423 research outputs found
Pl\"ucker Coordinates of the best-fit Stiefel Tropical Linear Space to a Mixture of Gaussian Distributions
In this research, we investigate a tropical principal component analysis
(PCA) as a best-fit Stiefel tropical linear space to a given sample over the
tropical projective torus for its dimensionality reduction and visualization.
Especially, we characterize the best-fit Stiefel tropical linear space to a
sample generated from a mixture of Gaussian distributions as the variances of
the Gaussians go to zero. For a single Gaussian distribution, we show that the
sum of residuals in terms of the tropical metric with the max-plus algebra over
a given sample to a fitted Stiefel tropical linear space converges to zero by
giving an upper bound for its convergence rate. Meanwhile, for a mixtures of
Gaussian distribution, we show that the best-fit tropical linear space can be
determined uniquely when we send variances to zero. We briefly consider the
best-fit topical polynomial as an extension for the mixture of more than two
Gaussians over the tropical projective space of dimension three. We show some
geometric properties of these tropical linear spaces and polynomials.Comment: To appear in Information Geometr
Tropical Principal Component Analysis and its Application to Phylogenetics
Principal component analysis is a widely-used method for the dimensionality
reduction of a given data set in a high-dimensional Euclidean space. Here we
define and analyze two analogues of principal component analysis in the setting
of tropical geometry. In one approach, we study the Stiefel tropical linear
space of fixed dimension closest to the data points in the tropical projective
torus; in the other approach, we consider the tropical polytope with a fixed
number of vertices closest to the data points. We then give approximative
algorithms for both approaches and apply them to phylogenetics, testing the
methods on simulated phylogenetic data and on an empirical dataset of
Apicomplexa genomes.Comment: 28 page
A module-theoretic approach to matroids
Speyer recognized that matroids encode the same data as a special class of
tropical linear spaces and Shaw interpreted tropically certain basic matroid
constructions; additionally, Frenk developed the perspective of tropical linear
spaces as modules over an idempotent semifield. All together, this provides
bridges between the combinatorics of matroids, the algebra of idempotent
modules, and the geometry of tropical linear spaces. The goal of this paper is
to strengthen and expand these bridges by systematically developing the
idempotent module theory of matroids. Applications include a geometric
interpretation of strong matroid maps and the factorization theorem; a
generalized notion of strong matroid maps, via an embedding of the category of
matroids into a category of module homomorphisms; a monotonicity property for
the stable sum and stable intersection of tropical linear spaces; a novel
perspective of fundamental transversal matroids; and a tropical analogue of
reduced row echelon form.Comment: 22 pages; v3 minor corrections/clarifications; to appear in JPA
Presentations of transversal valuated matroids
Once this article is published by the JLMS, the Elements version will need to bear a statement of the following form: "This is the accepted version of the following article: FULL CITE, which has been published in final form at [Link to final article]"Once this article is published by the JLMS, the Elements version will need to bear a statement of the following form: "This is the accepted version of the following article: FULL CITE, which has been published in final form at [Link to final article]
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