Blind Deconvolution using Convex Programming (1211.5608v3)

Abstract: We consider the problem of recovering two unknown vectors, $\boldsymbol{w}$ and $\boldsymbol{x}$, of length $L$ from their circular convolution. We make the structural assumption that the two vectors are members of known subspaces, one with dimension $N$ and the other with dimension $K$. Although the observed convolution is nonlinear in both $\boldsymbol{w}$ and $\boldsymbol{x}$, it is linear in the rank-1 matrix formed by their outer product $\boldsymbol{w}\boldsymbol{x}^*$. This observation allows us to recast the deconvolution problem as low-rank matrix recovery problem from linear measurements, whose natural convex relaxation is a nuclear norm minimization program. We prove the effectiveness of this relaxation by showing that for "generic" signals, the program can deconvolve $\boldsymbol{w}$ and $\boldsymbol{x}$ exactly when the maximum of $N$ and $K$ is almost on the order of $L$. That is, we show that if $\boldsymbol{x}$ is drawn from a random subspace of dimension $N$, and $\boldsymbol{w}$ is a vector in a subspace of dimension $K$ whose basis vectors are "spread out" in the frequency domain, then nuclear norm minimization recovers $\boldsymbol{w}\boldsymbol{x}^*$ without error. We discuss this result in the context of blind channel estimation in communications. If we have a message of length $N$ which we code using a random $L\times N$ coding matrix, and the encoded message travels through an unknown linear time-invariant channel of maximum length $K$, then the receiver can recover both the channel response and the message when $L\gtrsim N+K$, to within constant and log factors.

• The paper introduces a convex relaxation that reformulates blind deconvolution as a low-rank matrix recovery problem using nuclear norm minimization.
• The authors provide theoretical guarantees that generic, subspace-constrained signals can be exactly deconvolved, as supported by robust numerical experiments.
• This approach enhances blind channel estimation and signal separation in communications and imaging, paving the way for further advanced research.

Blind Deconvolution using Convex Programming

This paper tackles a challenging problem in signal processing—blind deconvolution—a process essential for signal separation when given their convolution. Authors Ahmed, Recht, and Romberg propose a novel approach that leverages convex programming, specifically nuclear norm minimization, to deconvolve two signals $w$ and $x$. Remarkably, the paper demonstrates the efficacy of this method under certain subspace constraints.

Problem Formulation

Blind deconvolution is the task of recovering two unknown vectors from their circular convolution. The paper hypothesizes that $w$ and $x$ belong to known subspaces with specific dimensions. The convolution can be recast as a low-rank matrix recovery over linear measurements, suitable for a convex programming approach using nuclear norm minimization.

Main Contributions

1. Convex Relaxation: The paper introduces a formulation of the deconvolution problem as a semidefinite program. By focusing on the rank-1 structure of the outer product matrix of $w$ and $x$, they propose nuclear norm minimization as a relaxation technique suitable for exact recovery, provided the signals are generic.
2. Theoretical Guarantees: The key result is that generic signals can be deconvolved accurately when the dimension constraints of their subspaces are appropriately defined. The techniques draw heavily on recent advances in low-rank matrix recovery from underdetermined systems.
3. Robustness in Signal Processing: Significant emphasis is put on the practical application of blind deconvolution for blind channel estimation in communications. The authors show that a message coded with a random matrix can be retrieved, alongside the channel response, if certain conditions are met regarding system dimensions.
4. Numerical Experiments: The authors provide substantial numerical demonstrations that detail exact recovery performance and robustness to noise, showcasing the proposed method’s efficacy under various conditions.

Implications and Future Work

The proposed method's ability to separate signals by solving an inverse problem signals a substantial improvement in the field of compressed sensing. This development has critical implications for fields requiring blind deconvolution, such as telecommunications (e.g., channel estimation), medical imaging, and image processing.

The theoretical groundwork laid here opens pathways for future research on robust algorithms that handle broader signal classes and complex noise conditions. Moreover, extending this framework to multi-dimensional signals promises enhancements in automated image deblurring and other higher-dimensional signal processing applications.

The paper's theoretical insights and convincing empirical evidence support its proposition of treating blind deconvolution problems as low-rank recovery tasks. This aligns with the broader trend in algorithms where exploiting problem structures yields efficient and effective solutions—a line of inquiry ripe for continued research and development.

Conclusion

The paper presents a detailed paper on the application of convex programming to the blind deconvolution problem, delivering theoretical and practical benefits. The advancement has the potential to influence a wide range of applications in communication systems and signal processing, marking a significant contribution to the field. The rigorous approach combining nuclear norm minimization with subspace assumptions provides a robust framework that could inspire novel methodologies and applications in diverse technological domains.