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August 2016 Geometric inference for general high-dimensional linear inverse problems
T. Tony Cai, Tengyuan Liang, Alexander Rakhlin
Ann. Statist. 44(4): 1536-1563 (August 2016). DOI: 10.1214/15-AOS1426


This paper presents a unified geometric framework for the statistical analysis of a general ill-posed linear inverse model which includes as special cases noisy compressed sensing, sign vector recovery, trace regression, orthogonal matrix estimation and noisy matrix completion. We propose computationally feasible convex programs for statistical inference including estimation, confidence intervals and hypothesis testing. A theoretical framework is developed to characterize the local estimation rate of convergence and to provide statistical inference guarantees. Our results are built based on the local conic geometry and duality. The difficulty of statistical inference is captured by the geometric characterization of the local tangent cone through the Gaussian width and Sudakov estimate.


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T. Tony Cai. Tengyuan Liang. Alexander Rakhlin. "Geometric inference for general high-dimensional linear inverse problems." Ann. Statist. 44 (4) 1536 - 1563, August 2016.


Received: 1 June 2015; Revised: 1 December 2015; Published: August 2016
First available in Project Euclid: 7 July 2016

zbMATH: 1357.62235
MathSciNet: MR3519932
Digital Object Identifier: 10.1214/15-AOS1426

Primary: 62J05 , 94B75

Keywords: conic geometry , convex relaxation , geometric functional analysis , High-dimensional statistics , linear inverse problems , statistical inference

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 4 • August 2016
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