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We consider the decomposition of a signal over an overcomplete set ofvectors. Minimization of the $\ell^1$-norm of the coefficient vector can oftenretrieve the sparsest solution (so-called "$\ell^1/\ell^0$-equivalence"), agenerally NP-hard task, and this fact has powered the field of compressedsensing. Wright et al.'s sparse representation-based classification (SRC)applies this relationship to machine learning, wherein the signal to bedecomposed represents the test sample and columns of the dictionary aretraining samples. We investigate the relationships between$\ell^1$-minimization, sparsity, and classification accuracy in SRC. Afterproving that the tractable, deterministic approach to verifying$\ell^1/\ell^0$-equivalence fundamentally conflicts with the high coherencebetween same-class training samples, we demonstrate that $\ell^1$-minimizationcan still recover the sparsest solution when the classes are well-separated.Further, using a nonlinear transform so that sparse recovery conditions may besatisfied, we demonstrate that approximate (not strict) equivalence is key tothe success of SRC.

The Use of Mutual Coherence to Prove $\ell^1/\ell^0$-Equivalence in Classification Problems