The use of mutual coherence to prove l1/l0-equivalence in classification problems (with C. Weaver), submitted for publication, 2019.

Abstract

We consider the decomposition of a signal over an overcomplete set of vectors. Minimization of the $\ell^1$-norm of the coefficient vector can often retrieve the sparsest solution (so-called "l1/l0-equivalence"), a generally NP-hard task, and this fact has powered the field of compressed sensing. Wright et al.'s sparse representation-based classification (SRC) applies this relationship to machine learning, wherein the signal to be decomposed represents the test sample and columns of the dictionary are training samples. We investigate the relationships between l1-minimization, sparsity, and classification accuracy in SRC. After proving that the tractable, deterministic approach to verifying l1/l0-equivalence fundamentally conflicts with the high coherence between same-class training samples, we demonstrate that l1-minimization can still recover the sparsest solution when the classes are well-separated. Further, using a nonlinear transform so that sparse recovery conditions may be satisfied, we demonstrate that approximate (not strict) equivalence is key to the success of SRC.

Keywords: sparse representation, representation-based classification, mutual coherence, compressed sensing

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