Optimal Non-Linear Models for Sparsity and Sampling

Given a set of vectors (the data) in a Hilbert space ℋ, we prove the existence of an optimal collection of subspaces minimizing the sum of the square of the distances between each vector and its closest subspace in the collection. This collection of subspaces gives the best sparse representation for...

Full description

Saved in:
Bibliographic Details
Published inThe Journal of fourier analysis and applications Vol. 14; no. 5-6; pp. 793 - 812
Main Authors Aldroubi, Akram, Cabrelli, Carlos, Molter, Ursula
Format Journal Article
LanguageEnglish
Published Boston SP Birkhäuser Verlag Boston 01.12.2008
Subjects
Abstract Harmonic Analysis
Approximations and Expansions
Fourier Analysis
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Partial Differential Equations
Signal,Image and Speech Processing
Frames
Sparsity
41A65
68P30
Sampling
42C15
Compressed sensing
94A20
Online AccessGet full text

Cover

More Information
Summary:Given a set of vectors (the data) in a Hilbert space ℋ, we prove the existence of an optimal collection of subspaces minimizing the sum of the square of the distances between each vector and its closest subspace in the collection. This collection of subspaces gives the best sparse representation for the given data, in a sense defined in the paper, and provides an optimal model for sampling in union of subspaces. The results are proved in a general setting and then applied to the case of low dimensional subspaces of ℝ N and to infinite dimensional shift-invariant spaces in L 2 (ℝ d ). We also present an iterative search algorithm for finding the solution subspaces. These results are tightly connected to the new emergent theories of compressed sensing and dictionary design, signal models for signals with finite rate of innovation, and the subspace segmentation problem.
ISSN:1069-5869
1531-5851
DOI:10.1007/s00041-008-9040-2