Sparsity estimation from compressive projections via sparse random matrices.

Area of Science:

• Signal Processing
• Information Theory
• Machine Learning

Background:

• Compressive sensing enables signal reconstruction from fewer samples than traditional methods.
• Estimating signal sparsity is crucial for efficient compressive sensing but often requires signal recovery.
• Existing sparsity estimation techniques face challenges in noisy environments and with non-exactly sparse signals.

Purpose of the Study:

• To develop strategies for estimating signal sparsity from compressive projections without signal recovery.
• To address both noise-free and noisy measurement scenarios.
• To extend the framework to non-exactly sparse signals.

Main Methods:

• A maximum likelihood (ML) approach using γ-sparsified random matrices.
• Exploiting mixture model properties where parameters depend on signal sparsity.
• Approximating the probability model with a two-component Gaussian mixture (2-GMM) for noisy settings, learned via expectation-maximization.

Main Results:

• Sufficient conditions for exact sparsity estimation in the noise-free case are established.
• The accuracy of the 2-GMM approximation in the noisy, large system limit is proven.
• Simulations demonstrate superior performance compared to state-of-the-art sparsity estimation methods.

Conclusions:

• The proposed ML-based framework effectively estimates signal sparsity from compressive measurements.
• The method is robust in noisy conditions and applicable to non-exactly sparse signals.
• The findings encourage further research into more generalized sparsity estimation frameworks.