Related papers: Upper Bounds on the Error of Sparse Vector and Low…
We consider high dimensional sparse regression, and develop strategies able to deal with arbitrary -- possibly, severe or coordinated -- errors in the covariance matrix $X$. These may come from corrupted data, persistent experimental…
Low-rank tensor approximation error bounds are proposed for the case of noisy input data that depend on low-rank representation type, rank and the dimensionality of the tensor. The bounds show that high-dimensional low-rank structured…
The recovery of the underlying low-rank structure of clean data corrupted with sparse noise/outliers is attracting increasing interest. However, in many low-level vision problems, the exact target rank of the underlying structure and the…
We study the problem of identifying the parameters of a linear system from its response to multiple unknown waveforms. We assume that the system response is a scaled superposition of time-delayed and frequency-shifted versions of the…
We study sparse recovery when observations come from mixed-quality sources: a small collection of high-quality measurements with small noise variance and a larger collection of lower-quality measurements with higher variance. For this…
We investigate the sparse recovery problem of reconstructing a high-dimensional non-negative sparse vector from lower dimensional linear measurements. While much work has focused on dense measurement matrices, sparse measurement schemes are…
We address the problem of recovering a sparse $n$-vector within a given subspace. This problem is a subtask of some approaches to dictionary learning and sparse principal component analysis. Hence, if we can prove scaling laws for recovery…
We study the problem of sparse tensor principal component analysis: given a tensor $\pmb Y = \pmb W + \lambda x^{\otimes p}$ with $\pmb W \in \otimes^p\mathbb{R}^n$ having i.i.d. Gaussian entries, the goal is to recover the $k$-sparse unit…
Designing computational experiments involving $\ell_1$ minimization with linear constraints in a finite-dimensional, real-valued space for receiving a sparse solution with a precise number $k$ of nonzero entries is, in general, difficult.…
Lower dimensional signal representation schemes frequently assume that the signal of interest lies in a single vector space. In the context of the recently developed theory of compressive sensing (CS), it is often assumed that the signal of…
We consider differentially private approximate singular vector computation. Known worst-case lower bounds show that the error of any differentially private algorithm must scale polynomially with the dimension of the singular vector. We are…
We explore algorithms and limitations for sparse optimization problems such as sparse linear regression and robust linear regression. The goal of the sparse linear regression problem is to identify a small number of key features, while the…
Learning optimal dictionaries for sparse coding has exposed characteristic sparse features of many natural signals. However, universal guarantees of the stability of such features in the presence of noise are lacking. Here, we provide very…
Originally developed for imputing missing entries in low rank, or approximately low rank matrices, matrix completion has proven widely effective in many problems where there is no reason to assume low-dimensional linear structure in the…
We consider the densest submatrix problem, which seeks the submatrix of fixed size of a given binary matrix that contains the most nonzero entries. This problem is a natural generalization of fundamental problems in combinatorial…
This paper considers a large class of problems where we seek to recover a low rank matrix and/or sparse vector from some set of measurements. While methods based on convex relaxations suffer from a (possibly large) estimator bias, and other…
This paper considers compressed sensing and affine rank minimization in both noiseless and noisy cases and establishes sharp restricted isometry conditions for sparse signal and low-rank matrix recovery. The analysis relies on a key…
We study the information-theoretic limits of exactly recovering the support of a sparse signal using noisy projections defined by various classes of measurement matrices. Our analysis is high-dimensional in nature, in which the number of…
We consider the problem of identifying the sparse principal component of a rank-deficient matrix. We introduce auxiliary spherical variables and prove that there exists a set of candidate index-sets (that is, sets of indices to the nonzero…
The problem of identifying sparse solutions for the link structure and dynamics of an unknown linear, time-invariant network is posed as finding sparse solutions x to Ax=b. If the sensing matrix A satisfies a rank condition, this problem…