2012/12/16 by Samet Oymak, Amin Jalali, Oymak, Samet +7 · 3 citations
Computer Science · Engineering · Mathematics · Medicine · Physics and Astronomy · #Advanced MRI Techniques and Applications #Advanced X-ray Imaging Techniques #Direction-of-Arrival Estimation Techniques #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques #cs.IT #math.IT #math.OC
paper · pdf · doi:10.48550/arxiv.1212.3753
38 pages, 9 figures
openalex publication_date 2012/12/16 · arxiv created 2014/07/25 · arxiv updated 2014/07/28 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
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., ℓ1 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 ℓ1 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.