Simultaneously Structured Models with Application to Sparse and Low-rank Matrices (1212.3753v3)

Abstract: The topic of recovery of a structured model given a small number of linear observations has been well-studied in recent years. Examples include recovering sparse or group-sparse vectors, low-rank matrices, and the sum of sparse and low-rank matrices, among others. In various applications in signal processing and machine learning, the model of interest is known to be structured in several ways at the same time, for example, a matrix that is simultaneously sparse and low-rank. Often norms that promote each individual structure are known, and allow for recovery using an order-wise optimal number of measurements (e.g., $\ell_1$ norm for sparsity, nuclear norm for matrix rank). Hence, it is reasonable to minimize a combination of such norms. We show that, surprisingly, if we use multi-objective optimization with these norms, then we can do no better, order-wise, than an algorithm that exploits only one of the present structures. This result suggests that to fully exploit the multiple structures, we need an entirely new convex relaxation, i.e. not one that is a function of the convex relaxations used for each structure. We then specialize our results to the case of sparse and low-rank matrices. We show that a nonconvex formulation of the problem can recover the model from very few measurements, which is on the order of the degrees of freedom of the matrix, whereas the convex problem obtained from a combination of the $\ell_1$ and nuclear norms requires many more measurements. This proves an order-wise gap between the performance of the convex and nonconvex recovery problems in this case. Our framework applies to arbitrary structure-inducing norms as well as to a wide range of measurement ensembles. This allows us to give performance bounds for problems such as sparse phase retrieval and low-rank tensor completion.

• The paper demonstrates that combined ℓ1 and nuclear norm strategies offer no better recovery guarantees than using a single norm.
• It shows that nonconvex formulations achieve near-optimal recovery from minimal measurements despite their computational challenges compared to convex methods.
• The research evaluates various measurement ensembles to provide critical theoretical benchmarks for high-dimensional signal recovery in practical applications.

An Expert Overview of "Simultaneously Structured Models with Application to Sparse and Low-rank Matrices"

The academic paper authored by Samew Oymak, Amin Jalali, Maryam Fazel, Yonina C. Eldar, and Babak Hassibi provides a rigorous investigation into the recovery of simultaneously structured models from sparse linear observations, with a particular focus on matrices exhibiting both sparse and low-rank characteristics. This research is positioned at the intersection of signal processing and machine learning, addressing a commonly encountered scenario where the target model is structured in multiple ways. This paper delivers insights into recovery methods and extends current theoretical understandings of this class of models.

Core Results and Findings

1. Performance Boundaries of Optimization Techniques: The paper delves deeply into the performance of multi-objective optimization strategies that exploit sparse and low-rank structures of matrices simultaneously. It reveals that using a combination of norms, specifically the ℓ<sub\>1</sub> norm for sparsity and the nuclear norm for low rank, yields no better theoretical recovery guarantees than utilizing a single norm. This revelation challenges prior assumptions about the advantages of using simultaneous norm optimization and underscores that such combined approaches do not reduce measurement requirements beyond employing just one of the structures efficiently.
2. Nonconvex vs. Convex Formulations: Notably, the paper demonstrates that a nonconvex approach can recover models from a minimal number of measurements, roughly corresponding to the matrix's degrees of freedom, much more effectively than the convex formulations. This revelation suggests that while nonconvex methods might theoretically outperform convex approaches, they remain largely intractable for practical purposes. However, this gap illuminates areas of potential advancements in optimization techniques which could improve practical feasibility.
3. Statistical and Theoretical Implications: The research extends its analyses to numerous measurement ensembles like Gaussian and subgaussian distributions and even tackles scenarios involving quadratic measurements, which is pertinent to phase retrieval problems. The theoretical benchmarks provided are critical for understanding the practical limits of current methodologies in high-dimensional signal recovery tasks.
4. Implications for Applications: The paper's implications are broad and apply to signal processing tasks such as compressed sensing, optical imaging, and various machine learning applications involving complex data structures like hyperspectral imaging and sparse tensor decomposition.

Implications for Future Research

The findings presented prompt several avenues for further investigation. For instance, the development of new algorithms that can efficiently handle the nonconvex optimization challenges could bridge the gap between theoretical possibilities and practical implementations. Moreover, exploring alternative norm combinations or developing new atomic norms that encapsulate multiple structures simultaneously could lead to more efficient recovery strategies.

Another line of inquiry lies in extending this framework to models beyond sparse and low-rank matrices, venturing into heterogeneous data structures common in modern data science and machine learning challenges. The research also lays foundational ground to paper the implications of their results in correlated sparsity models and advances in compressed sensing for broader applications.

Conclusion

This well-executed paper meticulously focuses on the recovery of simultaneously structured models from undersampled observations, providing crucial insights into the limitations and potentials of current optimization strategies. By establishing performance bounds and challenging existing notions, it paves the way for enriched jurisprudence and future explorations in signal processing and machine learning domains. The implications of this paper could see far-reaching applications in both academic research and industry applications, strengthening our approaches to handling complex, large-scale data efficiently and accurately.