Exact and stable recovery of sequences of signals with sparse increments via differential ℓ [subscript 1]-minimization
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Author(s)
Ba, Demba E.
Babadi, Behtash
Purdon, Patrick
Brown, Emery N.
Alternative Title
Exact and stable recovery of sequences of signals with sparse increments via differential ℓ1-minimization
Date Issued
December 2012
Journal
Advances in Neural Information Processing Systems 25 (NIPS 2012)
Publisher
Neural Information Processing Systems Foundation, Inc.
Citation
Ba, Demba, Behtash Babadi, Patrick Purdon, Emery Brown. "Exact and stable recovery of sequences of signals with sparse increments via differential ℓ1-minimization." Advances in Neural Information Processing Systems 25 (NIPS 2012), December 3-8, 2012, Lake Tahoe, Nevada, United States.
Version
Author's final manuscript
Abstract
We consider the problem of recovering a sequence of vectors, (Xk)[K over k=0], for which the increments X[subscript k] - X[subscript k-1] are S[subscript k]-sparse (with S[subscript k] typically smaller than S[subscript 1]), based on linear measurements (Y[subscript k] = A[subscript k]X[subscript k] + e[subscript k)[superscript K over k=1, where A[subscript k] and e[subscript k] denote the measurement matrix and noise, respectively. Assuming each A[subscript k] obeys the restricted isometry property (RIP) of a certain order--depending only on S[subscript k]--we show that in the absence of noise a convex program, which minimizes the weighted sum of the ℓ [subscript 1]-norm of successive differences subject to the linear measurement constraints, recovers the sequence (Xk)[K over k=1] exactly. This is an interesting result because this convex program is equivalent to a standard compressive sensing problem with a highly-structured aggregate measurement matrix which does not satisfy the RIP requirements in the standard sense, and yet we can achieve exact recovery. In the presence of bounded noise, we propose a quadratically-constrained convex program for recovery and derive bounds on the reconstruction error of the sequence. We supplement our theoretical analysis with simulations and an application to real video data. These further support the validity of the proposed approach for acquisition and recovery of signals with time-varying sparsity.
MIT Department
Massachusetts Institute of Technology. Department of Brain and Cognitive Sciences
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DOI of Published Version
http://papers.nips.cc/paper/4626-exact-and-stable-recovery-of-sequences-of-signals-with-sparse-increments-via-differential-_1-minimization
