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A classical problem that arises in numerous signal processing applications asks for the reconstruction of an unknown, $k$-sparse signal $x_0\in R^n$ from underdetermined, noisy, linear measurements $y=Ax_0+z\in R^m$. One standard approach…
This article considers recovery of signals that are sparse or approximately sparse in terms of a (possibly) highly overcomplete and coherent tight frame from undersampled data corrupted with additive noise. We show that the properly…
The problem of structured matrix estimation has been studied mostly under strong noise dependence assumptions. This paper considers a general framework of noisy low-rank-plus-sparse matrix recovery, where the noise matrix may come from any…
In this paper we study how the choice of loss functions of non-convex optimization problems affects their robustness and optimization landscape, through the study of noisy matrix sensing. In traditional regression tasks, mean squared error…
Compressive Sensing (CS) exploits the surprising fact that the information contained in a sparse signal can be preserved in a small number of compressive, often random linear measurements of that signal. Strong theoretical guarantees have…
Traditional deep neural nets (NNs) have shown the state-of-the-art performance in the task of classification in various applications. However, NNs have not considered any types of uncertainty associated with the class probabilities to…
In compressed sensing problems, $\ell_1$ minimization or Basis Pursuit was known to have the best provable phase transition performance of recoverable sparsity among polynomial-time algorithms. It is of great theoretical and practical…
$\ell_1$ minimization is often used for finding the sparse solutions of an under-determined linear system. In this paper we focus on finding sharp performance bounds on recovering approximately sparse signals using $\ell_1$ minimization,…
We study the problem of signal estimation from non-linear observations when the signal belongs to a low-dimensional set buried in a high-dimensional space. A rough heuristic often used in practice postulates that non-linear observations may…
Penalized estimation methods for diffusion processes and dependent data have recently gained significant attention due to their effectiveness in handling high-dimensional stochastic systems. In this work, we introduce an adaptive…
In this paper we present a linear programming solution for sign pattern recovery of a sparse signal from noisy random projections of the signal. We consider two types of noise models, input noise, where noise enters before the random…
Performance of regularized least-squares estimation in noisy compressed sensing is analyzed in the limit when the dimensions of the measurement matrix grow large. The sensing matrix is considered to be from a class of random ensembles that…
This paper investigates the state estimation problem for a class of complex networks, in which the dynamics of each node is subject to Gaussian noise, system uncertainties and nonlinearities. Based on a regularized least-squares approach,…
We analyze the asymptotic performance of sparse signal recovery from noisy measurements. In particular, we generalize some of the existing results for the Gaussian case to subgaussian and other ensembles. An achievable result is presented…
Accurate uncertainty quantification is a critical challenge in machine learning. While neural networks are highly versatile and capable of learning complex patterns, they often lack interpretability due to their ``black box'' nature. On the…
We consider the following signal recovery problem: given a measurement matrix $\Phi\in \mathbb{R}^{n\times p}$ and a noisy observation vector $c\in \mathbb{R}^{n}$ constructed from $c = \Phi\theta^* + \epsilon$ where $\epsilon\in…
The estimation of a precision matrix is a crucial problem in various research fields, particularly when working with high dimensional data. In such settings, the most common approach is to use the penalized maximum likelihood. The…
We study the problem of estimating low-rank matrices from linear measurements (a.k.a., matrix sensing) through nonconvex optimization. We propose an efficient stochastic variance reduced gradient descent algorithm to solve a nonconvex…
Deep neural networks (NNs) are powerful black box predictors that have recently achieved impressive performance on a wide spectrum of tasks. Quantifying predictive uncertainty in NNs is a challenging and yet unsolved problem. Bayesian NNs,…
In this paper, we consider a compressed sensing problem of reconstructing a sparse signal from an undersampled set of noisy linear measurements. The regularized least squares or least absolute shrinkage and selection operator (LASSO)…