2007/07/13 by Akram Aldroubi, Aldroubi, Akram, Carlos Cabrelli +3
Computer Science · Engineering · Mathematics · #41A65 #42C15 #68P30 #94A20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Image and Signal Denoising Methods #Sparse and Compressive Sensing Techniques #Structural Health Monitoring Techniques #math.CA #msc:41A65 #msc:42C15 #msc:68P30 #msc:94A20
paper · pdf · doi:10.48550/arxiv.0707.2008
20 pages, 2 figures. Final version of the paper. To appear in Journal of Fourier Analysis and Applications
openalex publication_date 2007/07/13 · arxiv created 2008/02/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 RN and to infinite dimensional shift-invariant spaces in L2(Rd). 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.