arXiv subjects

Swarnadip Ghosh

Publications and source records attributed to Swarnadip Ghosh.

8 recordsLinked to original sources

arXiv2024

Precise Asymptotics for Linear Mixed Models with Crossed Random Effects

We obtain an asymptotic normality result that reveals the precise asymptotic behavior of the maximum likelihood estimators of parameters for a very general class of linear mixed models containing cross random effects. In achieving the result, we overcome theoretical difficulties that arise from random effects being crossed as opposed to the simpler nested random effects case. Our new theory is for a class of Gaussian response linear mixed models which includes crossed random slopes that partner arbitrary multivariate predictor effects and does not require the cell counts to be balanced. Statistical utilities include confidence interval construction, Wald hypothesis test and sample size calculations.

math.ST↗

arXiv2023

Consistent and Scalable Composite Likelihood Estimation of Probit Models with Crossed Random Effects

Estimation of crossed random effects models commonly requires computational costs that grow faster than linearly in the sample size $N$, often as fast as $\Omega(N^{3/2})$, making them unsuitable for large data sets. For non-Gaussian responses, integrating out the random effects to get a marginal likelihood brings significant challenges, especially for high dimensional integrals where the Laplace approximation might not be accurate. We develop a composite likelihood approach to probit models that replaces the crossed random effects model with some hierarchical models that require only one-dimensional integrals. We show how to consistently estimate the crossed effects model parameters from the hierarchical model fits. We find that the computation scales linearly in the sample size. We illustrate the method on about five million observations from Stitch Fix where the crossed effects formulation would require an integral of dimension larger than $700{,}000$.

stat.ME↗

arXiv2023

Scalable solution to crossed random effects model with random slopes

The crossed random effects model is widely used, finding applications in various fields such as longitudinal studies, e-commerce, and recommender systems, among others. However, these models encounter scalability challenges, as the computational time for standard algorithms grows superlinearly with the number N of observations in the data set, commonly $\Omega(N^{3/2})$ or worse. Recent published works present scalable methods for crossed random effects in linear models and some generalized linear models, but those methods only allow for random intercepts. In this paper, we devise scalable algorithms for models that include random slopes. This addition brings substantial difficulty in estimating the random-effect covariance matrices in a scalable way. We address this issue by using a variational EM algorithm. Our proposed approach accommodates both diagonal covariance matrices and cases where no structure is assumed-a scenario common in fields such as psychology and neuroscience. In simulations, the proposed method is substantially faster than standard methods for large $N$. It is also more efficient than ordinary least squares which has a problem of greatly underestimating the sampling uncertainty in parameter estimates. We illustrate the new method on a MovieLens dataset, as well as a large data set (five million observations) from the online retailer Stitch Fix.

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arXiv2022

Efficient Estimation in Tensor Ising Models

The tensor Ising model is a discrete exponential family used for modeling binary data on networks with not just pairwise, but higher-order dependencies. A particularly important class of tensor Ising models are the tensor Curie-Weiss models, where all tuples of nodes of a particular order interact with the same intensity. The maximum likelihood estimator (MLE) is not explicit in this model, due to the presence of an intractable normalizing constant in the likelihood, and a computationally efficient alternative is to use the maximum pseudolikelihood estimator (MPLE). In this paper, we show that the MPLE is in fact as efficient as the MLE (in the Bahadur sense) in the $2$-spin model, and for all values of the null parameter above $\log 2$ in higher-order tensor models. Even if the null parameter happens to lie within the very small window between the threshold and $\log 2$, they are equally efficient unless the alternative parameter is large. Therefore, not only is the MPLE computationally preferable to the MLE, but also theoretically as efficient as the MLE over most of the parameter space. Our results extend to the more general class of Erdős-Rényi hypergraph Ising models, under slight sparsities too.

math.ST↗

arXiv2021

Scalable logistic regression with crossed random effects

The cost of both generalized least squares (GLS) and Gibbs sampling in a crossed random effects model can easily grow faster than $N^{3/2}$ for $N$ observations. Ghosh et al. (2020) develop a backfitting algorithm that reduces the cost to $O(N)$. Here we extend that method to a generalized linear mixed model for logistic regression. We use backfitting within an iteratively reweighted penalized least square algorithm. The specific approach is a version of penalized quasi-likelihood due to Schall (1991). A straightforward version of Schall's algorithm would also cost more than $N^{3/2}$ because it requires the trace of the inverse of a large matrix. We approximate that quantity at cost $O(N)$ and prove that this substitution makes an asymptotically negligible difference. Our backfitting algorithm also collapses the fixed effect with one random effect at a time in a way that is analogous to the collapsed Gibbs sampler of Papaspiliopoulos et al. (2020). We use a symmetric operator that facilitates efficient covariance computation. We illustrate our method on a real dataset from Stitch Fix. By properly accounting for crossed random effects we show that a naive logistic regression could underestimate sampling variances by several hundred fold.

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arXiv2021

Convergence rate of a collapsed Gibbs sampler for crossed random effects models

In this paper, we analyze the convergence rate of a collapsed Gibbs sampler for crossed random effects models. Our results apply to a substantially larger range of models than previous works, including models that incorporate missingness mechanism and unbalanced level data. The theoretical tools involved in our analysis include a connection between relaxation time and autoregression matrix, concentration inequalities, and random matrix theory.

stat.CO↗

arXiv2021

Feature Selection in High-dimensional Spaces Using Graph-Based Methods

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 from one of $K$ different distributions. Our algorithm can be applied in a completely nonparametric setup without any distributional assumptions on the data, and it aims at outputting those features in the data, that contribute the most to the overall distributional variation. At the heart of our method is the recursive application of distribution-free graph-based tests on subsets of the feature set, located at different depths of a hierarchical clustering tree constructed from the data. Our algorithm recovers all truly contributing features with high probability, while ensuring optimal control on false-discovery. We show the superior performance of our method over other existing ones through synthetic data, and demonstrate the utility of this method on several real-life datasets from the domains of climate change and biology, wherein our algorithm is not only able to detect known features expected to be associated with the underlying process, but also discovers novel targets that can be subsequently studied.

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arXiv2021

Backfitting for large scale crossed random effects regressions

Regression models with crossed random effect errors can be very expensive to compute. The cost of both generalized least squares and Gibbs sampling can easily grow as $N^{3/2}$ (or worse) for $N$ observations. Papaspiliopoulos et al. (2020) present a collapsed Gibbs sampler that costs $O(N)$, but under an extremely stringent sampling model. We propose a backfitting algorithm to compute a generalized least squares estimate and prove that it costs $O(N)$. A critical part of the proof is in ensuring that the number of iterations required is $O(1)$ which follows from keeping a certain matrix norm below $1-δ$ for some $δ>0$. Our conditions are greatly relaxed compared to those for the collapsed Gibbs sampler, though still strict. Empirically, the backfitting algorithm has a norm below $1-δ$ under conditions that are less strict than those in our assumptions. We illustrate the new algorithm on a ratings data set from Stitch Fix.

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