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Decision trees are a fundamental tool in machine learning for representing, classifying, and generalizing data. It is desirable to construct ``small'' decision trees, by minimizing either the \textit{size} ($s$) or the \textit{depth} $(d)$…

We investigate the extremal values of partial traces of matrix tensors under operator norm constraints. To evaluate these multi-linear quantities, we develop a comprehensive graphical formalism that encodes multi-leg partial traces, partial…

We study the minimax sample complexity of multicalibration in the batch setting. A learner observes $n$ i.i.d. samples from an unknown distribution and must output a (possibly randomized) predictor whose population multicalibration error,…

Stochastic Differential Equations (SDEs) are used as statistical models in many disciplines. However, intractable likelihood functions for SDEs make inference challenging, and we need to resort to simulation-based techniques to estimate and…

We explore how matrix bootstrap techniques can be used to constrain matrix and tensor models at finite $N$, where $N$ is the dimension of the matrix/tensor, taking a Gaussian model with a quartic interaction as example. For matrix models,…

In the context of undirected Gaussian graphical models, we introduce three estimators based on elastic net penalty for the underlying dependence graph. Our goal is to estimate the sparse precision matrix, from which to retrieve both the…

Consider an experiment involving a potentially small number of subjects. Some random variables are observed on each subject: a high-dimensional one called the "observed" random variable, and a one-dimensional one called the "outcome" random…

We consider the problem of estimating the number of distinct elements in a large data set (or, equivalently, the support size of the distribution induced by the data set) from a random sample of its elements. The problem occurs in many…

This work provides test error bounds for iterative fixed point methods on linear predictors -- specifically, stochastic and batch mirror descent (MD), and stochastic temporal difference learning (TD) -- with two core contributions: (a) a…

In statistical data assimilation one seeks the largest maximum of the conditional probability distribution $P(\mathbf{X},\mathbf{p}|\mathbf{Y})$ of model states, $\mathbf{X}$, and parameters,$\mathbf{p}$, conditioned on observations…

For real symmetric matrices that are accessible only through matrix vector products, we present Monte Carlo estimators for computing the diagonal elements. Our probabilistic bounds for normwise absolute and relative errors apply to Monte…

Large particle systems are often described by high-dimensional (linear) kinetic equations that are simulated using Monte Carlo methods for which the asymptotic convergence rate is independent of the dimensionality. Even though the…

We study some notions of negative dependence of a sampling scheme that can be used to derive variance bounds for the corresponding estimator or discrepancy bounds for the underlying random point set that are at least as good as the…

We address the inference problem concerning regression coefficients in a classical linear regression model using least squares estimates. The analysis is conducted under circumstances where network dependency exists across units in the…

In the last fifteen the subset sampling method has often been used in reliability problems as a tool for calculating small probabilities. This method is extrapolating from an initial Monte Carlo estimate for the probability content of a…

An algorithm is proposed that enables the imposition of shape constraints on regression curves, without requiring the constraints to be written as closed-form expressions, nor assuming the functional form of the loss function. This…

This paper considers the problem of matrix-variate logistic regression. It derives the fundamental error threshold on estimating low-rank coefficient matrices in the logistic regression problem by obtaining a lower bound on the minimax…

The framework of feedback graphs is a generalization of sequential decision-making with bandit or full information feedback. In this work, we study an extension where the directed feedback graph is stochastic, following a distribution…

There has been considerable recent interest in Bayesian modeling of high-dimensional networks via latent space approaches. When the number of nodes increases, estimation based on Markov Chain Monte Carlo can be extremely slow and show poor…

Bayesian networks are one of the most widely used classes of probabilistic models for risk management and decision support because of their interpretability and flexibility in including heterogeneous pieces of information. In any applied…