SIS (GSIS), and secondly, a novel stochastic search variable selection algorithm, termed as group informed variable selection algorithm (GiVSA), which uses the known group structure efficiently to explore the model space without discarding any covariate based on an initial screening. Screening consistency of GSIS, and theoretical mixing time of GiVSA are studied using the canonical path ensemble approach of Yang et al. (2016). Performance of the proposed prior with implementation of GSIS as well as GiVSA are validated using various simulated examples and a real data related to residential buildings.

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arXiv2022

Efficient Manifold and Subspace Approximations with Spherelets

In statistical dimensionality reduction, it is common to rely on the assumption that high dimensional data tend to concentrate near a lower dimensional manifold. There is a rich literature on approximating the unknown manifold, and on exploiting such approximations in clustering, data compression, and prediction. Most of the literature relies on linear or locally linear approximations. In this article, we propose a simple and general alternative, which instead uses spheres, an approach we refer to as spherelets. We develop spherical principal components analysis (SPCA), and provide theory on the convergence rate for global and local SPCA, while showing that spherelets can provide lower covering numbers and MSEs for many manifolds. Results relative to state-of-the-art competitors show gains in ability to accurately approximate manifolds with fewer components. Unlike most competitors, which simply output lower-dimensional features, our approach projects data onto the estimated manifold to produce fitted values that can be used for model assessment and cross validation. The methods are illustrated with applications to multiple data sets.

stat.ML↗

arXiv2021

Bayes Factor Asymptotics for Variable Selection in the Gaussian Process Framework

Although variable selection is one of the most popular areas of modern statistical research, much of its development has taken place in the classical paradigm compared to the Bayesian counterpart. Somewhat surprisingly, both the paradigms have focussed almost completely on linear models, in spite of the vast scope offered by the model liberation movement brought about by modern advancements in studying real, complex phenomena. In this article, we investigate general Bayesian variable selection in models driven by Gaussian processes, which allows us to treat linear, non-linear and nonparametric models, in conjunction with even dependent setups, in the same vein. We consider the Bayes factor route to variable selection, and develop a general asymptotic theory for the Gaussian process framework in the "large p, large n" settings even with p>>n, establishing almost sure exponential convergence of the Bayes factor under appropriately mild conditions. The fixed p setup is included as a special case. To illustrate, we apply our general result to variable selection in linear regression, Gaussian process model with squared exponential covariance function accommodating the covariates, and a first order autoregressive process with time-varying covariates. We also follow up our theoretical investigations with ample simulation experiments in the above regression contexts and variable selection in a real, riboflavin data consisting of 71 observations but 4088 covariates. For implementation of variable selection using Bayes factors, we develop a novel and effective general-purpose transdimensional, transformation based Markov chain Monte Carlo algorithm, which has played a crucial role in our simulated and real data applications.

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arXiv2019

Targeted Random Projection for Prediction from High-Dimensional Features

We consider the problem of computationally-efficient prediction with high dimensional and highly correlated predictors when accurate variable selection is effectively impossible. Direct application of penalization or Bayesian methods implemented with Markov chain Monte Carlo can be computationally daunting and unstable. A common solution is first stage dimension reduction through screening or projecting the design matrix to a lower dimensional hyper-plane. Screening is highly sensitive to threshold choice, while projections often have poor performance in very high-dimensions. We propose TArgeted Random Projection (TARP) to combine positive aspects of both strategies. TARP uses screening to order the inclusion probabilities of the features in the projection matrix used for dimension reduction, leading to data-informed sparsity. We provide theoretical support for a Bayesian predictive algorithm based on TARP, including statistical and computational complexity guarantees. Examples for simulated and real data applications illustrate gains relative to a variety of competitors.

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arXiv2019

Estimating densities with nonlinear support using Fisher-Gaussian kernels

Current tools for multivariate density estimation struggle when the density is concentrated near a nonlinear subspace or manifold. Most approaches require choice of a kernel, with the multivariate Gaussian by far the most commonly used. Although heavy-tailed and skewed extensions have been proposed, such kernels cannot capture curvature in the support of the data. This leads to poor performance unless the sample size is very large relative to the dimension of the data. This article proposes a novel generalization of the Gaussian distribution, which includes an additional curvature parameter. We refer to the proposed class as Fisher-Gaussian (FG) kernels, since they arise by sampling from a von Mises-Fisher density on the sphere and adding Gaussian noise. The FG density has an analytic form, and is amenable to straightforward implementation within Bayesian mixture models using Markov chain Monte Carlo. We provide theory on large support, and illustrate gains relative to competitors in simulated and real data applications.

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arXiv2017

Targeted Random Projection for Prediction from High-Dimensional Features

We consider the problem of computationally-efficient prediction from high dimensional and highly correlated predictors in challenging settings where accurate variable selection is effectively impossible. Direct application of penalization or Bayesian methods implemented with Markov chain Monte Carlo can be computationally daunting and unstable. Hence, some type of dimensionality reduction prior to statistical analysis is in order. Common solutions include application of screening algorithms to reduce the regressors, or dimension reduction using projections of the design matrix. The former approach can be highly sensitive to threshold choice in finite samples, while the later can have poor performance in very high-dimensional settings. We propose a TArgeted Random Projection (TARP) approach that combines positive aspects of both strategies to boost performance. In particular, we propose to use information from independent screening to order the inclusion probabilities of the features in the projection matrix used for dimension reduction, leading to data-informed sparsity. We provide theoretical support for a Bayesian predictive algorithm based on TARP, including both statistical and computational complexity guarantees. Examples for simulated and real data applications illustrate gains relative to a variety of competitors.

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arXiv2015

A Mixture of g-priors for Variable Selection when the Number of Regressors Grows with the Sample Size

We consider variable selection problem in linear regression using mixture of $g$-priors. A number of mixtures are proposed in the literature which work well, especially when the number of regressors $p$ is fixed. In this paper, we propose a mixture of $g$-priors suitable for the case when $p$ grows with the sample size $n$. We study the performance of the method based on the proposed prior when $p=O(n^b),~0<b<1$. Along with model selection consistency, we also investigate the performance of the proposed prior when the true model does not belong to the model space considered. We find conditions under which the proposed prior is consistent in appropriate sense when normal linear models are considered. Further, we consider the case with non-normal errors in the regression model and study the performance of the model selection procedure. We also compare the performance of the proposed prior with that of several other mixtures available in the literature, both theoretically and using simulated data sets.

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