[Paper Review] Coherence and RIP Analysis for Greedy Algorithms in Compressive Sensing.

Mingrui Yang, Frank de Hoog|arXiv (Cornell University)|Jul 8, 2013

Sparse and Compressive Sensing Techniques22 references5 citations

TL;DR

This paper introduces 2-coherence, a new dictionary coherence measure that strengthens the link between mutual coherence and restricted isometry constants (RIC), enabling improved recovery guarantees for greedy algorithms in compressive sensing. It establishes a tighter RIC bound for orthogonal matching pursuit (OMP) and introduces OMPT—a thresholding-enhanced variant of OMP—proving it achieves identical recovery performance to OMP under the same conditions, enhancing practical feasibility.

ABSTRACT

In this paper we define a new coherence index, named 2-coherence, of a given dictionary and study its relationship with the traditional mutual coherence and the restricted isometry constant. By exploring this relationship, we obtain more general results on sparse signal reconstruction using greedy algorithms in the compressive sensing (CS) framework. In particular, we obtain an improved bound over the best known results on the restricted isometry constant for successful recovery of sparse signals using orthogonal matching pursuit (OMP). We also initialized a study of a thresholding type greedy algorithm named orthogonal matching pursuit with thresholding (OMPT), which is more feasible in practice than OMP. We analyze its performance in CS framework for both noiseless and noisy cases in terms of coherence indices and the restricted isometry constant. We show that given the same assumptions as required for OMP, it achieves exactly the same reconstruction performance as OMP. Index Terms Compressive sensing, mutual coherence, 2-coherence, restricted isometry property, orthogonal matching pursuit (OMP), orthogonal matching pursuit with thresholding (OMPT). I.

Motivation & Objective

• To define and analyze a new coherence measure, 2-coherence, to strengthen theoretical links between coherence and restricted isometry properties in compressive sensing.
• To improve existing recovery bounds for orthogonal matching pursuit (OMP) by deriving a tighter restricted isometry constant (RIC) condition using 2-coherence.
• To propose and analyze a practical variant of OMP, orthogonal matching pursuit with thresholding (OMPT), to enhance feasibility in real-world applications.
• To establish that OMPT achieves the same theoretical recovery performance as OMP under identical assumptions, validating its practical utility.

Proposed method

• Introduces 2-coherence as a new measure of dictionary coherence, defined based on pairwise correlations between atoms, extending the concept of mutual coherence.
• Analyzes the relationship between 2-coherence, mutual coherence, and the restricted isometry constant (RIC), deriving theoretical bounds that connect these quantities.
• Applies the derived coherence-RIC relationships to improve the RIC threshold for successful sparse signal recovery via OMP, yielding a tighter bound than prior results.
• Proposes OMPT as a greedy algorithm that applies thresholding to select atoms, reducing computational cost while preserving theoretical recovery guarantees.
• Uses coherence-based and RIC-based analysis to prove that OMPT achieves the same recovery performance as OMP under the same conditions.
• Employs theoretical analysis in both noiseless and noisy compressive sensing settings to evaluate the performance of OMPT using coherence indices and RIC.

Experimental results

Research questions

• RQ1 How does the newly defined 2-coherence relate to traditional mutual coherence and the restricted isometry constant (RIC)?
• RQ2 Can the relationship between 2-coherence and RIC lead to improved recovery bounds for OMP in sparse signal reconstruction?
• RQ3 Does the proposed OMPT algorithm maintain the same theoretical recovery performance as OMP while improving practical feasibility?
• RQ4 What are the coherence-based and RIC-based performance guarantees for OMPT in both noiseless and noisy compressive sensing scenarios?

Key findings

• The paper derives a tighter restricted isometry constant (RIC) bound for successful sparse signal recovery using OMP than previously known results, improving theoretical recovery guarantees.
• 2-coherence is formally defined and shown to provide a more refined characterization of dictionary structure than mutual coherence alone.
• The relationship between 2-coherence, mutual coherence, and RIC is analytically established, enabling stronger theoretical connections in compressive sensing.
• OMPT is introduced as a practical variant of OMP that incorporates thresholding, reducing computational overhead without sacrificing performance.
• Under the same assumptions required for OMP, OMPT achieves identical recovery performance, validating its theoretical equivalence to OMP.
• Theoretical analysis confirms that OMPT maintains stable recovery in both noiseless and noisy compressive sensing settings using coherence and RIC-based metrics.