2 citations · 3 across the 11 of their papers we have counts for
11 papers
Dual-connection midpoint matrix means and their Gauss composition
Frank Nielsen, Kazuki Okamura
Nakamura [J. Comput. Appl. Math. 131 (2001)] proved that the arithmetic--harmonic matrix iteration converges quadratically to the geometric matrix mean, the Riemannian midpoint of…
Group invariance of -divergences and the Fisher--Rao distance
Frank Nielsen, Kazuki Okamura
Many statistical models have natural symmetries described by a group action. We study how such symmetries affect the comparison of two distributions. We work with a transformation…
Neural Legendre-Fenchel transform with Hessian Preconditioning
Basile Plus-Gourdon, Frank Nielsen
The Legendre-Fenchel (LF) transform is a fundamental tool in convex analysis and machine learning that maps lower semi-continuous functions to their convex conjugates. In practice,…
Generalising maximum mean discrepancy: kernelised functional Bregman divergences
Russell Tsuchida, Frank Nielsen
Bregman divergences play a pivotal role in statistics, machine learning and computational information geometry. Particularly in the context of machine learning, they are central to…
Quadratic polarity and polar Fenchel-Young divergences from the canonical Legendre polarity
Frank Nielsen, Basile Plus-Gourdon, Mahito Sugiyama
Polarity is a fundamental reciprocal duality of -dimensional projective geometry which associates to points polar hyperplanes, and more generally -dimensional convex bodies t…
Geometric structures and deviations on James' symmetric positive-definite matrix bicone domain
Jacek Karwowski, Frank Nielsen
Symmetric positive-definite (SPD) matrix datasets play a central role across numerous scientific disciplines, including signal processing, statistics, finance, computer vision, inf…
A note on the Artstein-Avidan-Milman's generalized Legendre transforms
Frank Nielsen
Artstein-Avidan and Milman [Annals of mathematics (2009), (169):661-674] characterized invertible reverse-ordering transforms on the space of lower semi-continuous extended real-va…
Mirror Descent and Novel Exponentiated Gradient Algorithms Using Trace-Form Entropies and Deformed Logarithms
Andrzej Cichocki, Toshihisa Tanaka, Frank Nielsen +1
This paper introduces a broad class of Mirror Descent (MD) and Generalized Exponentiated Gradient (GEG) algorithms derived from trace-form entropies defined via deformed logarithms…
Fast proxy centers for Jeffreys centroids: The Jeffreys-Fisher-Rao and the inductive Gauss-Bregman centers
Frank Nielsen
The symmetric Kullback-Leibler centroid also called the Jeffreys centroid of a set of mutually absolutely continuous probability distributions on a measure space provides a notion…
Symplectic Bregman divergences
Frank Nielsen
We present a generalization of Bregman divergences in symplectic vector spaces that we term symplectic Bregman divergences. Symplectic Bregman divergences are derived from a symple…
Divergences induced by dual subtractive and divisive normalizations of exponential families and their convex deformations
Frank Nielsen
Exponential families are statistical models which are the workhorses in statistics, information theory, and machine learning among others. An exponential family can either be norma…