arXiv subjects

Keiji Miura

Publications and source records attributed to Keiji Miura.

10 recordsLinked to original sources

arXiv2026

Tropical Fermat-Weber Polytropes

We study the geometry of tropical Fermat--Weber points, that is, optimal solutions to a location problem over a projective space using a dissimilarity measure derived from the tropical metric. It is well-known that for a given sample, such points are not necessarily unique, and we show that the set of all possible Fermat--Weber points forms a polytrope. This follows from the fact that our location problem turns out to be dual to a particular minimum-cost flow problem, and we describe the polytrope of optimal locations in the terminology of tropical geometry. We also provide a simple gradient descent algorithm that converges to the Fermat--Weber polytrope.

math.CO↗

arXiv2024

Tropical Decision Boundaries for Neural Networks Are Robust Against Adversarial Attacks

We introduce a simple, easy to implement, and computationally efficient tropical convolutional neural network architecture that is robust against adversarial attacks. We exploit the tropical nature of piece-wise linear neural networks by embedding the data in the tropical projective torus in a single hidden layer which can be added to any model. We study the geometry of its decision boundary theoretically and show its robustness against adversarial attacks on image datasets using computational experiments.

cs.LG↗

arXiv2023

Tropical Geometric Tools for Machine Learning: the TML package

In the last decade, developments in tropical geometry have provided a number of uses directly applicable to problems in statistical learning. The TML package is the first R package which contains a comprehensive set of tools and methods used for basic computations related to tropical convexity, visualization of tropically convex sets, as well as supervised and unsupervised learning models using the tropical metric under the max-plus algebra over the tropical projective torus. Primarily, the TML package employs a Hit and Run Markov chain Monte Carlo sampler in conjunction with the tropical metric as its main tool for statistical inference. In addition to basic computation and various applications of the tropical HAR sampler, we also focus on several supervised and unsupervised methods incorporated in the TML package including tropical principal component analysis, tropical logistic regression and tropical kernel density estimation.

stat.ML↗

arXiv2023

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.

cs.DM↗

arXiv2023

Tropical Density Estimation of Phylogenetic Trees

Much evidence from biological theory and empirical data indicates that, gene tree, phylogenetic trees reconstructed from different genes (loci), do not have to have exactly the same tree topologies. Such incongruence between gene trees might be caused by some ``unusual'' evolutionary events, such as meiotic sexual recombination in eukaryotes or horizontal transfers of genetic material in prokaryotes. However, most of gene trees are constrained by the tree topology of its species tree, that is, the phylogenetic tree of a given species following their evolutionary history. In order to discover ``outlying'' gene trees which do not follow the ``main distribution(s)'' of trees, we propose to apply the ``tropical metric'' with the max-plus algebra from tropical geometry to a non-parametric estimation of gene trees over the space of phylogenetic trees. In this research we apply the ``tropical metric,'' a well-defined metric over the space of phylogenetic trees under the max-plus algebra, to non-parametric estimation of gene trees distribution over the tree space. Kernel density estimator (KDE) is one of the most popular non-parametric estimation of a distribution from a given sample, and we propose an analogue of the classical KDE in the setting of tropical geometry with the tropical metric which measures the length of an intrinsic geodesic between trees over the tree space. We estimate the probability of an observed tree by empirical frequencies of nearby trees, with the level of influence determined by the tropical metric. Then, with simulated data generated from the multispecies coalescent model, we show that the non-parametric estimation of gene tree distribution using the tropical metric performs better than one using the Billera-Holmes-Vogtmann (BHV) metric developed by Weyenberg et al. in terms of computational times and accuracy. We then apply it to Apicomplexa data.

q-bio.PE↗

arXiv2023

Maximum Inscribed and Minimum Enclosing Tropical Balls of Tropical Polytopes and Applications to Volume Estimation and Uniform Sampling

We consider a minimum enclosing and maximum inscribed tropical balls for any given tropical polytope over the tropical projective torus in terms of the tropical metric with the max-plus algebra. We show that we can obtain such tropical balls via linear programming. Then we apply minimum enclosing and maximum inscribed tropical balls of any given tropical polytope to estimate the volume of and sample uniformly from the tropical polytope.

math.CO↗

arXiv2023

Plücker 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.

math.CO↗

arXiv2022

Tropical Support Vector Machines: Evaluations and Extension to Function Spaces

Support Vector Machines (SVMs) are one of the most popular supervised learning models to classify using a hyperplane in an Euclidean space. Similar to SVMs, tropical SVMs classify data points using a tropical hyperplane under the tropical metric with the max-plus algebra. In this paper, first we show generalization error bounds of tropical SVMs over the tropical projective torus. While the generalization error bounds attained via Vapnik-Chervonenkis (VC) dimensions in a distribution-free manner still depend on the dimension, we also show numerically and theoretically by extreme value statistics that the tropical SVMs for classifying data points from two Gaussian distributions as well as empirical data sets of different neuron types are fairly robust against the curse of dimensionality. Extreme value statistics also underlie the anomalous scaling behaviors of the tropical distance between random vectors with additional noise dimensions. Finally, we define tropical SVMs over a function space with the tropical metric.

cs.LG↗

arXiv2022

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.

math.CO↗

arXiv2004

Pulse-coupled resonate-and-fire models

We analyze two pulse-coupled resonate-and-fire neurons. Numerical simulation reveals that an anti-phase state is an attractor of this model. We can analytically explain the stability of anti-phase states by means of a return map of firing times, which we propose in this paper. The resultant stability condition turns out to be quite simple. The phase diagram based on our theory shows that there are two types of anti-phase states. One of these cannot be seen in coupled integrate-and-fire models and is peculiar to resonate-and-fire models. The results of our theory coincide with those of numerical simulations.

q-bio.NC↗