[Paper Review] Blind Compressed Sensing

Sivan Gleichman, Yonina C. Eldar|arXiv (Cornell University)|Feb 12, 2010

Sparse and Compressive Sensing Techniques4 citations

TL;DR

This paper introduces blind compressed sensing (BCS), a framework that enables recovery of sparse signals from compressed measurements without prior knowledge of the sparsity basis. By imposing structural constraints on the unknown basis—such as membership in a known finite set or sparsity within a dictionary—the method achieves unique recovery with high probability using Gaussian measurement matrices, matching standard compressed sensing performance when signals are sufficiently sparse.

ABSTRACT

The fundamental principle underlying compressed sensing is that a signal, which is sparse under some basis representation, can be recovered from a small number of linear measurements. However, prior knowledge of the sparsity basis is essential for the recovery process. This work introduces the concept of blind compressed sensing, which avoids the need to know the sparsity basis in both the sampling and the recovery process. We suggest three possible constraints on the sparsity basis that can be added to the problem in order to make its solution unique. For each constraint we prove conditions for uniqueness, and suggest a simple method to retrieve the solution. Under the uniqueness conditions, and as long as the signals are sparse enough, we demonstrate through simulations that without knowing the sparsity basis our methods can achieve results similar to those of standard compressed sensing, which relay on prior knowledge of the sparsity basis. This offers a general sampling and reconstruction system that fits all sparse signals, regardless of the sparsity basis, under the conditions and constraints presented in this work.

Motivation & Objective

• To address the limitation in compressed sensing that requires prior knowledge of the sparsity basis for signal recovery.
• To develop a unified sampling and reconstruction framework applicable to any sparse signal, regardless of the underlying sparsity basis.
• To prove theoretical conditions under which blind recovery is uniquely possible despite the absence of basis knowledge.
• To propose practical algorithms for blind recovery under specific structural constraints on the sparsity basis.
• To demonstrate via simulations that BCS achieves performance comparable to standard compressed sensing when signals are sufficiently sparse.

Proposed method

• Proposes three constraints to ensure uniqueness in blind compressed sensing: (1) the sparsity basis belongs to a known finite set of bases, (2) the basis is a sparse combination of a given dictionary’s columns, and (3) structured sparsity across multiple signals.
• Treats the first constraint as a series of standard compressed sensing problems, enabling recovery via standard CS solvers.
• For the second constraint, formulates the problem as standard compressed sensing or as dictionary learning with a sparse dictionary, and compares both approaches.
• Uses Gaussian random measurement matrices, proving they satisfy the uniqueness conditions with probability one under the proposed constraints.
• Employs a two-stage recovery process: first estimating the sparsity basis under constraints, then reconstructing the signal using standard sparse recovery algorithms.
• Proves that under the constraints, the measurement matrix must have full spark and avoid inter-block diagonal structure to ensure unique recovery.

Experimental results

Research questions

• RQ1 Under what conditions can a sparse signal be uniquely recovered from compressed measurements without prior knowledge of the sparsity basis?
• RQ2 Can blind compressed sensing achieve performance comparable to standard compressed sensing when the true sparsity basis is unknown?
• RQ3 What structural constraints on the sparsity basis ensure unique recovery in the blind setting?
• RQ4 How does the number of measurements and signals affect the success of blind recovery under the proposed constraints?
• RQ5 Can Gaussian random measurement matrices guarantee unique recovery in blind compressed sensing under the proposed constraints?

Key findings

• Blind compressed sensing achieves signal recovery performance comparable to standard compressed sensing when signals are sufficiently sparse and the sparsity basis satisfies the proposed constraints.
• With probability one, Gaussian random measurement matrices satisfy the uniqueness conditions for blind recovery under the proposed constraints.
• The number of measurements required for unique recovery in blind compressed sensing is equal to the ambient dimension of the signal when no constraints are applied, highlighting the necessity of structural assumptions.
• For the finite set of bases constraint, the method reduces to solving multiple standard compressed sensing problems, enabling efficient recovery using existing solvers.
• For the sparse basis constraint, the problem can be framed as dictionary learning with a sparse dictionary, and both standard CS and DL approaches yield similar recovery performance.
• Simulations confirm that under the structural constraints and sufficient sparsity, blind compressed sensing recovers signals with reconstruction accuracy close to that of standard compressed sensing with known sparsity bases.