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Universality in Learning from Linear Measurements

Abbasi, Ehsan and Salehi, Fariborz and Hassibi, Babak (2019) Universality in Learning from Linear Measurements. . (Unpublished) http://resolver.caltech.edu/CaltechAUTHORS:20190628-083750345

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Abstract

We study the problem of recovering a structured signal from independently and identically drawn linear measurements. A convex penalty function f(⋅) is considered which penalizes deviations from the desired structure, and signal recovery is performed by minimizing f(⋅) subject to the linear measurement constraints. The main question of interest is to determine the minimum number of measurements that is necessary and sufficient for the perfect recovery of the unknown signal with high probability. Our main result states that, under some mild conditions on f(⋅) and on the distribution from which the linear measurements are drawn, the minimum number of measurements required for perfect recovery depends only on the first and second order statistics of the measurement vectors. As a result, the required of number of measurements can be determining by studying measurement vectors that are Gaussian (and have the same mean vector and covariance matrix) for which a rich literature and comprehensive theory exists. As an application, we show that the minimum number of random quadratic measurements (also known as rank-one projections) required to recover a low rank positive semi-definite matrix is 3nr, where n is the dimension of the matrix and r is its rank. As a consequence, we settle the long standing open question of determining the minimum number of measurements required for perfect signal recovery in phase retrieval using the celebrated PhaseLift algorithm, and show it to be 3n.


Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/1906.08396arXivDiscussion Paper
Record Number:CaltechAUTHORS:20190628-083750345
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20190628-083750345
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:96808
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:28 Jun 2019 17:11
Last Modified:28 Jun 2019 17:11

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