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61
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 time-data 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.
Multivariate Pólya-Schur classification problems in the Weyl algebra
, 2008
"... A multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts. We classify all linear partial differential operators in the Weyl algebra An that preserve stability. An important tool that we develop in the process is the higher dimensional generalizat ..."
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Cited by 29 (9 self)
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A multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts. We classify all linear partial differential operators in the Weyl algebra An that preserve stability. An important tool that we develop in the process is the higher dimensional generalization of Pólya-Schur’s notion of multiplier sequence. We characterize all multivariate multiplier sequences as well as those of finite order. Next, we establish a multivariate extension of the Cauchy-Poincaré interlacing theorem and prove a natural analog of the Lax conjecture for real stable polynomials in two variables. Using the latter we describe all operators in A1 that preserve univariate hyperbolic polynomials by means of determinants and homogenized symbols. Our methods also yield homotopical properties for symbols of linear stability preservers and a duality theorem showing that an operator in An preserves stability if and only if its Fischer-Fock adjoint does. These are powerful multivariate extensions of the classical Hermite-Poulain-Jensen theorem, Pólya’s curve theorem and Schur-Maló-Szegö composition theorems. Examples, applications to strict stability preservers and further directions are also discussed.
Hyperbolic polynomials approach to van der Waerden and Schrijver-Valiant like Conjectures: sharper bounds, simpler proofs and algorithmic applications
- Proc. STOC 2006, preprint math.CO/0510452
"... Let p(x1,..., xn) = p(X), X ∈ Rn be a homogeneous polynomial of degree n in n real variables, e = (1, 1,.., 1) ∈ Rn be a vector of all ones. Such polynomial p is called e-hyperbolic if for all real vectors X ∈ Rn the univariate polynomial equation P(te − X) = 0 has all real roots λ1(X) ≥... ≥ λn ..."
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Cited by 28 (8 self)
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Let p(x1,..., xn) = p(X), X ∈ Rn be a homogeneous polynomial of degree n in n real variables, e = (1, 1,.., 1) ∈ Rn be a vector of all ones. Such polynomial p is called e-hyperbolic if for all real vectors X ∈ Rn the univariate polynomial equation P(te − X) = 0 has all real roots λ1(X) ≥... ≥ λn(X). The number of nonzero roots |{i: λi(X) ̸ = 0} | is called Rankp(X). A e-hyperbolic polynomial p is called POS-hyperbolic if roots of vectors X ∈ Rn + with nonnegative coordinates are also nonnegative (the orthant Rn + belongs to the hyperbolic cone) and p(e)> 0. Below {e1,..., en} stands for the canonical orthogonal basis in Rn. The main results states that if p(x1, x2,..., xn) is a POS-hyperbolic (homogeneous) polynomial of degree n, Rankp(ei) = Ri and p(x1, x2,..., xn) ≥ ∏ 1≤i≤n xi; xi> 0, 1 ≤ i ≤ n, then the following inequality holds ∂n p(0,..., 0) ≥
OBSTRUCTIONS TO DETERMINANTAL REPRESENTABILITY
, 2010
"... There has recently been ample interest in the question of which sets can be represented by linear matrix inequalities (LMIs). A necessary condition is that the set is rigidly convex, and it has been conjectured that rigid convexity is also sufficient. To this end Helton and Vinnikov conjectured th ..."
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Cited by 24 (2 self)
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There has recently been ample interest in the question of which sets can be represented by linear matrix inequalities (LMIs). A necessary condition is that the set is rigidly convex, and it has been conjectured that rigid convexity is also sufficient. To this end Helton and Vinnikov conjectured that any real zero polynomial admits a determinantal representation with symmetric matrices. We disprove this conjecture. By relating the question of finding LMI representations to the problem of determining whether a polymatroid is representable over the complex numbers, we find a real zero polynomial such that no power of it admits a determinantal representation. The proof uses recent results of Wagner and Wei on matroids with the half-plane property, and the polymatroids associated to hyperbolic polynomials introduced by Gurvits.
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.
Interior-point methods for optimization
, 2008
"... This article describes the current state of the art of interior-point methods (IPMs) for convex, conic, and general nonlinear optimization. We discuss the theory, outline the algorithms, and comment on the applicability of this class of methods, which have revolutionized the field over the last twen ..."
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Cited by 18 (0 self)
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This article describes the current state of the art of interior-point methods (IPMs) for convex, conic, and general nonlinear optimization. We discuss the theory, outline the algorithms, and comment on the applicability of this class of methods, which have revolutionized the field over the last twenty years.
Exposed faces of semidefinite representable sets
"... Abstract. A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite. Motivated by the fact that diagonal LMIs define polyhedra, the solution set of an LMI is called a spectrahedron. Linear images ..."
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Cited by 18 (5 self)
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Abstract. A linear matrix inequality (LMI) is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite. Motivated by the fact that diagonal LMIs define polyhedra, the solution set of an LMI is called a spectrahedron. Linear images of spectrahedra are called semidefinite representable sets. Part of the interest in spectrahedra and semidefinite representable sets arises from the fact that one can efficiently optimize linear functions on them by semidefinite programming, like one can do on polyhedra by linear programming. It is known that every face of a spectrahedron is exposed. This is also true in the general context of rigidly convex sets. We study the same question for semidefinite representable sets. Lasserre proposed a moment matrix method to construct semidefinite representations for certain sets. Our main result is that this method can only work if all faces of the considered set are exposed. This necessary condition complements sufficient conditions recently proved by Lasserre, Helton and Nie.
Invariance and Efficiency of Convex Representations
, 2005
"... We consider two notions for the representations of convex cones: G-representation and lifted-G-representation. The former represents a convex cone as a slice of another; the latter allows in addition, the usage of auxiliary variables in the representation. We first study the basic properties of thes ..."
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Cited by 11 (3 self)
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We consider two notions for the representations of convex cones: G-representation and lifted-G-representation. The former represents a convex cone as a slice of another; the latter allows in addition, the usage of auxiliary variables in the representation. We first study the basic properties of these representations. We show that some basic properties of convex cones are invariant under one notion of representation but not the other. In particular, we prove that lifted-G-representation is closed under duality when the representing cone is self-dual. We also prove that strict complementarity of a convex optimization problem in conic form is preserved under G-representations. Then we move to study efficiency measures for representations. We evaluate the representations of homogeneous convex cones based on the “smoothness” of the transformations mapping the central path of the representation to the central path of the represented optimization problem.
Detecting rigid convexity of bivariate polynomials
, 2009
"... Given a polynomial x ∈ R n ↦ → p(x) in n = 2 variables, a symbolic-numerical algorithm is first described for detecting whether the connected component of the plane sublevel set P = {x: p(x) ≥ 0} containing the origin is rigidly convex, or equivalently, whether it has a linear matrix inequality (LM ..."
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Cited by 11 (5 self)
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Given a polynomial x ∈ R n ↦ → p(x) in n = 2 variables, a symbolic-numerical algorithm is first described for detecting whether the connected component of the plane sublevel set P = {x: p(x) ≥ 0} containing the origin is rigidly convex, or equivalently, whether it has a linear matrix inequality (LMI) representation, or equivalently, if polynomial p(x) is hyperbolic with respect to the origin. The problem boils down to checking whether a univariate polynomial matrix is positive semidefinite, an optimization problem that can be solved with eigenvalue decomposition. When the variety C = {x: p(x) = 0} is an algebraic curve of genus zero, a second algorithm based on Bézoutians is proposed to detect whether P has an LMI representation and to build such a representation from a rational parametrization of C. Finally, some extensions to positive genus curves and to the case n> 2 are mentioned.
