564 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
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
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