Related papers: An efficiency upper bound for inverse covariance e…
Multivariate Gaussian is often used as a first approximation to the distribution of high-dimensional data. Determining the parameters of this distribution under various constraints is a widely studied problem in statistics, and is often…
This paper establishes optimal convergence rates for estimation of structured covariance operators of Gaussian processes. We study banded operators with kernels that decay rapidly off-the-diagonal and $L^q$-sparse operators with an…
We study the fundamental problem of estimating the mean of a $d$-dimensional distribution with covariance $\Sigma \preccurlyeq \sigma^2 I_d$ given $n$ samples. When $d = 1$, \cite{catoni} showed an estimator with error $(1+o(1)) \cdot…
For a tall $n\times d$ matrix $A$ and a random $m\times n$ sketching matrix $S$, the sketched estimate of the inverse covariance matrix $(A^\top A)^{-1}$ is typically biased: $E[(\tilde A^\top\tilde A)^{-1}]\ne(A^\top A)^{-1}$, where…
Estimation of a high dimensional precision matrix is a critical problem to many areas of statistics including Gaussian graphical models and inference on high dimensional data. Working under the structural assumption of sparsity, we propose…
The density ratio is an important metric for evaluating the relative likelihood of two probability distributions, with extensive applications in statistics and machine learning. However, existing estimation theories for density ratios often…
When inferring parameters from a Gaussian-distributed data set by computing a likelihood, a covariance matrix is needed that describes the data errors and their correlations. If the covariance matrix is not known a priori, it may be…
We consider computationally-efficient estimation of population parameters when observations are subject to missing data. In particular, we consider estimation under the realizable contamination model of missing data in which an $\epsilon$…
In Gaussian graphical models, the likelihood equations must typically be solved iteratively. We investigate two algorithms: A version of iterative proportional scaling which avoids inversion of large matrices, and an algorithm based on…
Diffusion models can generate a variety of high-quality images by modeling complex data distributions. Trained diffusion models can also be very effective image priors for solving inverse problems. Most of the existing diffusion-based…
Let $X$ be a symmetric, isotropic random vector in $\mathbb{R}^m$ and let $X_1...,X_n$ be independent copies of $X$. We show that under mild assumptions on $\|X\|_2$ (a suitable thin-shell bound) and on the tail-decay of the marginals…
We propose a nonconvex estimator for joint multivariate regression and precision matrix estimation in the high dimensional regime, under sparsity constraints. A gradient descent algorithm with hard thresholding is developed to solve the…
We consider a general class of regression models with normally distributed covariates, and the associated nonconvex problem of fitting these models from data. We develop a general recipe for analyzing the convergence of iterative algorithms…
We study the optimal sample complexity of variable selection in linear regression under general design covariance, and show that subset selection is optimal while under standard complexity assumptions, efficient algorithms for this problem…
Covariance estimation becomes challenging in the regime where the number p of variables outstrips the number n of samples available to construct the estimate. One way to circumvent this problem is to assume that the covariance matrix is…
Gaussian time-series models are often specified through their spectral density. Such models present several computational challenges, in particular because of the non-sparse nature of the covariance matrix. We derive a fast approximation of…
We study the problem of high-dimensional covariance estimation under the constraint that the partial correlations are nonnegative. The sign constraints dramatically simplify estimation: the Gaussian maximum likelihood estimator is well…
We provide optimal lower bounds for two well-known parameter estimation (also known as statistical estimation) tasks in high dimensions with approximate differential privacy. First, we prove that for any $\alpha \le O(1)$, estimating the…
High dimensional covariance estimation and graphical models is a contemporary topic in statistics and machine learning having widespread applications. An important line of research in this regard is to shrink the extreme spectrum of the…
The goal of this paper is to show that a single robust estimator of the mean of a multivariate Gaussian distribution can enjoy five desirable properties. First, it is computationally tractable in the sense that it can be computed in a time…