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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 heuristic,(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)
The classical compressed sensing problem is to find the sparsest solution to an underdetermined system of linear equations. A good convex approximation to this problem is to minimize the &#x2113;<inf>1</inf> norm subject to affine constraints. The Iterative Reweighted Least Squares (IRLSp) algorithm (0 &#38;#60; p &#x2264; 1), has been proposed as a method(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)
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)
Recovering sparse vectors and low-rank matrices from noisy linear measurements has been the focus of much recent research. Various reconstruction algorithms have been studied, including &#x2113;<inf>1</inf> and nuclear norm minimization as well as &#x2113;<inf>p</inf> minimization with p &#38;#60; 1. These algorithms are known to succeed if certain(More)
We propose clustering samples given their pairwise similarities by factorizing the similarity matrix into the product of a cluster probability matrix and its transpose. We propose a rotation-based algorithm to compute this left-stochastic decomposition (LSD). Theoretical results link the LSD clustering method to a soft kernel k-means clustering, give(More)