[Paper Review] On partial sparse recovery

Afonso S. Bandeira, Katya Scheinberg+1 more|arXiv (Cornell University)|Apr 9, 2013

Sparse and Compressive Sensing TechniquesEngineering14 references17 citations

TL;DR

This paper introduces analogues of the null space property (NSP) and restricted isometry property (RIP) for partial sparse recovery, where only a subset of the solution vector is assumed to be sparse. It proves these new conditions are sufficient for exact and robust recovery of partially sparse signals using ℓ₁-minimization on the sparse part, and shows that Gaussian random matrices can guarantee recovery with fewer measurements than classical compressed sensing when part of the support is known in advance.

ABSTRACT

We consider the problem of recovering a partially sparse solution of an underdetermined system of linear equations by minimizing the $\ell_1$-norm of the part of the solution vector which is known to be sparse. Such a problem is closely related to a classical problem in Compressed Sensing where the $\ell_1$-norm of the whole solution vector is minimized. We introduce analogues of restricted isometry and null space properties for the recovery of partially sparse vectors and show that these new properties are implied by their original counterparts. We show also how to extend recovery under noisy measurements to the partially sparse case.

Motivation & Objective

• Address the problem of recovering partially sparse solutions in underdetermined linear systems where only a subset of variables is expected to be sparse.
• Develop new theoretical conditions—partial NSP and partial RIP—specifically tailored for ℓ₁-minimization on the sparse part of the solution.
• Demonstrate that these new conditions are implied by classical NSP and RIP, ensuring their validity under standard compressed sensing assumptions.
• Extend the analysis to noisy measurements, providing error bounds for partially sparse recovery under uncertainty.
• Show that partial sparsity enables recovery with significantly fewer measurements than classical compressed sensing, especially when the sparse part is small.

Proposed method

• Propose a modified ℓ₁-minimization problem that minimizes only the ℓ₁-norm of the unknown sparse part of the solution vector, while satisfying the linear measurement constraints.
• Introduce the partial null space property (partial NSP) as a necessary and sufficient condition for exact recovery of partially sparse vectors.
• Define the partial restricted isometry property (partial RIP) for matrices, characterizing how well the measurement matrix preserves ℓ₂-norms of partially sparse vectors.
• Prove that partial NSP and partial RIP are implied by their classical counterparts, ensuring that standard random matrices (e.g., Gaussian) satisfy the new conditions.
• Derive error bounds for the partially sparse recovery problem under noisy measurements, relating the recovery error to the noise level and matrix constants.
• Analyze Gaussian random matrices and derive a high-probability bound on the number of measurements required for partial RIP, showing a reduction in required measurements compared to full sparsity.

Experimental results

Research questions

• RQ1 Can we establish sufficient conditions for exact recovery of partially sparse vectors using ℓ₁-minimization restricted to the sparse part?
• RQ2 How do the new partial NSP and partial RIP conditions relate to classical NSP and RIP in terms of strength and implications?
• RQ3 Can we extend the theoretical guarantees of compressed sensing to the case where part of the support is known in advance?
• RQ4 What is the minimal number of measurements required to ensure high-probability partial sparse recovery using Gaussian random matrices?
• RQ5 How does the error in partially sparse recovery scale with noise levels and matrix properties?

Key findings

• The partial null space property (partial NSP) is both necessary and sufficient for exact recovery of partially sparse vectors via ℓ₁-minimization on the sparse part.
• The partial restricted isometry property (partial RIP) is sufficient for robust recovery under noisy measurements, and its constants can be bounded using the classical RIP constants.
• For Gaussian random matrices, the number of measurements required for partial sparse recovery is asymptotically smaller than in classical compressed sensing when the sparse part size is small, specifically requiring only O(s + log(N−r)) measurements when s−r = O(1).
• The error bounds for noisy recovery show that the reconstruction error in the dense part is controlled by the noise level and the ℓ₂-error in the sparse part, with explicit constants derived from the matrix’s partial RIP constants.
• The probability of failure in partial RIP satisfaction decays polynomially with N, ensuring high-probability recovery for sufficiently large k.
• The proposed framework allows for significant reductions in measurement requirements when prior knowledge about the support of the sparse part is available, making it particularly useful in applications like image reconstruction and Hessian recovery.