We consider a problem of considerable practical interest: the recovery of a data matrix from a sampling of its entries. Suppose that we observe m entries selected uniformly at random from a matrix M. Can we complete the matrix and recover the entries that we have not seen?

We show that one can perfectly recover most low-rank matrices from what appears to be an incomplete set of entries. We prove that if the number m of sampled entries obeys $$m\ge C\,n^{1.2}r\log n$$ for some positive numerical constant C, then with very high probability, most n×n matrices of rank r can be perfectly recovered by solving a simple convex optimization program. This program finds the matrix with minimum nuclear norm that fits the data. The condition above assumes that the rank is not too large. However, if one replaces the 1.2 exponent with 1.25, then the result holds for all values of the rank. Similar results hold for arbitrary rectangular matrices as well. Our results are connected with the recent literature on compressed sensing, and show that objects other than signals and images can be perfectly reconstructed from very limited information.

/pdf/exact-matrix-completion-via-convex-optimization-3nmwb4bacp.pdf

Exact Matrix Completion via Convex Optimization

The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard because it contains vector cardinality minimization as a special case.

In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum-rank solution can be recovered by solving a convex optimization problem, namely, the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is sufficiently large.

The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this preexisting concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to minimizing the nuclear norm and illustrate our results with numerical examples.

https://faculty.washington.edu/mfazel/low-rank-v2.pdf

Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

Suppose that one observes an incomplete subset of entries selected from a low-rank matrix. When is it possible to complete the matrix and recover the entries that have not been seen? We demonstrate that in very general settings, one can perfectly recover all of the missing entries from most sufficiently large subsets by solving a convex programming problem that finds the matrix with the minimum nuclear norm agreeing with the observed entries. The techniques used in this analysis draw upon parallels in the field of compressed sensing, demonstrating that objects other than signals and images can be perfectly reconstructed from very limited information.

/pdf/exact-matrix-completion-via-convex-optimization-15hkb1t9u7.pdf

Exact matrix completion via convex optimization

This paper presents control and coordination algorithms for groups of vehicles. The focus is on autonomous vehicle networks performing distributed sensing tasks where each vehicle plays the role of a mobile tunable sensor. The paper proposes gradient descent algorithms for a class of utility functions which encode optimal coverage and sensing policies. The resulting closed-loop behavior is adaptive, distributed, asynchronous, and verifiably correct.

Coverage control for mobile sensing networks

We present model reduction methods with guaranteed error bounds for systems represented by a Linear Fractional Transformation (LFT) on a repeated scalar uncertainty structure. These reduction methods can be interpreted either as doing state order reduction for multi-dimensionalsystems, or as uncertainty simplification in the case of uncertain systems, and are based on finding solutions to a pair of Linear Matrix Inequalities (LMIs). A related necessary and sufficient condition for the exact reducibility of stable uncertain systems is also presented.

Model Reduction of Multi-Dimensional and Uncertain Systems

Model reduction methods are presented for systems represented by a linear fractional transformation on a repeated scalar uncertainty structure. These methods involve a complete generalization of balanced realizations, balanced Gramians, and balanced truncation model reduction with guaranteed error bounds, based on solutions to a pair of linear matrix inequalities which generalize Lyapunov equations. The resulting reduction methods immediately apply to uncertainty simplification and state order reduction in the case of uncertain systems but also may be interpreted as state order reduction for multidimensional systems.

/pdf/model-reduction-of-multidimensional-and-uncertain-systems-1q7z5drpey.pdf

Model reduction of multidimensional and uncertain systems

Synthesis of Place Transition Nets for Simulation and Control of Manufacturing Systems

This paper studies a distributed continuous-time bi-virus model in which two competing viruses spread over a network consisting of multiple groups of individuals. Limiting behaviors of the network are characterized by analyzing the equilibria of the system and their stability. Specifically, when the two viruses spread over possibly different directed infection graphs, the system may have the following: first, a unique equilibrium, the healthy state, which is globally stable, implying that both viruses will eventually be eradicated, second, two equilibria including the healthy state and a dominant virus state, which is almost globally stable, implying that one virus will pervade the entire network causing a single-virus epidemic while the other virus will be eradicated, or third, at least three equilibria including the healthy state and two dominant virus states, depending on certain conditions on the healing and infection rates. When the two viruses spread over the same directed infection graph, the system may have zero or infinitely many coexisting epidemic equilibria, which represents the pervasion of the two viruses. Sensitivity properties of some nontrivial equilibria are investigated in the context of a decentralized control technique, and an impossibility result is given for a certain type of distributed feedback controller.

Analysis and Control of a Continuous-Time Bi-Virus Model

Error-bounds are developed for balanced truncation of linear time-varying systems, leading to an extension of the "twice the sum of the tail" formulas, well known in the time-invariant case. The approach relies on an operator-theoretic framework for analysis of linear time-varying systems. This provides a multivariable notion of frequency for such systems, which are thus characterized by rational functions of many complex variables, allowing the problem to be formulated in the linear-fractional framework. Using a time-varying version of standard necessary conditions for reduced-order modeling, based on convex operator inequalities, we show that these error-bounds for balanced truncation are related to the closest possible reduced-order modeling error in a sense which parallels the time-invariant case.

Carolyn L. Beck

Papers

Model Reduction of Multi-Dimensional and Uncertain Systems

Model reduction of multidimensional and uncertain systems

Synthesis of Place Transition Nets for Simulation and Control of Manufacturing Systems

Analysis and Control of a Continuous-Time Bi-Virus Model

Error-bounds for balanced model-reduction of linear time-varying systems