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We study the fundamental problems of Gaussian mean estimation and linear regression with Gaussian covariates in the presence of Huber contamination. Our main contribution is the design of the first sample near-optimal and almost linear-time…
We study the problem of corrupted sensing, a generalization of compressed sensing in which one aims to recover a signal from a collection of corrupted or unreliable measurements. While an arbitrary signal cannot be recovered in the face of…
This work considers distributed sensing and transmission of sporadic random samples. Lower bounds are derived for the reconstruction error of a single normally or uniformly-distributed finite-dimensional vector imperfectly measured by a…
This paper determines to within a single measurement the minimum number of measurements required to successfully reconstruct a signal drawn from a Gaussian mixture model in the low-noise regime. The method is to develop upper and lower…
We address the problem of how to achieve optimal inference in distributed quantile regression without stringent scaling conditions. This is challenging due to the non-smooth nature of the quantile regression (QR) loss function, which…
We examine the problem of selecting a small set of linear measurements for reconstructing high-dimensional signals. Well-established methods for optimizing such measurements include principal component analysis (PCA), independent component…
Algorithmic robust statistics has traditionally focused on the contamination model where a small fraction of the samples are arbitrarily corrupted. We consider a recent contamination model that combines two kinds of corruptions: (i) small…
In this paper we study the quantization stage that is implicit in any compressed sensing signal acquisition paradigm. We propose using Sigma-Delta quantization and a subsequent reconstruction scheme based on convex optimization. We prove…
Recovery of the sparsity pattern (or support) of an unknown sparse vector from a limited number of noisy linear measurements is an important problem in compressed sensing. In the high-dimensional setting, it is known that recovery with a…
We consider the problem of sparse normal means estimation in a distributed setting with communication constraints. We assume there are $M$ machines, each holding $d$-dimensional observations of a $K$-sparse vector $\mu$ corrupted by…
A new framework of compressive sensing (CS), namely statistical compressive sensing (SCS), that aims at efficiently sampling a collection of signals that follow a statistical distribution and achieving accurate reconstruction on average, is…
We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and…
In the setting of entangled single-sample distributions, the goal is to estimate some common parameter shared by a family of $n$ distributions, given one single sample from each distribution. This paper studies mean estimation for entangled…
Sparse linear regression with ill-conditioned Gaussian random designs is widely believed to exhibit a statistical/computational gap, but there is surprisingly little formal evidence for this belief, even in the form of examples that are…
Compressive sensing is a technique to sample signals well below the Nyquist rate using linear measurement operators. In this paper we present an algorithm for signal reconstruction given such a set of measurements. This algorithm…
We describe a general technique that yields the first {\em Statistical Query lower bounds} for a range of fundamental high-dimensional learning problems involving Gaussian distributions. Our main results are for the problems of (1) learning…
Consider the problem of reconstructing a multidimensional signal from an underdetermined set of measurements, as in the setting of compressed sensing. Without any additional assumptions, this problem is ill-posed. However, for signals such…
Recent advances in quantized compressed sensing and high-dimensional estimation have shown that signal recovery is even feasible under strong non-linear distortions in the observation process. An important characteristic of associated…
In this paper we derive information theoretic performance bounds to sensing and reconstruction of sparse phenomena from noisy projections. We consider two settings: output noise models where the noise enters after the projection and input…
We study the problem of approximately recovering signals on a manifold from one-bit linear measurements drawn from either a Gaussian ensemble, partial circulant ensemble, or bounded orthonormal ensemble and quantized using Sigma-Delta or…