[Paper Review] From Blind deconvolution to Blind Super-Resolution through convex programming

Augustin Cosse|arXiv (Cornell University)|Sep 26, 2017

Sparse and Compressive Sensing Techniques40 references3 citations

TL;DR

This paper establishes that blind deconvolution and blind super-resolution can be solved via nuclear norm minimization using convex programming, provided a small number of input signals live in known low-dimensional subspaces. The key result shows exact recovery is possible with high probability when the ambient dimension $ L \gtrsim K^{3/2} \mu_m^2 $ and the number of inputs $ N \gtrsim K^{1/2} \mu_h^2 $, up to log factors, by constructing a certificate via a Neumann series and applying concentration bounds to control its terms.

ABSTRACT

This paper discusses the recovery of an unknown signal $x\in \mathbb{R}^L$ through the result of its convolution with an unknown filter $h \in \mathbb{R}^L$. This problem, also known as blind deconvolution, has been studied extensively by the signal processing and applied mathematics communities, leading to a diversity of proofs and algorithms based on various assumptions on the filter and its input. Sparsity of this filter, or in contrast, non vanishing of its Fourier transform are instances of such assumptions. The main result of this paper shows that blind deconvolution can be solved through nuclear norm relaxation in the case of a fully unknown channel, as soon as this channel is probed through a few $N \gtrsim μ^2_m K^{1/2}$ input signals $x_n = C_n m_n$, $n=1,\ldots,N,$ that are living in known $K$-dimensional subspaces $C_n$ of $\mathbb{R}^L$. This result holds with high probability on the genericity of the subspaces $C_n$ as soon as $L\gtrsim K^{3/2}$ and $N\gtrsim K^{1/2}$ up to log factors. Our proof system relies on the construction of a certificate of optimality for the underlying convex program. This certificate expands as a Neumann series and is shown to satisfy the conditions for the recovery of the matrix encoding the unknowns by controlling the terms in this series. An incidental consequence of the result of this paper, following from the lack of assumptions on the filter, is that nuclear norm relaxation can be extended from blind deconvolution to blind super-resolution, as soon as the unknown ideal low pass filter has a sufficiently large support compared to the ambient dimension $L$. Numerical experiments supporting the theory as well as its application to blind super-resolution are provided.

Motivation & Objective

• To establish conditions under which blind deconvolution can be solved via convex programming without prior assumptions on the filter's sparsity or incoherence.
• To extend the applicability of nuclear norm minimization from blind deconvolution to blind super-resolution by removing restrictive assumptions on the filter.
• To provide a theoretical guarantee for exact recovery of both the unknown signal and filter using a minimal number of probing signals in low-dimensional subspaces.
• To develop a certificate-based proof framework using a Neumann series expansion to verify optimality of the convex relaxation.

Proposed method

• The method formulates blind deconvolution as a low-rank matrix recovery problem by lifting the signal and filter into a matrix variable, then minimizing the nuclear norm to promote low rank.
• A certificate of optimality is constructed as a Neumann series expansion, which is shown to satisfy the subgradient condition for exact recovery.
• The first two terms of the Neumann series are bounded using subexponential Bernstein inequality and a matrix version of the Rosenthal-Pinelis inequality for U-statistics.
• The remaining higher-order terms are controlled via a general argument based on concentration and decoupling techniques for U-statistics.
• The proof relies on coherence parameters $ \mu_m $ and $ \mu_h $, which quantify the spread of energy in the input signals and the filter's Fourier transform, respectively.
• The framework is extended to blind super-resolution by noting that a sufficiently spread-out low-pass filter supports recovery under the same convex relaxation.

Experimental results

Research questions

• RQ1 Can blind deconvolution be solved via convex programming without assuming sparsity or incoherence of the filter?
• RQ2 What is the minimal number of input signals required to recover both the signal and filter in a blind deconvolution setup?
• RQ3 Can nuclear norm minimization be extended to blind super-resolution under general conditions on the filter?
• RQ4 How can a certificate of optimality be constructed and verified in the absence of prior knowledge on the filter or input?
• RQ5 What role do coherence parameters $ \mu_m $ and $ \mu_h $ play in determining the sample complexity of recovery?

Key findings

• Exact recovery of both the unknown signal and filter is possible with high probability when $ L \gtrsim K^{3/2} \mu_m^2 $ and $ N \gtrsim K^{1/2} \mu_h^2 $, up to logarithmic factors.
• The method requires only a few input signals living in known $ K $-dimensional subspaces, significantly reducing the number of measurements needed.
• The proof constructs a certificate via a Neumann series, with the first two terms bounded using subexponential and Rosenthal-type inequalities.
• The second-order term is decomposed into univariate and cross-term contributions, each bounded using decoupling and U-statistic techniques.
• The approach removes the need for sparsity or incoherence assumptions, unlike prior work, and applies to blind super-resolution when the filter's Fourier support is sufficiently large.
• Numerical experiments confirm the theoretical predictions and demonstrate the method's effectiveness in blind super-resolution applications.