Measuring Less, Recovering More: Distribution-Aware Weighted ℓ1 Analysis

2026-05-01

Raziyeh Takbiri

Sajad Daei *

Mikael Skoglund

Gabor Fodor

DOI：10.1016/j.sigpro.2026.110673

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Recovering signals that are sparse in a transform domain–so-called analysis-sparse vectors–from undersampled measurements is a fundamental problem in compressed sensing. In many practical scenarios, such as imaging, geophysics, and communications, the distribution of transform coefficients is known in advance, providing valuable prior information. Weighted ℓ1 analysis offers a natural framework to exploit this knowledge, yet a principled method for choosing the weights has remained unsolved. In this work, we propose a distribution-aware framework for weight design. Using convex geometric tools, we establish a computable and nearly tight upper bound on the expected number of Gaussian measurements needed for exact recovery. This bound depends only on two accessible summaries of the prior in the analysis domain: marginal support probabilities and expected signs. Minimizing it yields a near-optimal weight vector that is stable, scale-invariant, and applicable a priori, without iterative tuning. Our theory accurately predicts recovery thresholds and shows how prior knowledge can be converted into provable measurement savings. Extensive simulations confirm substantial reductions in the number of required measurements and reconstruction error relative to unweighted ℓ1 analysis, with the largest improvements observed for coherent and redundant operators. Overall, this work establishes a rigorous and practical recipe for leveraging distributional priors in analysis-based compressed sensing.

Keywords:

Compressed sensing

analysis sparsity

weighted ℓ1 analysis

statistical dimension

prior distributions

Measuring Less, Recovering More: Distribution-Aware Weighted ℓ1 Analysis

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