Results 1 
4 of
4
Simultaneously Structured Models with Application to Sparse and Lowrank Matrices
, 2014
"... The topic of recovery of a structured model given a small number of linear observations has been wellstudied in recent years. Examples include recovering sparse or groupsparse vectors, lowrank matrices, and the sum of sparse and lowrank matrices, among others. In various applications in signal p ..."
Abstract

Cited by 41 (5 self)
 Add to MetaCart
The topic of recovery of a structured model given a small number of linear observations has been wellstudied in recent years. Examples include recovering sparse or groupsparse vectors, lowrank matrices, and the sum of sparse and lowrank 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 lowrank. Often norms that promote each individual structure are known, and allow for recovery using an orderwise optimal number of measurements (e.g., `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 multiobjective optimization with these norms, then we can do no better, orderwise, 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 lowrank 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 `1 and nuclear norms requires many more measurements. This proves an orderwise gap between the performance of the convex and nonconvex recovery problems in this case. Our framework applies to arbitrary structureinducing 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 lowrank tensor completion.
Square deal: Lower bounds and improved relaxations for tensor recovery
 CoRR
"... Recovering a lowrank tensor from incomplete information is a recurring problem in signal processing and machine learning. The most popular convex relaxation of this problem minimizes the sum of the nuclear norms of the unfoldings of the tensor. We show that this approach can be substantially subopt ..."
Abstract

Cited by 22 (0 self)
 Add to MetaCart
(Show Context)
Recovering a lowrank tensor from incomplete information is a recurring problem in signal processing and machine learning. The most popular convex relaxation of this problem minimizes the sum of the nuclear norms of the unfoldings of the tensor. We show that this approach can be substantially suboptimal: reliably recovering a Kway tensor of length n and Tucker rank r from Gaussian measurements requires Ω(rnK−1) observations. In contrast, a certain (intractable) nonconvex formulation needs only O(rK+nrK) observations. We introduce a very simple, new convex relaxation, which partially bridges this gap. Our new formulation succeeds with O(rbK/2cndK/2e) observations. While these results pertain to Gaussian measurements, simulations strongly suggest that the new norm also outperforms the sum of nuclear norms for tensor completion from a random subset of entries. Our lower bound for the sumofnuclearnorms model follows from a new result on recovering signals with multiple sparse structures (e.g. sparse, low rank), which perhaps surprisingly demonstrates the significant suboptimality of the commonly used recovery approach via minimizing the sum of individual sparsity inducing norms (e.g. l1, nuclear norm). Our new formulation for lowrank tensor recovery however opens the possibility in reducing the sample complexity by exploiting several structures jointly. 1
Name Title Institution Email
"... Abstract — The objective of this project is the development of advanced methods for the processing of multienergy Xray data collected in limited view geometries. Such data holds the promise of providing much improved capability relative to single and dual energy systems for mapping materials throu ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract — The objective of this project is the development of advanced methods for the processing of multienergy Xray data collected in limited view geometries. Such data holds the promise of providing much improved capability relative to single and dual energy systems for mapping materials throughout a piece of luggage and, hence, more accurately identifying threat items. Complicating this task are two issues. First, current ideas for deployable multienergy systems are all of the limited view variety. Thus, existing image formation methods based on Fourier techniques are no longer applicable. Second, traditional methods that process data on an energybyenergy basis fail to fully exploit the interenergy information content that is embedded within these data. The approach pursued here is based on the use of sophisticated iterative processing methods for forming images that make use of and extend recently developed ideas from multilinear algebra. These techniques consider the full spatialspectral structure of the data. Within the context of full view, angle decimated data processing of our prior work has demonstrated the clear benefi ts of this approach. This project is focused on extending these ideas to the limited view case. The initial year of this project has been devoted to the development of a computational forward model for a limited view system that is based on a prototype system developed by our industrial collaborators at