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In this research, we investigate a tropical principal component analysis(PCA) as a best-fit Stiefel tropical linear space to a given sample over thetropical projective torus for its dimensionality reduction and visualization.Especially, we characterize the best-fit Stiefel tropical linear space to asample generated from a mixture of Gaussian distributions as the variances ofthe Gaussians go to zero. For a single Gaussian distribution, we show that thesum of residuals in terms of the tropical metric with the max-plus algebra overa given sample to a fitted Stiefel tropical linear space converges to zero bygiving an upper bound for its convergence rate. Meanwhile, for a mixtures ofGaussian distribution, we show that the best-fit tropical linear space can bedetermined uniquely when we send variances to zero. We briefly consider thebest-fit topical polynomial as an extension for the mixture of more than twoGaussians over the tropical projective space of dimension three. We show somegeometric properties of these tropical linear spaces and polynomials.

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