Improved RIP Analysis of Orthogonal Matching Pursuit

Orthogonal Matching Pursuit (OMP) has long been considered a powerful heuristic for attacking compressive sensing problems; however, its theoretical development is, unfortunately, somewhat lacking. This paper presents an improved Restricted Isometry Property (RIP) based performance guarantee for T-sparse signal reconstruction that asymptotically approaches the conjectured lower bound given in Davenport et al. We also further extend the state-of-the-art by deriving reconstruction error bounds for the case of general non-sparse signals subjected to measurement noise. We then generalize our results to the case of K-fold Orthogonal Matching Pursuit (KOMP). We finish by presenting an empirical analysis suggesting that OMP and KOMP outperform other compressive sensing algorithms in average case scenarios. This turns out to be quite surprising since RIP analysis (i.e. worst case scenario) suggests that these matching pursuits should perform roughly T^0.5 times worse than convex optimization, CoSAMP, and Iterative Thresholding.

💡 Research Summary

This paper revisits the theoretical foundations of Orthogonal Matching Pursuit (OMP), a widely used greedy algorithm in compressive sensing, and provides substantially stronger guarantees based on the Restricted Isometry Property (RIP). The authors first point out that existing OMP analyses either rely on coherence measures or on very restrictive RIP conditions such as δ_T < 1/√T, which are far from the conjectured lower bound and rarely satisfied by practical measurement matrices.

In the preliminaries, the standard notation is introduced together with two lemmas that capture the near‑unitary behavior of sub‑matrices of a RIP matrix (Lemma 1) and an extension of RIP bounds to non‑sparse signals (Lemma 2). These lemmas form the backbone of the subsequent proofs.

The core contribution appears in Theorem 1, where the authors prove that OMP exactly recovers any T‑sparse signal provided the measurement matrix Φ satisfies the RIP condition

  δ_T < 1 / √(T + 1).

This bound is asymptotically tight: it approaches the lower bound conjectured by Davenport et al. (2010) and improves upon earlier results (e.g., δ_T < 1 / (3√T) in