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Vladislav Voroninski

Publications and source records attributed to Vladislav Voroninski.

22 records · Page 2Linked to original sources
arXiv2014

A strong restricted isometry property, with an application to phaseless compressed sensing

The many variants of the restricted isometry property (RIP) have proven to be crucial theoretical tools in the fields of compressed sensing and matrix completion. The study of extending compressed sensing to accommodate phaseless measurements naturally motivates a strong notion of restricted isometry property (SRIP), which we develop in this paper. We show that if $A \in \mathbb{R}^{m\times n}$ satisfies SRIP and phaseless measurements $|Ax_0| = b$ are observed about a $k$-sparse signal $x_0 \in \mathbb{R}^n$, then minimizing the $\ell_1$ norm subject to $ |Ax| = b $ recovers $x_0$ up to multiplication by a global sign. Moreover, we establish that the SRIP holds for the random Gaussian matrices typically used for standard compressed sensing, implying that phaseless compressed sensing is possible from $O(k \log (n/k))$ measurements with these matrices via $\ell_1$ minimization over $|Ax| = b$. Our analysis also yields an erasure robust version of the Johnson-Lindenstrauss Lemma.

cs.IT↗
arXiv2012

Sparse Signal Recovery from Quadratic Measurements via Convex Programming

In this paper we consider a system of quadratic equations | |^2 = b_j, j = 1, ..., m, where x in R^n is unknown while normal random vectors z_j in R_n and quadratic measurements b_j in R are known. The system is assumed to be underdetermined, i.e., m < n. We prove that if there exists a sparse solution x, i.e., at most k components of x are non-zero, then by solving a convex optimization program, we can solve for x up to a multiplicative constant with high probability, provided that k <= O((m/log n)^(1/2)). On the other hand, we prove that k <= O(log n (m)^(1/2)) is necessary for a class of naive convex relaxations to be exact.

cs.IT↗
arXiv2011

PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming

Suppose we wish to recover a signal x in C^n from m intensity measurements of the form | |^2, i = 1, 2,..., m; that is, from data in which phase information is missing. We prove that if the vectors z_i are sampled independently and uniformly at random on the unit sphere, then the signal x can be recovered exactly (up to a global phase factor) by solving a convenient semidefinite program---a trace-norm minimization problem; this holds with large probability provided that m is on the order of n log n, and without any assumption about the signal whatsoever. This novel result demonstrates that in some instances, the combinatorial phase retrieval problem can be solved by convex programming techniques. Finally, we also prove that our methodology is robust vis a vis additive noise.

cs.IT↗