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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(More)
Several problems arising in control system analysis and design, such as reduced order controller synthesis , involve minimizing the rank of a matrix variable subject to linear matrix inequality (LMI) constraints. Except in some special cases, solving this rank minimization problem (globally) is very difficult. One simple and surprisingly effective(More)
We present a heuristic for minimizing the rank of a positive semidefinite matrix over a convex set. We use the logarithm of the determinant as a smooth approximation for rank, and locally minimize this function to obtain a sequence of trace minimization problems. We then present a lemma that relates the rank of any general matrix to that of a corresponding(More)
The problem of minimizing the rank of a matrix subject to affine constraints has applications in several areas including machine learning, and is known to be NP-hard. A tractable relaxation for this problem is nuclear norm (or trace norm) minimization, which is guaranteed to find the minimum rank matrix under suitable assumptions. In this paper, we propose(More)
We consider the problem of portfolio selection, with transaction costs and constraints on exposure to risk. Linear transaction costs, bounds on the variance of the return, and bounds on different shortfall probabilities are efficiently handled by convex optimization methods. For such problems, the globally optimal portfolio can be computed very rapidly.(More)
for his contributions to the theory and algorithms for large-scale optimization. Abstract. We introduce a flexible optimization framework for nuclear norm minimization of matrices with linear structure, including Hankel, Toeplitz and moment structures, and catalog applications from diverse fields under this framework. We discuss various first-order methods(More)
— In this tutorial paper, we consider the problem of minimizing the rank of a matrix over a convex set. The Rank Minimization Problem (RMP) arises in diverse areas such as control, system identification, statistics and signal processing, and is known to be computationally NP-hard. We give an overview of the problem, its interpretations, applications, and(More)
Recovering structured models (e.g., sparse or group-sparse vectors, low-rank matrices) given a few linear observations have been well-studied recently. In various applications in signal processing and machine learning, the model of interest is structured in several ways, for example, a matrix that is simultaneously sparse and low rank. Often norms that(More)
—Inspired by the theory of compressed sensing and employing random channel access, we propose a distributed energy-efficient sensor network scheme denoted by Random Access Compressed Sensing (RACS). The proposed scheme is suitable for long-term deployment of large underwater networks , in which saving energy and bandwidth is of crucial importance. During(More)
—The matrix rank minimization problem consists of finding a matrix of minimum rank that satisfies given convex constraints. It is NP-hard in general and has applications in control , system identification, and machine learning. Reweighted trace minimization has been considered as an iterative heuristic for this problem. In this paper, we analyze the(More)