Analysis of The Ratio of ℓ_1 and ℓ_2 Norms in Compressed Sensing
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We first propose a novel criterion that guarantees that an s-sparse signal is the local minimizer of the ℓ_1/ℓ_2 objective; our criterion is interpretable and useful in practice. We also give the first uniform recovery condition using a geometric characterization of the null space of the measurement matrix, and show that this condition is easily satisfied for a class of random matrices. We also present analysis on the stability of the procedure when noise pollutes data. Numerical experiments are provided that compare ℓ_1/ℓ_2 with some other popular non-convex methods in compressed sensing. Finally, we propose a novel initialization approach to accelerate the numerical optimization procedure. We call this initialization approach support selection, and we demonstrate that it empirically improves the performance of existing ℓ_1/ℓ_2 algorithms.
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