Typical l1-recovery limit of sparse vectors represented by concatenations of random orthogonal matrices

Yoshiyuki Kabashima, Mikko Vehkaperä, Saikat Chatterjee

Research output: Contribution to journalArticleScientificpeer-review

8 Citations (Scopus)

Abstract

We consider the problem of recovering an N-dimensional sparse vector x from its linear transformation y=Dx of M (<N) dimensions. Minimization of the l1-norm of x under the constraint y=Dx is a standard approach for the recovery problem, and earlier studies report that the critical condition for typically successful l1-recovery is universal over a variety of randomly constructed matrices D. To examine the extent of the universality, we focus on the case in which D is provided by concatenating T=N/M matrices O 1,O2,...,OT drawn uniformly according to the Haar measure on the M×M orthogonal matrices. By using the replica method in conjunction with the development of an integral formula to handle the random orthogonal matrices, we show that the concatenated matrices can result in better recovery performance than that predicted by the universality when the density of non-zero signals is not uniform among the T matrix modules. The universal condition is reproduced for the special case of uniform non-zero signal densities. Extensive numerical experiments support the theoretical predictions.

Original languageEnglish
Article numberP12003
JournalJournal of Statistical Mechanics: Theory and Experiment
Volume2012
Issue number12
DOIs
Publication statusPublished - 1 Dec 2012
MoE publication typeA1 Journal article-refereed

Fingerprint

Orthogonal matrix
Concatenation
Random Matrices
Recovery
recovery
Universality
matrices
Replica Method
Haar Measure
L1-norm
Integral Formula
M-matrix
Linear transformation
Numerical Experiment
linear transformations
Module
Prediction
replicas
norms
modules

Keywords

  • error correcting codes
  • source and channel coding
  • statistical inference

Cite this

@article{7ad74978a049440fa53219ae7410462f,
title = "Typical l1-recovery limit of sparse vectors represented by concatenations of random orthogonal matrices",
abstract = "We consider the problem of recovering an N-dimensional sparse vector x from its linear transformation y=Dx of M (<N) dimensions. Minimization of the l1-norm of x under the constraint y=Dx is a standard approach for the recovery problem, and earlier studies report that the critical condition for typically successful l1-recovery is universal over a variety of randomly constructed matrices D. To examine the extent of the universality, we focus on the case in which D is provided by concatenating T=N/M matrices O 1,O2,...,OT drawn uniformly according to the Haar measure on the M×M orthogonal matrices. By using the replica method in conjunction with the development of an integral formula to handle the random orthogonal matrices, we show that the concatenated matrices can result in better recovery performance than that predicted by the universality when the density of non-zero signals is not uniform among the T matrix modules. The universal condition is reproduced for the special case of uniform non-zero signal densities. Extensive numerical experiments support the theoretical predictions.",
keywords = "error correcting codes, source and channel coding, statistical inference",
author = "Yoshiyuki Kabashima and Mikko Vehkaper{\"a} and Saikat Chatterjee",
year = "2012",
month = "12",
day = "1",
doi = "10.1088/1742-5468/2012/12/P12003",
language = "English",
volume = "2012",
journal = "Journal of Statistical Mechanics: Theory and Experiment",
issn = "1742-5468",
publisher = "Institute of Physics IOP",
number = "12",

}

Typical l1-recovery limit of sparse vectors represented by concatenations of random orthogonal matrices. / Kabashima, Yoshiyuki; Vehkaperä, Mikko; Chatterjee, Saikat.

In: Journal of Statistical Mechanics: Theory and Experiment, Vol. 2012, No. 12, P12003, 01.12.2012.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Typical l1-recovery limit of sparse vectors represented by concatenations of random orthogonal matrices

AU - Kabashima, Yoshiyuki

AU - Vehkaperä, Mikko

AU - Chatterjee, Saikat

PY - 2012/12/1

Y1 - 2012/12/1

N2 - We consider the problem of recovering an N-dimensional sparse vector x from its linear transformation y=Dx of M (<N) dimensions. Minimization of the l1-norm of x under the constraint y=Dx is a standard approach for the recovery problem, and earlier studies report that the critical condition for typically successful l1-recovery is universal over a variety of randomly constructed matrices D. To examine the extent of the universality, we focus on the case in which D is provided by concatenating T=N/M matrices O 1,O2,...,OT drawn uniformly according to the Haar measure on the M×M orthogonal matrices. By using the replica method in conjunction with the development of an integral formula to handle the random orthogonal matrices, we show that the concatenated matrices can result in better recovery performance than that predicted by the universality when the density of non-zero signals is not uniform among the T matrix modules. The universal condition is reproduced for the special case of uniform non-zero signal densities. Extensive numerical experiments support the theoretical predictions.

AB - We consider the problem of recovering an N-dimensional sparse vector x from its linear transformation y=Dx of M (<N) dimensions. Minimization of the l1-norm of x under the constraint y=Dx is a standard approach for the recovery problem, and earlier studies report that the critical condition for typically successful l1-recovery is universal over a variety of randomly constructed matrices D. To examine the extent of the universality, we focus on the case in which D is provided by concatenating T=N/M matrices O 1,O2,...,OT drawn uniformly according to the Haar measure on the M×M orthogonal matrices. By using the replica method in conjunction with the development of an integral formula to handle the random orthogonal matrices, we show that the concatenated matrices can result in better recovery performance than that predicted by the universality when the density of non-zero signals is not uniform among the T matrix modules. The universal condition is reproduced for the special case of uniform non-zero signal densities. Extensive numerical experiments support the theoretical predictions.

KW - error correcting codes

KW - source and channel coding

KW - statistical inference

UR - http://www.scopus.com/inward/record.url?scp=84871207413&partnerID=8YFLogxK

U2 - 10.1088/1742-5468/2012/12/P12003

DO - 10.1088/1742-5468/2012/12/P12003

M3 - Article

AN - SCOPUS:84871207413

VL - 2012

JO - Journal of Statistical Mechanics: Theory and Experiment

JF - Journal of Statistical Mechanics: Theory and Experiment

SN - 1742-5468

IS - 12

M1 - P12003

ER -