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 $$ \delta_{K + 1} \geq \frac{1}{\sqrt{K+\frac{1}{4}}}. $$Comment: To appear in IEEE Transactions on Signal Processin

A modified multiple OLS (m2OLS) algorithm for signal recovery in compressive sensing

“…• For the noisy case, the conditions given by (14) and (30), along with the conditions given by (13), and (27) are sufficient. Of these, we have already seen that (27) implies (13).…”

“…• For the noisy case, the conditions given by (14) and (30), along with the conditions given by (13), and (27) are sufficient. Of these, we have already seen that (27) implies (13). On the other hand, it is easy to check that the numerator of the RHS of (30) is larger than that of the RHS of (14).…”

A Near-Optimal Condition for Exact Sparse Recovery with Orthogonal Least Squares

“…We note that our result in Lemma 4 outperforms this result in the following aspects: i) Under the same RIP condition (δ K+1 ≤ 1 2 ), a tighter upper bound of max j∈Ω\T | r k , φ j | can be established using Lemma 4. By applying [6,Lemma 4]…”

“…Clearly, the bound in (22) is tighter than that in (21) by the factor of 2 √ 3 . ii) In [6], by putting…”

“…There have been previous efforts to analyze a recovery guarantee of the OLS algorithm [3], [6]- [8]. So far, the best result states that OLS guarantees the exact reconstruction under [6, Theorem 1]…”