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On sparse reconstruction from Fourier and Gaussian measurements
This paper improves upon best‐known guarantees for exact reconstruction of a sparse signal f from a small universal sample of Fourier measurements by showing that there exists a set of frequencies Ω such that one can exactly reconstruct every r‐sparse signal f of length n from its frequencies in Ω, using the convex relaxation.
Hanson-Wright inequality and sub-gaussian concentration
In this expository note, we give a modern proof of Hanson-Wright inequality for quadratic forms in sub-gaussian random variables.We deduce a useful concentration inequality for sub-gaussian random
Random Vectors in the Isotropic Position
Abstract Letybe a random vector in R n, satisfying E y⊗y=id. LetMbe a natural number and lety1, …, yMbe independent copies ofy. We study the question of approximation of the identity operator by
Smallest singular value of a random rectangular matrix
We prove an optimal estimate of the smallest singular value of a random sub‐Gaussian matrix, valid for all dimensions. For an N × n matrix A with independent and identically distributed sub‐Gaussian
Reconstruction From Anisotropic Random Measurements
A reduction principle is proved showing that the RE condition can be guaranteed by checking the restricted isometry on a certain family of low-dimensional subspaces, and this principle allows us to establish theRE condition for several broad classes of random matrices with dependent entries.
Non-asymptotic theory of random matrices: extreme singular values
This survey addresses the non-asymptotic theory of extreme singular values of random matrices with independent entries and focuses on recently developed geometric methods for estimating the hard edge ofrandom matrices (the smallest singular value).
Sampling from large matrices: An approach through geometric functional analysis
The result for the cut-norm yields a slight improvement on the best-known sample complexity for an approximation algorithm for MAX-2CSP problems, and the law of large numbers for operator-valued random variables for Banach spaces is used.
Invertibility of random matrices: norm of the inverse
Let A be an n × n matrix, whose entries are independent copies of a centered random variable satisfying the subgaussian tail estimate. We prove that the operator norm of A-1 does not exceed Cn3 / 2
Sparse reconstruction by convex relaxation: Fourier and Gaussian measurements
The first guarantees for universal measurements (i.e. which work for all sparse functions) with reasonable constants are proved, based on the technique of geometric functional analysis and probability in Banach spaces.