[Paper Review] RIP-like Properties in Subsampled Blind Deconvolution

Kiryung Lee, Marius Junge|arXiv (Cornell University)|Nov 19, 2015

Sparse and Compressive Sensing Techniques12 references16 citations

TL;DR

This paper establishes near-optimal performance guarantees for subsampled blind deconvolution by introducing and analyzing RIP-like properties tailored to the nonconvex spectral flatness prior. It proves that with i.i.d. random dictionaries, stable recovery of sparse signals is achievable at sample complexity $ m = O((\mu_1 s_2 + \mu_2 s_1)\log^5 n) $, enabling provable recovery via a practical algorithm under near-optimal sampling rates.

ABSTRACT

We derive near optimal performance guarantees for subsampled blind deconvolution. Blind deconvolution is an ill-posed bilinear inverse problem and additional subsampling makes the problem even more challenging. Sparsity and spectral flatness priors on unknown signals are introduced to overcome these difficulties. While being crucial for deriving desired near optimal performance guarantees, unlike the sparsity prior with a nice union-of-subspaces structure, the spectral flatness prior corresponds to a nonconvex cone structure, which is not preserved by elementary set operations. This prohibits the operator arising in subsampled blind deconvolution from satisfying the standard restricted isometry property (RIP) at near optimal sample complexity, which motivated us to study other RIP-like properties. Combined with the performance guarantees derived using these RIP-like properties in a companion paper, we show that subsampled blind deconvolution is provably solved at near optimal sample complexity by a practical algorithm.

Motivation & Objective

• To address the challenge of stable recovery in subsampled blind deconvolution, where standard restricted isometry property (RIP) fails due to nonconvex spectral flatness priors.
• To derive RIP-like properties that capture the structure of the measurement operator in blind deconvolution under spectral flatness and sparsity constraints.
• To establish provable performance guarantees at near-optimal sample complexity for a practical reconstruction algorithm.
• To show that random dictionaries with i.i.d. complex Gaussian entries yield these RIP-like properties with high probability.

Proposed method

• Introduce a lifted formulation of blind deconvolution as a low-rank matrix recovery problem from linear measurements.
• Define a new class of RIP-like properties—specifically, the $({\mathcal{S}}, {\mathcal{S}}', \delta)$-restricted angle-preserving property (RAP)—to handle the nonconvex cone structure from spectral flatness.
• Use Gaussian concentration of measure to bound the deviation of bilinear forms over structured sets of rank-1 matrices.
• Analyze the Frobenius norm, spectral norm, and entropy net complexity of the sets $\Delta_{\mathrm{R}}$ and $\Delta_{\mathrm{L}}$ to control the concentration of the measurement operator.
• Prove that for i.i.d. random dictionaries with $ CN(0,1/n) $ entries, the required RIP-like properties hold with high probability at near-optimal sample complexity.
• Leverage companion results to show that a practical algorithm achieves stable recovery under these derived properties.

Experimental results

Research questions

• RQ1 Can standard RIP-based recovery guarantees be applied to subsampled blind deconvolution with spectral flatness priors?
• RQ2 What RIP-like properties are necessary and sufficient to ensure stable recovery when the spectral flatness prior induces a nonconvex cone structure?
• RQ3 At what sample complexity can stable recovery be guaranteed for blind deconvolution with sparse and spectrally flat signals?
• RQ4 How do i.i.d. random dictionaries affect the RIP-like properties of the measurement operator in this setting?
• RQ5 Can a practical algorithm achieve near-optimal sample complexity for blind deconvolution under these conditions?

Key findings

• The measurement operator in subsampled blind deconvolution fails to satisfy the standard restricted isometry property (RIP) at near-optimal sample complexity due to the nonconvex spectral flatness prior.
• The paper introduces and establishes the $({\mathcal{S}}, {\mathcal{S}}', \delta)$-RAP as a viable RIP-like alternative that captures the necessary structure for stable recovery.
• With high probability, i.i.d. random dictionaries with $ CN(0,1/n) $ entries ensure the required RIP-like properties hold at sample complexity $ m = O((\mu_1 s_2 + \mu_2 s_1)\log^5 n) $.
• This sample complexity is near-optimal and enables provable stable recovery of sparse signals via a practical algorithm.
• The spectral flatness prior, though nonconvex, is crucial for achieving near-optimal performance and enables the derivation of new RIP-like properties.
• The results extend to subgaussian and partial Fourier sensing matrices, suggesting broader applicability beyond the i.i.d. case.