Pl\"ucker Coordinates of the best-fit Stiefel Tropical Linear Space to a Mixture of Gaussian Distributions

TL;DR

This paper explores tropical PCA to find the best-fit tropical linear space for data from Gaussian mixtures, analyzing convergence properties and geometric features in tropical projective space.

Contribution

It characterizes the best-fit tropical linear space for Gaussian mixtures as variances approach zero and extends to tropical polynomials for multiple Gaussians.

Findings

Residuals converge to zero for single Gaussian as variance decreases.

Unique best-fit tropical linear space exists for Gaussian mixtures with small variances.

Geometric properties of tropical linear spaces and polynomials are analyzed.

Abstract

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.