Rapid, Robust, and Reliable Blind Deconvolution via Nonconvex Optimization

2016/06/15 by Xiaodong Li, Shuyang Ling, Li, Xiaodong +5 · 5 citations
Computer Science · Engineering · Mathematics · #Advanced Image Processing Techniques #FOS: Computer and information sciences #Information Theory (cs.IT) #Photoacoustic and Ultrasonic Imaging #Sparse and Compressive Sensing Techniques #cs.IT #math.IT

paper · pdf · doi:10.48550/arxiv.1606.04933

arxiv created 2016/06/15 · openalex publication_date 2016/06/15 · arxiv updated 2016/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the question of reconstructing two signals f and g from their convolution y = f∗ g. This problem, known as \em blind deconvolution, pervades many areas of science and technology, including astronomy, medical imaging, optics, and wireless communications. A key challenge of this intricate non-convex optimization problem is that it might exhibit many local minima. We present an efficient numerical algorithm that is guaranteed to recover the exact solution, when the number of measurements is (up to log-factors) slightly larger than the information-theoretical minimum, and under reasonable conditions on f and g. The proposed regularized gradient descent algorithm converges at a geometric rate and is provably robust in the presence of noise. To the best of our knowledge, our algorithm is the first blind deconvolution algorithm that is numerically efficient, robust against noise, and comes with rigorous recovery guarantees under certain subspace conditions. Moreover, numerical experiments do not only provide empirical verification of our theory, but they also demonstrate that our method yields excellent performance even in situations beyond our theoretical framework.

Citations

• A Unified Framework for Identifiability Analysis in Bilinear Inverse Problems with Applications to Subspace and Sparsity Models
• Low-rank Solutions of Linear Matrix Equations via Procrustes Flow
• PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming
• Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guarantees
• Guarantees of Riemannian Optimization for Low Rank Matrix Recovery
• A Simpler Approach to Matrix Completion
• Gradient Descent Converges to Minimizers
• Convergence Analysis for Rectangular Matrix Completion Using Burer-Monteiro Factorization and Gradient Descent
• Nuclear norm penalization and optimal rates for noisy low rank matrix\n completion

Cited by

• Regularized Overestimated Newton
• Phase retrieval with rank d measurements -- descending algorithms phase transitions
• Optimal spectral initializers impact on phase retrieval phase transitions -- an RDT view
• Phase transition of descending phase retrieval algorithms
• Structured Local Optima in Sparse Blind Deconvolution

Related

• Regularized Gradient Descent: A Nonconvex Recipe for Fast Joint Blind Deconvolution and Demixing
• Convex Sparse Blind Deconvolution
• Composite optimization for robust blind deconvolution
• Robust Blind Deconvolution via Mirror Descent
• A Nonconvex Approach for Exact and Efficient Multichannel Sparse Blind Deconvolution