The Annals of Statistics
- Ann. Statist.
- Volume 44, Number 4 (2016), 1536-1563.
Geometric inference for general high-dimensional linear inverse problems
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.
Ann. Statist., Volume 44, Number 4 (2016), 1536-1563.
Received: June 2015
Revised: December 2015
First available in Project Euclid: 7 July 2016
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Cai, T. Tony; Liang, Tengyuan; Rakhlin, Alexander. Geometric inference for general high-dimensional linear inverse problems. Ann. Statist. 44 (2016), no. 4, 1536--1563. doi:10.1214/15-AOS1426. https://projecteuclid.org/euclid.aos/1467894707
- “Geometric inference for general high-dimensional linear inverse problems”. Due to space constraints, we have relegated remaining proofs to the Supplement , where details of proof for Lemmas 2–4, Theorem 6 and Corollary 1–5 are included.