An Optimal Restricted Isometry Condition for Exact Sparse Recovery with Orthogonal Least Squares
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The orthogonal least squares (OLS) algorithm is popularly used in sparse recovery, subset selection, and function approximation. In this paper, we analyze the performance guarantee of OLS. Specifically, we show that if a sampling matrix Φ has unit ℓ_2-norm columns and satisfies the restricted isometry property (RIP) of order K+1 with δ_K+1 <C_K = 1/√(K), K=1, 1/√(K+1/4), K=2, 1/√(K+1/16), K=3, 1/√(K), K > 4, then OLS exactly recovers any K-sparse vector x from its measurements y = Φx in K iterations. Furthermore, we show that the proposed guarantee is optimal in the sense that OLS may fail the recovery under δ_K+1> C_K. Additionally, we show that if the columns of a sampling matrix are ℓ_2-normalized, then the proposed condition is also an optimal recovery guarantee for the orthogonal matching pursuit (OMP) algorithm. Also, we establish a recovery guarantee of OLS in the more general case where a sampling matrix might not have unit ℓ_2-norm columns. Moreover, we analyze the performance of OLS in the noisy case. Our result demonstrates that under a suitable constraint on the minimum magnitude of nonzero elements in an input signal, the proposed RIP condition ensures OLS to identify the support exactly.
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