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The Power of Convex Relaxation: NearOptimal Matrix Completion
, 2009
"... This paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries. This is known as the matrix completion problem, and comes up in a great number of applications, including the famous Netflix Prize and other similar questions in collaborative filtering. In ..."
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Cited by 356 (7 self)
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to recover a matrix of rank r exactly by any method whatsoever (the information theoretic limit). More importantly, the paper shows that, under certain incoherence assumptions on the singular vectors of the matrix, recovery is possible by solving a convenient convex program as soon as the number of entries
Statistical Limits of Convex Relaxations
"... Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this paper, we study the statistical limits of convex relaxations. ..."
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Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this paper, we study the statistical limits of convex relaxations
Algorithms for simultaneous sparse approximation. Part II: Convex relaxation
, 2004
"... Abstract. A simultaneous sparse approximation problem requests a good approximation of several input signals at once using different linear combinations of the same elementary signals. At the same time, the problem balances the error in approximation against the total number of elementary signals th ..."
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Cited by 365 (5 self)
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problems. The second part of the paper develops another algorithmic approach called convex relaxation, and it provides theoretical results on the performance of convex relaxation for simultaneous sparse approximation. Date: Typeset on March 17, 2005. Key words and phrases. Greedy algorithms, Orthogonal
Convex relaxations for permutation problems
"... Abstract. Seriation seeks to reconstruct a linear order between variables using unsorted, pairwise similarity information. It has direct applications in archeology and shotgun gene sequencing, for example. We write seriation as an optimization problem by proving the equivalence between the seriatio ..."
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Cited by 6 (2 self)
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the seriation and combinatorial 2SUM problems on similarity matrices (2SUM is a quadratic minimization problem over permutations). The seriation problem can be solved exactly by a spectral algorithm in the noiseless case, and we derive several convex relaxations for 2SUM to improve the robustness
On convex relaxations of quadrilinear terms
, 2009
"... The best known method to find exact or at least εapproximate solutions to polynomial programming problems is the spatial BranchandBound algorithm, which rests on computing lower bounds to the value of the objective function to be minimized on each region that it explores. These lower bounds are o ..."
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Cited by 11 (8 self)
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are often computed by solving convex relaxations of the original program. Although convex envelopes are explicitly known (via linear inequalities) for bilinear and trilinear terms on arbitrary boxes, such a description is unknown, in general, for multilinear terms of higher order. In this paper, we study
Convex Relaxations for Subset Selection
, 2010
"... We use convex relaxation techniques to produce lower bounds on the optimal value of subset selection problems and generate good approximate solutions. We then explicitly bound the quality of these relaxations by studying the approximation ratio of sparse eigenvalue relaxations. Our results are used ..."
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Cited by 4 (0 self)
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We use convex relaxation techniques to produce lower bounds on the optimal value of subset selection problems and generate good approximate solutions. We then explicitly bound the quality of these relaxations by studying the approximation ratio of sparse eigenvalue relaxations. Our results are used
Convex Relaxations and Integrality Gaps
"... We discuss the effectiveness of linear and semidefinite relaxations in approximating the optimum for combinatorial optimization problems. Various hierarchies of these relaxations, such as the ones defined by Lovász and Schrijver [47], Sherali and Adams [55] and Lasserre [42] generate increasingly st ..."
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Cited by 20 (0 self)
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We discuss the effectiveness of linear and semidefinite relaxations in approximating the optimum for combinatorial optimization problems. Various hierarchies of these relaxations, such as the ones defined by Lovász and Schrijver [47], Sherali and Adams [55] and Lasserre [42] generate increasingly
Convex relaxation of combinatorial penalties
, 2011
"... In this paper, we propose an unifying view of several recently proposed structured sparsityinducing norms. We consider the situation of a model simultaneously (a) penalized by a setfunction defined on the support of the unknown parameter vector which represents prior knowledge on supports, and (b) r ..."
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Cited by 13 (9 self)
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) regularized in ℓpnorm. We show that the natural combinatorial optimization problems obtained may be relaxed into convex optimization problems and introduce a notion, the lower combinatorial envelope of a setfunction, that characterizes the tightness of our relaxations. We moreover establish links with norms
Convex relaxations of latent variable training
 In Advances in Neural Information Processing Systems 20
, 2007
"... We investigate a new, convex relaxation of an expectationmaximization (EM) variant that approximates a standard objective while eliminating local minima. First, a cautionary result is presented, showing that any convex relaxation of EM over hidden variables must give trivial results if any dependen ..."
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Cited by 13 (6 self)
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We investigate a new, convex relaxation of an expectationmaximization (EM) variant that approximates a standard objective while eliminating local minima. First, a cautionary result is presented, showing that any convex relaxation of EM over hidden variables must give trivial results if any
Direct convex relaxations of sparse svm
 in ICML ’07: Proceedings of the 24th international conference on Machine learning
"... Although support vector machines (SVMs) for binary classification give rise to a decision rule that only relies on a subset of the training data points (support vectors), it will in general be based on all available features in the input space. We propose two direct, novel convex relaxations of a no ..."
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Cited by 25 (0 self)
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Although support vector machines (SVMs) for binary classification give rise to a decision rule that only relies on a subset of the training data points (support vectors), it will in general be based on all available features in the input space. We propose two direct, novel convex relaxations of a
Results 1  10
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