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

Petrified Geobacillus thermoglucosidasius colony to strontianite

When biomass of the thermophilic bacteria Geobacillus thermoglucosidasius is brought into contact with a hydrogel containing sodium acetate and strontium, the biomass petrifies and hardens, becoming a mineralized thin film after incubation at 60˚C for 72 h. Analysis by energy dispersive X-ray and X-ray diffraction shows that the mineralized thin film is strontianite. This is the first report of biomass completely changing to strontianite. Strontianite of thermophilic bacterial origin may be formed in the hydrothermal oligotrophic environment of the deep subsurface. DOI: http://dx.doi.org/10.5281/zenodo.114670

Tropical neural networks and its applications to classifying phylogenetic trees

Deep neural networks show great success when input vectors are in an Euclidean space. However, those classical neural networks show a poor performance when inputs are phylogenetic trees, which can be written as vectors in the tropical projective torus. Here we propose tropical embedding to transform a vector in the tropical projective torus to a vector in the Euclidean space via the tropical metric. We introduce a tropical neural network where the first layer is a tropical embedding layer and the following layers are the same as the classical ones. We prove that this neural network with the tropical metric is a universal approximator and we derive a backpropagation rule for deep neural networks. Then we provide TensorFlow 2 codes for implementing a tropical neural network in the same fashion as the classical one, where the weights initialization problem is considered according to the extreme value statistics. We apply our method to empirical data including sequences of hemagglutinin for influenza virus from New York. Finally we show that a tropical neural network can be interpreted as a generalization of a tropical logistic regression

Slope Traversal Experiments with Slip Compensation Control for Lunar/Planetary Exploration Rover

2008 IEEE International Conference on Robotics and Automation, Pasadena, CA, USA, May 19-23, 200

Hit and Run Sampling from Tropically Convex Sets

In this paper we propose Hit and Run (HAR) sampling from a tropically convex set. The key ingredient of HAR sampling from a tropically convex set is sampling uniformly from a tropical line segment over the tropical projective torus, which runs linearly in its computational time complexity. We show that this HAR sampling method samples uniformly from a tropical polytope which is the smallest tropical convex set of finitely many vertices. Finally, we apply this novel method to any given distribution using Metropolis-Hasting filtering over a tropical polytope