Rapid, Robust, and Reliable Blind Deconvolution via Nonconvex Optimization

Abstract

We study the question of reconstructing two signals $f$ and $g$ from their convolution $y=f\ast g$. 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.

References in corpus (11)

• A Simpler Approach to Matrix Completion
• Self-Calibration and Biconvex Compressive Sensing
• Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guarantees
• Low-rank Solutions of Linear Matrix Equations via Procrustes Flow
• Gradient Descent Converges to Minimizers
• Convergence Analysis for Rectangular Matrix Completion Using Burer-Monteiro Factorization and Gradient Descent
• A Unified Framework for Identifiability Analysis in Bilinear Inverse Problems with Applications to Subspace and Sparsity Models
• Blind Deconvolution Meets Blind Demixing: Algorithms and Performance Bounds
• Guarantees of Riemannian Optimization for Low Rank Matrix Recovery
• Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow
• Identifiability in Blind Deconvolution with Subspace or Sparsity Constraints

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• Regularized Gradient Descent: A Nonconvex Recipe for Fast Joint Blind Deconvolution and Demixing
• Through the Haze: a Non-Convex Approach to Blind Gain Calibration for Linear Random Sensing Models
• Structured Local Optima in Sparse Blind Deconvolution
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• BranchHull: Convex bilinear inversion from the entrywise product of signals with known signs
• Nonconvex Low-Rank Matrix Recovery with Arbitrary Outliers via Median-Truncated Gradient Descent
• Empirical Chaos Processes and Blind Deconvolution
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• Spectral Compressed Sensing via Projected Gradient Descent
• From Blind deconvolution to Blind Super-Resolution through convex programming
• Short-Message Communication and FIR System Identification using Huffman Sequences