Source-linked AI summary
Rapid, Robust, and Reliable Blind Deconvolution via Nonconvex Optimization
Xiaodong Li, Shuyang Ling, Thomas Strohmer, Ke Wei
TL;DR
Blind deconvolution seeks to recover two unknown signals from their convolution despite ill-posedness and nonconvex optimization difficulties. The paper proposes regularized gradient descent with a carefully chosen initialization, obtaining geometric convergence, noise robustness, and recovery guarantees under subspace conditions. Its measurement requirement is only slightly above the information-theoretic minimum, up to logarithmic factors.
Problem
Blind deconvolution asks how to recover two unknown signals from their convolution, but is ill-posed without additional assumptions and difficult because its objective is highly nonconvex.
Method
The paper uses regularized gradient descent initialized by a carefully chosen guess to solve a nonconvex blind deconvolution objective under subspace conditions.
Results
The algorithm converges geometrically, is robust to noise, and succeeds with a number of measurements only slightly above the information-theoretic minimum, up to log-factors.
Takeaways & Limitations
The method provides an efficient nonconvex alternative with rigorous recovery guarantees and measurement requirements roughly comparable to the cited convex approach.
Takeaways & Limitations
The guarantees rely on subspace conditions, including a Gaussian signal-subspace matrix in the theoretical analysis.
Abstract
from arXiv · showhide
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.
1 Introduction
Blind deconvolution asks how to recover two unknown signals from their convolution, a problem arising across science and engineering. The paper develops a nonconvex approach intended to combine efficient computation, rigorous recovery guarantees, and robustness to noise.
- Problem motivation: Blind deconvolution reconstructs unknown signals f and g from their convolution y=f∗g.The problem appears in astronomy, medical imaging, optics, and communications engineering.
- Problem motivation: Without additional assumptions, blind deconvolution is ill-posed; known-subspace assumptions make the problem mathematically tractable but still highly nonconvex.Even under subspace constraints, the objective may contain many local minima.
- Paper contribution: The proposed regularized gradient descent algorithm is numerically efficient, converges geometrically, and has rigorous convergence guarantees.The method is designed to recover the exact solution under stated subspace conditions.
- Paper contribution: The method requires only slightly more than the information-theoretic minimum number of measurements, up to logarithmic factors.Its measurement requirement is roughly comparable to the convex approach in [1], while experiments indicate a smaller requirement.
- Related approach: The paper follows a two-step nonconvex strategy: construct a good initial guess, then use gradient descent to converge to the true solution.This strategy is motivated by prior work on phase retrieval and matrix completion.
- Related approach: The method incorporates a penalty term and neighborhood regions around the true solution to characterize a basin of attraction.These design ideas are attributed to prior work on nonconvex matrix completion.
2 Problem setup
The paper models blind deconvolution with structured signals and additive noise, then reformulates convolution in an equivalent Fourier-domain bilinear model. Lifting converts this nonlinear model into a linear operator framework, while regularization addresses the resulting nonconvex optimization challenge.
- Model setup: The model observes y=f∗g+n, with unknown blurring function f, signal g, and additive noise n.The convolution is circular, and f is treated as the blurring function while g is the signal of interest.
- Signal assumptions: The blurring function is assumed to lie in a known low-dimensional subspace, represented by f=Bh, with only the first K entries potentially nonzero.This corresponds to compact support or sufficiently fast decay of f.
- Signal assumptions: The signal of interest satisfies g=Cx, where C is a known L×N matrix; theory assumes C is Gaussian, while experiments also test wavelet and Hadamard-type subspaces.Wavelet subspaces are associated with images and Hadamard-type matrices with communications.
- Signal assumptions: The noise is modeled as additive white complex Gaussian noise scaled by the true signal norms, with σ^-2 serving as a signal-to-noise ratio measure.The noise model is used for both theoretical analysis and numerical purposes.
- Equivalent formulation: A Fourier-domain reformulation expresses the convolution model using the unitary DFT and a matrix B formed from its first K columns.This equivalent representation is more convenient for analysis and computation.
- Optimization formulation: Lifting converts the bilinear recovery problem into a linear measurement model involving a rank-one matrix, increasing the solution-space dimension.The lifted operator is defined from columns of B* and A*.
- Optimization formulation: Directly minimizing the nonlinear least-squares objective is difficult because the problem is highly nonconvex and gradient-based methods can become trapped in local minima.Convex relaxation avoids this issue but requires solving an expensive semidefinite program.
3 Algorithm and main result
The paper combines a carefully constructed initialization with regularized gradient descent to recover blind deconvolution solutions under subspace and incoherence conditions. The method has geometric convergence, noise robustness, and measurement requirements close to the information-theoretic or convex benchmarks.
- Conditions: The framework assumes known subspaces and controls incoherence between the true signal and measurement-matrix rows, while accounting for scaling nonuniqueness.The solution is identifiable only up to (αh0, α^-1x0), and the neighborhoods prevent unstable norm imbalance.
- Algorithm: The algorithm first constructs an initialization inside a basin of attraction, then uses regularized gradient descent to keep iterates within that basin.The basin controls distance, incoherence, and numerical scaling around the true solution.
- Algorithm: The regularizer replaces difficult projection onto the distance and incoherence neighborhoods while enforcing that iterates remain controlled.Its penalty combines terms for the distance neighborhood with a truncated-quadratic term related to incoherence.
- Main results: With a suitable initialization, gradient descent converges geometrically to the true solution in the noiseless case and to a nearby solution under additive noise.The convergence guarantee holds with high probability under the theorem’s measurement and parameter conditions.
- Main results: The required measurements are roughly comparable to the convex approach up to log-factors, while experiments indicate the non-convex method can require fewer measurements.The convergence rate is determined by ηω, and parameter choices create a tradeoff between sampling complexity or estimation quality and computation time.
- Main results: As the number of measurements grows, the noisy objective behaves increasingly like its noiseless counterpart, reducing the heuristic effect of noise on recovery.The iterates converge to an equivalence-class element, with deviation governed by additive noise.
4 Numerical simulations
The experiments evaluate recovery probability as measurement count and incoherence vary, comparing regularized gradient descent (regGrad), gradient descent (Grad), and nuclear norm minimization (NNM). regGrad and Grad generally outperform NNM, while regGrad gains an advantage for highly incoherent signals.
- 4.1 Number of measurements vs size of signals: The experiments compare regGrad, Grad, and NNM using simulated blind deconvolution data.regGrad minimizes the loss plus regularization, Grad uses the loss alone, and NNM is the nuclear norm method from.
- 4.1 Number of measurements vs size of signals: Figure 1 varies L/(K + N) and records the probability of successful recovery across 50 random tests for Gaussian and partial Hadamard measurement matrices.Gaussian tests span L from K + N to 4(K + N), while partial Hadamard tests use L = 2^s for 6 ≤ s ≤ 10.
- 4.1 Number of measurements vs size of signals: regGrad and Grad require significantly fewer measurements than NNM to achieve successful recovery with high probability.The comparison uses recovery probabilities plotted against the measurement count in Figure 1.
incoherence.
The experiments evaluate recovery probability, noise robustness, measurement efficiency, and image deblurring performance for Algorithm 2 under several sensing and subspace settings.
- Numerical experiments: At least 10% higher successful-recovery probability is observed for regGrad than Grad when L ≥ 6(K + N).The comparison uses 100 random tests for each measurement count.
- Robustness to additive noise: Recovery error scales linearly with noise level, while increasing L reduces relative reconstruction error linearly on a log-log scale.The robustness experiments use additive Gaussian noise across multiple noise levels and measurement counts.
- Image deblurring: For MRI deblurring, the image is modeled in a Haar-wavelet subspace and the blur kernel in a low-frequency Fourier subspace.The MRI image is approximately sparse in the Haar basis, while the kernel is supported on a low-frequency region.
- Image deblurring: With known subspaces, the reconstructed MRI image is visually indistinguishable from the ground truth; with unknown subspaces, Algorithm 2 still returns a reasonable reconstruction.The unknown-subspace experiment uses 35,000 wavelet coefficients and assumes only that the kernel support lies within a small box.
- Image deblurring: The unknown-subspace MRI reconstruction leaves room for improvement, motivating a possible additional total-variation penalty.The authors leave this extension for future research.
5 Proof of the main theorem
The proof establishes convergence by combining local regularity, local RIP, local smoothness, and robustness conditions within a neighborhood around the ground truth. These conditions keep gradient-descent iterates in the convergence region and support the theorem’s guarantees.
- Proof architecture: Four local conditions organize the proof: regularity gives objective decrease, RIP transfers it to error decrease, smoothness gives the convergence rate, and robustness handles additive noise.The proof architecture explicitly assigns each condition a distinct role.
- Local smoothness: The local smoothness constant scales as O(d0(1 + σ2)(K + N) log2 L), constraining the stepsize and linear convergence rate.The stepsize is chosen below the reciprocal of this constant.
- Neighborhood structure: The sublevel region N_eF lies inside the local amplitude and incoherence regions, and under the stated conditions its intersection with N_ε lies in a smaller neighborhood around the ground truth.This containment is used to control the iterates locally.
- Neighborhood structure: N_eF may contain several disconnected regions because the objective is nonconvex, so the containment argument does not imply that all of N_eF lies in N_ε.The proof identifies only one or more components as potentially contained in the target neighborhood.
- Neighborhood structure: A line segment entirely contained in N_eF, with one endpoint in N_ε, remains in N_d0 ∩ N_μ ∩ N_ε.This geometric fact supports the local descent argument.
- Gradient-descent invariance: With a suitable stepsize, gradient descent preserves membership in N_ε ∩ N_eF, while the local regularity and smoothness conditions control descent and convergence.The proof combines the invariant-region argument with the modified descent lemma.
- Theorem conclusion: The theorem is concluded under local regularity, local RIP, robustness, and local smoothness, whose validity is justified in subsequent subsections.The result is presented as the culmination of the four-condition proof framework.
6 Appendix
The appendix supplies the complex-variable descent argument and probabilistic tools used in the main proof. It adapts a descent lemma to constrained complex iterates and records concentration results for random variables and matrices.
- Complex-variable descent: Lemma 6.1 provides a descent inequality for continuously differentiable real-valued functions of complex variables within N_ε ∩ N_eF.The admissible update segment must remain inside the same local region.
- Complex-variable descent: The proof of Lemma 6.1 applies the chain rule and the Fundamental Theorem of Calculus to the real-valued function along an update path.This adapts the standard descent-lemma proof to complex variables.
- Matrix concentration: Theorem 6.2 gives an operator-norm tail bound for sums of independent centered random matrices controlled by a matrix ψ1 norm and a variance parameter.The bound includes the matrix dimensions and a failure-probability parameter.
- Probabilistic tools: The appendix records a tail bound for random variables with exponential tails and standard properties of complex Gaussian vectors.These results support probabilistic estimates used throughout the proofs.