[Paper Review] Simultaneously Structured Models with Application to Sparse and Low-rank Matrices

Samet Oymak, Amin Jalali+3 more|arXiv (Cornell University)|Dec 16, 2012

Sparse and Compressive Sensing Techniques67 references16 citations

TL;DR

This paper investigates the recovery of matrices that are simultaneously sparse and low-rank, showing that standard convex relaxations combining ℓ₁ and nuclear norms fail to exploit the joint structure, requiring order-wise more measurements than the theoretical minimum. It proves a fundamental gap between convex and nonconvex approaches, demonstrating that nonconvex formulations can recover such matrices from only 𝒪(rk) measurements—matching the degrees of freedom—while convex methods require Ω(min{k², rn}) measurements.

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.

Motivation & Objective

• To understand the fundamental limits of convex optimization in recovering signals with multiple simultaneous structures, such as sparsity and low-rankness.
• To investigate whether combining standard norms (e.g., ℓ₁ and nuclear norm) for multiple structures leads to optimal measurement complexity.
• To establish theoretical lower bounds on the number of measurements required for successful recovery using multi-objective convex relaxation.
• To demonstrate that convex combinations of individual structure-promoting norms cannot achieve the optimal sample complexity when multiple structures coexist.
• To propose a framework applicable to a wide range of measurement ensembles, including matrix completion and Gaussian measurements, for analyzing jointly structured models.

Proposed method

• Proposes a general framework for modeling signals with multiple simultaneous structures using atomic norms and convex hulls of structured atoms.
• Analyzes the performance of multi-objective convex optimization using a combination of individual structure-inducing norms (e.g., ℓ₁ for sparsity, nuclear norm for low-rank).
• Derives lower bounds on the number of measurements required for successful recovery via such convex relaxations, showing they are order-wise no better than using only one structure.
• Introduces a nonconvex formulation that exploits the joint structure and proves it can recover matrices from 𝒪(rk) measurements, matching the degrees of freedom.
• Applies the framework to diverse measurement ensembles, including subsampled standard basis (matrix completion), Gaussian, subgaussian, and quadratic measurements.
• Uses duality and geometric arguments to characterize the Pareto front of the multi-objective problem, showing that the joint structure is not captured by norm combinations.

Experimental results

Research questions

• RQ1 Can convex optimization combining ℓ₁ and nuclear norms achieve optimal sample complexity for jointly sparse and low-rank matrices?
• RQ2 What is the minimum number of generic linear measurements required to recover a matrix that is both sparse and low-rank using convex relaxation?
• RQ3 Is there a fundamental gap between the performance of convex and nonconvex formulations in recovering simultaneously structured matrices?
• RQ4 Can a new convex relaxation be constructed that fully exploits multiple structures, rather than relying on combinations of individual norms?
• RQ5 How do the results generalize across different measurement ensembles, such as matrix completion and Gaussian measurements?

Key findings

• The number of measurements required for successful recovery using a convex combination of ℓ₁ and nuclear norms is lower bounded by Ω(min{k², rn}), which is order-wise no better than using only one of the two structures.
• A nonconvex formulation can recover jointly sparse and low-rank matrices from 𝒪(rk) measurements, matching the degrees of freedom of the model.
• This establishes an order-wise gap between convex and nonconvex recovery: convex methods require significantly more measurements than the information-theoretic minimum.
• The framework applies to a wide range of measurement ensembles, including matrix completion, Gaussian, subgaussian, and quadratic measurements, with performance bounds derived accordingly.
• The results suggest that to fully exploit multiple structures, a new convex relaxation—not a combination of existing norms—is required.
• For the sparse and low-rank matrix recovery problem, the convex approach using ℓ₁ + nuclear norm requires Ω(min{k², rn}) measurements, while the true degrees of freedom are 𝒪(rk), indicating a fundamental inefficiency.