Robust Spectral Compressed Sensing via Structured Matrix Completion (1304.8126v5)

Abstract: The paper explores the problem of \emph{spectral compressed sensing}, which aims to recover a spectrally sparse signal from a small random subset of its $n$ time domain samples. The signal of interest is assumed to be a superposition of $r$ multi-dimensional complex sinusoids, while the underlying frequencies can assume any \emph{continuous} values in the normalized frequency domain. Conventional compressed sensing paradigms suffer from the basis mismatch issue when imposing a discrete dictionary on the Fourier representation. To address this issue, we develop a novel algorithm, called \emph{Enhanced Matrix Completion (EMaC)}, based on structured matrix completion that does not require prior knowledge of the model order. The algorithm starts by arranging the data into a low-rank enhanced form exhibiting multi-fold Hankel structure, and then attempts recovery via nuclear norm minimization. Under mild incoherence conditions, EMaC allows perfect recovery as soon as the number of samples exceeds the order of $r\log^{4}n$, and is stable against bounded noise. Even if a constant portion of samples are corrupted with arbitrary magnitude, EMaC still allows exact recovery, provided that the sample complexity exceeds the order of $r^{2}\log^{3}n$. Along the way, our results demonstrate the power of convex relaxation in completing a low-rank multi-fold Hankel or Toeplitz matrix from minimal observed entries. The performance of our algorithm and its applicability to super resolution are further validated by numerical experiments.

• The paper introduces the EMaC algorithm that transforms spectral compressed sensing into a structured matrix completion problem using multi-fold Hankel matrices.
• The paper establishes exact recovery guarantees under sample complexities of rlog⁴(n) and r²log³(n), demonstrating strong robustness against noise and outliers.
• The paper demonstrates that EMaC outperforms traditional methods, offering promising applications in super resolution, imaging, and wireless communications.

Overview of Robust Spectral Compressed Sensing via Structured Matrix Completion

The research presented in the paper tackles the challenging domain of spectral compressed sensing, with a focus on recovering spectrally sparse signals from limited time-domain samples. The authors address a key pitfall in conventional compressed sensing, which typically hinges on a discrete dictionary representation, leading to basis mismatch when dealing with continuous frequency domains. To circumvent this, the authors propose the Enhanced Matrix Completion (EMaC) algorithm, which relies on structured matrix completion methodologies without needing prior information about model order.

The EMaC algorithm cleverly arranges data into a low-rank, multi-fold Hankel matrix and uses nuclear norm minimization for signal recovery. The robustness of the algorithm is demonstrated under conditions where samples exceed the threshold of $r\log^4n$, with stability against bound noises. Additionally, even when faced with a portion of samples corrupted by high-magnitude noise, EMaC achieves perfect recovery if the sample complexity exceeds $r^2\log^3n$.

Numerical experiments further corroborate the algorithm's efficiency, presenting advantages over traditional compressed sensing methods. This positions EMaC as a viable tool for super resolution tasks and general structured matrix completion needs, exemplifying the adeptness of convex relaxation techniques in handling minimal data entries.

Key Findings

1. Algorithm Novelty: EMaC is developed by transforming spectral sampling into a matrix completion problem using multi-fold Hankel arrangements. This is innovative because it does not require predetermined frequency estimations, a common requirement in traditional harmonic retrieval methods.
2. Sample & Noise Robustness: The theoretical performance is proven under mild incoherence conditions, enabling exact recovery from a minimal number of samples. Its resilience against bounded noise and substantial outliers significantly outperforms traditional methods, which often require precise noise models and are sensitive to deviations.
3. Broader Implications: The work extends beyond harmonic retrieval, providing insight into Hankel matrix completion achieving near-theoretical limits. This holds implications for control systems, NLP, and vision areas needing structural matrix recovery.

Practical and Theoretical Implications

From a practical perspective, the EMaC algorithm reduces the intractable requirements of spectral sensing hardware and is particularly relevant in medical imaging and wireless communications, where reduced complexity and cost are crucial. Theoretically, it highlights the power of convex optimization in solving matrix completion problems when faced with minimal and corrupted data, suggesting new avenues for research in AI and signal processing.

Future Directions

Future research could explore optimizing the computational complexity of EMaC, making it scalable for larger datasets. Enhanced theoretical frameworks might disclose new horizons in structured matrix recovery and compressed sensing, further obliterating existing bounds and adaptations for linear projections within the compressed sensing paradigm.

In closing, this research makes significant strides in overcoming basis mismatch challenges inherent in conventional compressed sensing, establishing the groundwork for more adaptable, efficient algorithms capable of handling the unpredictability and sparsity of modern data-driven domains.