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43
The Convex Geometry of Linear Inverse Problems
, 2010
"... In applications throughout science and engineering one is often faced with the challenge of solving an illposed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constr ..."
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Cited by 189 (20 self)
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In applications throughout science and engineering one is often faced with the challenge of solving an illposed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constrained structurally so that they only have a few degrees of freedom relative to their ambient dimension. This paper provides a general framework to convert notions of simplicity into convex penalty functions, resulting in convex optimization solutions to linear, underdetermined inverse problems. The class of simple models considered are those formed as the sum of a few atoms from some (possibly infinite) elementary atomic set; examples include wellstudied cases such as sparse vectors (e.g., signal processing, statistics) and lowrank matrices (e.g., control, statistics), as well as several others including sums of a few permutations matrices (e.g., ranked elections, multiobject tracking), lowrank tensors (e.g., computer vision, neuroscience), orthogonal matrices (e.g., machine learning), and atomic measures (e.g., system identification). The convex programming formulation is based on minimizing the norm induced by the convex hull of the atomic set; this norm is referred to as the atomic norm. The facial
Linear vs. Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds
, 2012
"... We solve a 20year old problem posed by Yannakakis and prove that there exists no polynomialsize linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope an ..."
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Cited by 51 (11 self)
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We solve a 20year old problem posed by Yannakakis and prove that there exists no polynomialsize linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between oneway quantum communication protocols and semidefinite programming reformulations of LPs.
Computational and Statistical Tradeoffs via Convex Relaxation
, 2012
"... In modern data analysis, one is frequently faced with statistical inference problems involving massive datasets. Processing such large datasets is usually viewed as a substantial computational challenge. However, if data are a statistician’s main resource then access to more data should be viewed as ..."
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Cited by 45 (1 self)
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In modern data analysis, one is frequently faced with statistical inference problems involving massive datasets. Processing such large datasets is usually viewed as a substantial computational challenge. However, if data are a statistician’s main resource then access to more data should be viewed as an asset rather than as a burden. In this paper we describe a computational framework based on convex relaxation to reduce the computational complexity of an inference procedure when one has access to increasingly larger datasets. Convex relaxation techniques have been widely used in theoretical computer science as they give tractable approximation algorithms to many computationally intractable tasks. We demonstrate the efficacy of this methodology in statistical estimation in providing concrete timedata tradeoffs in a class of denoising problems. Thus, convex relaxation offers a principled approach to exploit the statistical gains from larger datasets to reduce the runtime of inference algorithms.
Lifts of convex sets and cone factorizations
 Mathematics of OR
"... Abstract. In this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or “lift ” of the convex set is especially useful if the cone admits an efficient algorithm for linear op ..."
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Cited by 21 (8 self)
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Abstract. In this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or “lift ” of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices. We show that the existence of a lift of a convex set to a cone is equivalent to the existence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual. This generalizes a theorem of Yannakakis that established a connection between polyhedral lifts of a polytope and nonnegative factorizations of its slack matrix. Symmetric lifts of convex sets can also be characterized similarly. When the cones live in a family, our results lead to the definition of the rank of a convex set with respect to this family. We present results about this rank in the context of cones of positive semidefinite matrices. Our methods provide new tools for understanding cone lifts of convex sets. 1.
Positive polynomials and projections of spectrahedra
, 2010
"... This work is concerned with different aspects of spectrahedra and their projections, sets that are important in semidefinite optimization. We prove results on the limitations of so called Lasserre and theta body relaxation methods for semialgebraic sets and varieties. As a special case we obtain th ..."
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Cited by 18 (1 self)
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This work is concerned with different aspects of spectrahedra and their projections, sets that are important in semidefinite optimization. We prove results on the limitations of so called Lasserre and theta body relaxation methods for semialgebraic sets and varieties. As a special case we obtain the main result of [17] on nonexposed faces. We also solve the open problems from that work. We further prove some helpful facts which can not be found in the existing literature, for example that the closure of a projection of a spectrahedron is again such a projection. We give a unified account of several results on convex hulls of curves and images of polynomial maps. We finally prove a Positivstellensatz for projections of spectrahedra, which exceeds the known results that only work for basic closed semialgebraic sets.
Discriminants and Nonnegative Polynomials
, 2010
"... For a semialgebraic set K in R n, let Pd(K) = {f ∈ R[x]≤d: f(u) ≥ 0 ∀ u ∈ K} be the cone of polynomials in x ∈ R n of degrees at most d that are nonnegative on K. This paper studies the geometry of its boundary ∂Pd(K). When K = R n and d is even, we show that its boundary ∂Pd(K) lies on the irredu ..."
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Cited by 14 (3 self)
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For a semialgebraic set K in R n, let Pd(K) = {f ∈ R[x]≤d: f(u) ≥ 0 ∀ u ∈ K} be the cone of polynomials in x ∈ R n of degrees at most d that are nonnegative on K. This paper studies the geometry of its boundary ∂Pd(K). When K = R n and d is even, we show that its boundary ∂Pd(K) lies on the irreducible hypersurface defined by the discriminant ∆(f) of f. When K = {x ∈ R n: g1(x) = · · · = gm(x) = 0} is a real algebraic variety, we show that ∂Pd(K) lies on the hypersurface defined by the discriminant ∆(f, g1,..., gm) of f, g1,...,gm. When K is a general semialgebraic set, we show that ∂Pd(K) lies on a union of hypersurfaces defined by the discriminantal equations. Explicit formulae for the degrees of these hypersurfaces and discriminants are given. We also prove that typically Pd(K) does not have a barrier of type − log ϕ(f) when ϕ(f) is required to be a polynomial, but such a barrier exits if ϕ(f) is allowed to be semialgebraic. Some illustrating examples are shown.
Convex hulls of algebraic sets
, 2010
"... This article describes a method to compute successive convex approximations of the convex hull of a set of points in Rn that are the solutions to a system of polynomial equations over the reals. The method relies on sums of squares of polynomials and the dual theory of moment matrices. The main fea ..."
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Cited by 10 (1 self)
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This article describes a method to compute successive convex approximations of the convex hull of a set of points in Rn that are the solutions to a system of polynomial equations over the reals. The method relies on sums of squares of polynomials and the dual theory of moment matrices. The main feature of the technique is that all computations are done modulo the ideal generated by the polynomials defining the set to the convexified. This work was motivated by questions raised by Lovász concerning extensions of the theta body of a graph to arbitrary real algebraic varieties, and hence the relaxations described here are called theta bodies. The convexification process can be seen as an incarnation of Lasserre’s hierarchy of convex relaxations of a semialgebraic set in Rn. When the defining ideal is real radical the results become especially nice. We provide several examples of the method and discuss convergence issues. Finite convergence, especially after the first step of the method, can be described explicitly for finite point sets.