Multivariate Gaussians, semidefinite matrix completion, and convex algebraic geometry

Sturmfels, Bernd; Uhler, Caroline

doi:10.1007/s10463-010-0295-4

Cited by 61 publications

(104 citation statements)

References 23 publications

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“…We study the exceptional cases and give a complete semi-algebraic description of the exceptional families of extreme points in terms of convex duality (normal cones) and a computational way of getting a list of potentially exceptional strata from the algebraic boundary of the dual. This proves an assertion made by Sturmfels and Uhler in [17] Proposition 2.4.…”

Section: Let Z Be An Irreducible Component Of the Zariski Closure Of supporting

confidence: 83%

“…This class includes prominent families such as the moment matrices of probability distributions and the highly symmetric orbitopes. It does not include examples such as hyperbolicity cones and spectrahedra, which have received attention from applications of semi-definite programming in polynomial optimisation, see [2] and [19], and statistics of Gaussian graphical models, see [17].…”

Section: Introductionmentioning

confidence: 99%

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Algebraic boundaries of convex semi-algebraic sets

Sinn

2015

Res Math Sci

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We study the algebraic boundary of a convex semi-algebraic set via duality in convex and algebraic geometry. We generalise the correspondence of facets of a polytope with the vertices of the dual polytope to general semi-algebraic convex sets. In this case, exceptional families of extreme points might exist and we characterise them semi-algebraically. We also give a strategy for computing a complete list of exceptional families, given the algebraic boundary of the dual convex set.

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Section: Let Z Be An Irreducible Component Of the Zariski Closure Of supporting

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Algebraic boundaries of convex semi-algebraic sets

Sinn

2015

Res Math Sci

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“…In this presentation we consider mathematical procedures for estimating these sequence-dependent parameter sets. While we describe our results within the specific context of the cgDNA model, the maximum entropy approach to enforcing a prescribed sparsity pattern in the stiffness (or precision) matrix in a Gaussian is potentially also of wider interest [3,6,10,36,38]. For example, in the specific application fields of numerical weather prediction and data assimilation, both sparse covariance and sparse inverse covariance (or precision) matrix estimates are adopted using other techniques such as tapering [2,5,11,33,37].…”

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confidence: 99%

“…Moreover, the maximum absolute entropy fit can be constructed using a simple, local inversion algorithm [12], whereas the relative entropy fit requires numerical optimization techniques. We note that in the absolute entropy case, the maximization problem reduces to a matrix completion problem that has been previously studied [3,6,10,14,19,22,36,38]. Furthermore, although we only consider means and covariances in this work, higher-order moments are also of interest, and it is understood that these could be accommodated in the maximum absolute entropy approach in a natural way [16,17,18].…”

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confidence: 99%

“…Estimation via maximum absolute entropy. Both absolute and relative entropies have been widely employed in various parameter estimation methods in statistics [3,6,10,36,38] and statistical mechanics [16,17,18]. For instance, the maximization of the absolute entropy D abs (ρ m ) over a class of model densities ρ m yields a best-fit density ρ within the class; such a density is a best-fit in the sense that it maximizes entropy, and can be understood as being a most uniform (or least biased) density in the class.…”

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confidence: 99%

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Absolute versus Relative Entropy Parameter Estimation in a Coarse-Grain Model of DNA

González¹,

Pasi²,

Petkevičiūtė³

et al. 2017

Multiscale Model. Simul.

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Abstract. Maximum entropy procedures for estimating coarse-grain parameters from molecular dynamics (MD) simulation data are considered within the specific context of the sequence-dependent cgDNA rigid-base model of DNA. We describe a quite general approach that exploits a maximum absolute entropy principle to fit an observed matrix of covariances subject to the constraint of only allowing a prescribed sparsity pattern of nearest-neighbor interactions in the free energy. We also allow indefinite local stiffness-matrix parameter blocks that nevertheless always generate a positive-definite model stiffness matrix. Beginning from a database of atomic-resolution MD simulations of a library of short DNA oligomers in explicit solvent, these procedures deliver a complete parameter set for the cgDNA model. Due to the intrinsic linear structure of DNA and the convergence characteristics of the MD time series data, the maximum absolute entropy parameter set yields significantly improved predictions of persistence lengths, when compared to a previous parameter set that was fit to the same MD data, but using a maximum relative entropy fitting principle and local stiffness-matrix parameter blocks that were constrained to be semidefinite.Key words. entropy principles, parameter estimation, molecular modeling, statistical mechanics AMS subject classifications. 62H12, 82B05, 82D60, 15A83, 15A09 DOI. 10.1137/16M10860911. Introduction. An important problem in molecular biology is to understand how the mechanical properties of DNA depend on the sequence of bases along its two backbones. Properties that influence bending, twisting, shearing, and stretching in different directions along and across the two strands are believed to be essential in various biological processes such as DNA looping [34], nucleosome positioning [35], and other DNA-protein interactions, and gene regulation [26], all of which depend on the probability of DNA to adopt various three-dimensional configurations [4]. Consequently models at differing length scales are needed to quantify how the mechanical properties of DNA depend upon its sequence. The intermediate scales of a few tens to a few hundreds of base pairs are of particular biological interest. The study of sequence-dependent effects at such scales requires the development of specialized coarse-grain models, because, with contemporary computational resources, all-atom molecular dynamics (MD) simulations at these lengths remain intensive, particularly given the large number of possible sequences, while sequence-dependent behavior is

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Maximum Likelihood Estimates for Gaussian Mixtures Are Transcendental

Améndola

Drton

Sturmfels

2016

Lecture Notes in Computer Science

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Gaussian mixture models are central to classical statistics, widely used in the information sciences, and have a rich mathematical structure. We examine their maximum likelihood estimates through the lens of algebraic statistics. The MLE is not an algebraic function of the data, so there is no notion of ML degree for these models. The critical points of the likelihood function are transcendental, and there is no bound on their number, even for mixtures of two univariate Gaussians.

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Multivariate Gaussians, semidefinite matrix completion, and convex algebraic geometry

Cited by 61 publications

References 23 publications

Algebraic boundaries of convex semi-algebraic sets

Algebraic boundaries of convex semi-algebraic sets

Absolute versus Relative Entropy Parameter Estimation in a Coarse-Grain Model of DNA

Maximum Likelihood Estimates for Gaussian Mixtures Are Transcendental

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