Fast $$(1+\epsilon )$$ -Approximation of the Löwner Extremal Matrices of High-Dimensional Symmetric Matrices

Nielsen, Frank; Nock, Richard

doi:10.1007/978-3-319-47058-0_6

Cited by 2 publications

(4 citation statements)

References 19 publications

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“…We describe two techniques that improve over this naive method: (1) an approximation technique using an exact formula for polytopal Hilbert geometries, and (2) an exact formula for the Birkhoff's projective metric using a whitening transformation for covariance matrices. 2 using half-vectorization of off-diagonal elements of the matrix [84]. Then compute the convex hull of s sample correlation matrix points p 1 , .…”

Section: Methodsmentioning

confidence: 99%

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Clustering in Hilbert simplex geometry

Nielsen,

Sun

2017

Preprint

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Clustering categorical distributions in the probability simplex is a fundamental task met in many applications dealing with normalized histograms. Traditionally, the differential-geometric structures of the probability simplex have been used either by (i) setting the Riemannian metric tensor to the Fisher information matrix of the categorical distributions, or (ii) defining the dualistic information-geometric structure induced by a smooth dissimilarity measure, the Kullback-Leibler divergence. In this work, we introduce for this clustering task a novel computationally-friendly framework for modeling the probability simplex termed Hilbert simplex geometry. In the Hilbert simplex geometry, the distance function is described by a polytope. We discuss the pros and cons of those different statistical modelings, and benchmark experimentally these geometries for center-based k-means and k-center clusterings. We show that Hilbert metric in the probability simplex satisfies the property of information monotonicity. Furthermore, since a canonical Hilbert metric distance can be defined on any bounded convex subset of the Euclidean space, we also consider Hilbert's projective geometry of the elliptope of correlation matrices and study its clustering performances.

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Section: Methodsmentioning

confidence: 99%

“…Let P + denote the pointed cone of positive semi-definite matrices. Cone P + defines a partial order (called Löwner ordering [84]):…”

Section: Methodsmentioning

confidence: 99%

Clustering in Hilbert simplex geometry

Nielsen,

Sun

2017

Preprint

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“…When the minimization is performed with respect to the Fröbenius distance, we can solve this problem using techniques of Euclidean computational geometry [ 33 , 47 ] by vectorizing the PSD matrices

into corresponding vectors

such that

, where

vectorizes a matrix by stacking its column vectors. In fact, since the matrices are symmetric, it is enough to half-vectorize the matrices:

, where

, see [ 50 ].…”

Section: The Smallest Enclosing Ball In the Spd Manifold And In Thmentioning

confidence: 99%

“…A real matrix

is said symmetric positive-definite (SPD) if and only if

for all

with

. This positive-definiteness property is written

, where ≻ denotes the partial Löwner ordering [ 50 ]. Let

be the space of real symmetric positive-definite matrices [ 10 , 44 , 51 , 52 ] of dimension

.…”

Section: Introductionmentioning

confidence: 99%

The Siegel–Klein Disk: Hilbert Geometry of the Siegel Disk Domain

Nielsen

2020

Entropy

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We study the Hilbert geometry induced by the Siegel disk domain, an open-bounded convex set of complex square matrices of operator norm strictly less than one. This Hilbert geometry yields a generalization of the Klein disk model of hyperbolic geometry, henceforth called the Siegel–Klein disk model to differentiate it from the classical Siegel upper plane and disk domains. In the Siegel–Klein disk, geodesics are by construction always unique and Euclidean straight, allowing one to design efficient geometric algorithms and data structures from computational geometry. For example, we show how to approximate the smallest enclosing ball of a set of complex square matrices in the Siegel disk domains: We compare two generalizations of the iterative core-set algorithm of Badoiu and Clarkson (BC) in the Siegel–Poincaré disk and in the Siegel–Klein disk: We demonstrate that geometric computing in the Siegel–Klein disk allows one (i) to bypass the time-costly recentering operations to the disk origin required at each iteration of the BC algorithm in the Siegel–Poincaré disk model, and (ii) to approximate fast and numerically the Siegel–Klein distance with guaranteed lower and upper bounds derived from nested Hilbert geometries.

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Fast $$(1+\epsilon )$$ -Approximation of the Löwner Extremal Matrices of High-Dimensional Symmetric Matrices

Cited by 2 publications

References 19 publications

Clustering in Hilbert simplex geometry

Clustering in Hilbert simplex geometry

The Siegel–Klein Disk: Hilbert Geometry of the Siegel Disk Domain

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