Nearly Optimal Bounds for Orthogonal Least Squares

Abstract: In this paper, we study the orthogonal least squares (OLS) algorithm for sparse recovery. On the one hand, we show that if the sampling matrix $\mathbf{A}$ satisfies the restricted isometry property (RIP) of order $K + 1$ with isometry constant $$ \delta_{K + 1} < \frac{1}{\sqrt{K+1}}, $$ then OLS exactly recovers the support of any $K$-sparse vector $\mathbf{x}$ from its samples $\mathbf{y} = \mathbf{A} \mathbf{x}$ in $K$ iterations. On the other hand, we show that OLS may not be able to recover the support of a $K$-sparse vector $\mathbf{x}$ in $K$ iterations for some $K$ if $$ \delta_{K + 1} \geq \frac{1}{\sqrt{K+\frac{1}{4}}}. $$