arXiv subjects

Ruriko Yoshida

Publications and source records attributed to Ruriko Yoshida.

At least 37 records · Page 2Linked to original sources

arXiv2023

Imputing phylogenetic trees using tropical polytopes over the space of phylogenetic trees

When we apply comparative phylogenetic analyses to genome data, it is a well-known problem and challenge that some of given species (or taxa) often have missing genes. In such a case, we have to impute a missing part of a gene tree from a sample of gene trees. In this short paper we propose a novel method to infer a missing part of a phylogenetic tree using an analogue of a classical linear regression in the setting of tropical geometry. In our approach, we consider a tropical polytope, a convex hull with respect to the tropical metric closest to the data points. We show a condition that we can guarantee that an estimated tree from our method has at most four Robinson-Foulds (RF) distance from the ground truth and computational experiments with simulated data show our method works well.

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

Connecting Tables with Allowing Negative Cell Counts

It is well-known that computing a Markov basis for a discrete loglinear model is very hard in general. Thus, we focus on connecting tables in a fiber via a subset of a Markov basis and in this paper, we consider connecting tables if we allow cell counts in each tale to be $-1$. In this paper we show that if a subset of a Markov basis connects all tables in the fiber which contains a table with all ones, then moves in this subset connect tables in the fiber if we allow cell counts to be $-1$. In addition, we show that in some cases under the no-three-way interaction model, we can connect tables by all basic moves of $2 \times 2 \times 2$ minors with allowing $X_{ijk} \geq -1$. We then apply this Markov Chain Monte Carlo (MCMC) scheme to an empirical data on Naval officer and enlisted population. Our computational experiments show it works well and we end with the conjecture on the no-three-way interaction model.

stat.ME↗

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↗

arXiv2022

Tropical Geometry of Phylogenetic Tree Space: A Statistical Perspective

Phylogenetic trees are the fundamental mathematical representation of evolutionary processes in biology. They are also objects of interest in pure mathematics, such as algebraic geometry and combinatorics, due to their discrete geometry. Although they are important data structures, they face the significant challenge that sets of trees form a non-Euclidean phylogenetic tree space, which means that standard computational and statistical methods cannot be directly applied. In this work, we explore the statistical feasibility of a pure mathematical representation of the set of all phylogenetic trees based on tropical geometry for both descriptive and inferential statistics, and unsupervised and supervised machine learning. Our exploration is both theoretical and practical. We show that the tropical geometric phylogenetic tree space endowed with a generalized Hilbert projective metric exhibits analytic, geometric, and topological properties that are desirable for theoretical studies in probability and statistics and allow for well-defined questions to be posed. We illustrate the statistical feasibility of the tropical geometric perspective for phylogenetic trees with an example of both a descriptive and inferential statistical task. Moreover, this approach exhibits increased computational efficiency and statistical performance over the current state-of-the-art, which we illustrate with a real data example on seasonal influenza. Our results demonstrate the viability of the tropical geometric setting for parametric statistical and probabilistic studies of sets of phylogenetic trees.

math.MG↗

arXiv2022

SARS-CoV-2 Dissemination using a Network of the United States Counties

During 2020 and 2021, severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) transmission has been increasing amongst the world's population at an alarming rate. Reducing the spread of SARS-CoV-2 and other diseases that are spread in similar manners is paramount for public health officials as they seek to effectively manage resources and potential population control measures such as social distancing and quarantines. By analyzing the United States' county network structure, one can model and interdict potential higher infection areas. County officials can provide targeted information, preparedness training, as well as increase testing in these areas. While these approaches may provide adequate countermeasures for localized areas, they are inadequate for the holistic United States. We solve this problem by collecting coronavirus disease 2019 (COVID-19) infections and deaths from the Center for Disease Control and Prevention{\color{black},} and {\color{black} adjacency between all counties obtained} from the United States Census Bureau. Generalized network autoregressive (GNAR) time series models have been proposed as an efficient learning algorithm for networked datasets. This work fuses network science and operations research techniques to univariately model COVID-19 cases, deaths, and current survivors across the United States' county network structure.

cs.SI↗

arXiv2022

Tropical Geometric Variation of Phylogenetic Tree Shapes

We study the behavior of phylogenetic tree shapes in the tropical geometric interpretation of tree space. Tree shapes are formally referred to as tree topologies; a tree topology can also be thought of as a tree combinatorial type, which is given by the tree's branching configuration and leaf labeling. We use the tropical line segment as a framework to define notions of variance as well as invariance of tree topologies: we provide a combinatorial search theorem that describes all tree topologies occurring along a tropical line segment, as well as a setting under which tree topologies do not change along a tropical line segment. Our study is motivated by comparison to the moduli space endowed with a geodesic metric proposed by Billera, Holmes, and Vogtmann (referred to as BHV space); we consider the tropical geometric setting as an alternative framework to BHV space for sets of phylogenetic trees. We give an algorithm to compute tropical line segments which is lower in computational complexity than the fastest method currently available for BHV geodesics and show that its trajectory behaves more subtly: while the BHV geodesic traverses the origin for vastly different tree topologies, the tropical line segment bypasses it.

math.CO↗

arXiv2021

Solving reward-collecting problems with UAVs: a comparison of online optimization and Q-learning

Uncrewed autonomous vehicles (UAVs) have made significant contributions to reconnaissance and surveillance missions in past US military campaigns. As the prevalence of UAVs increases, there has also been improvements in counter-UAV technology that makes it difficult for them to successfully obtain valuable intelligence within an area of interest. Hence, it has become important that modern UAVs can accomplish their missions while maximizing their chances of survival. In this work, we specifically study the problem of identifying a short path from a designated start to a goal, while collecting all rewards and avoiding adversaries that move randomly on the grid. We also provide a possible application of the framework in a military setting, that of autonomous casualty evacuation. We present a comparison of three methods to solve this problem: namely we implement a Deep Q-Learning model, an $\varepsilon$-greedy tabular Q-Learning model, and an online optimization framework. Our computational experiments, designed using simple grid-world environments with random adversaries showcase how these approaches work and compare them in terms of performance, accuracy, and computational time.

cs.LG↗

arXiv2021

Tree Topologies along a Tropical Line Segment

Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.

math.CO↗

arXiv2020

Tropical Data Science

Phylogenomics is a new field which applies to tools in phylogenetics to genome data. Due to a new technology and increasing amount of data, we face new challenges to analyze them over a space of phylogenetic trees. Because a space of phylogenetic trees with a fixed set of labels on leaves is not Euclidean, we cannot simply apply tools in data science. In this paper we survey some new developments of machine learning models using tropical geometry to analyze a set of phylogenetic trees over a tree space.

math.CO↗

arXiv2020

Tropical Support Vector Machine and its Applications to Phylogenomics

Most data in genome-wide phylogenetic analysis (phylogenomics) is essentially multidimensional, posing a major challenge to human comprehension and computational analysis. Also, we can not directly apply statistical learning models in data science to a set of phylogenetic trees since the space of phylogenetic trees is not Euclidean. In fact, the space of phylogenetic trees is a tropical Grassmannian in terms of max-plus algebra. Therefore, to classify multi-locus data sets for phylogenetic analysis, we propose tropical support vector machines (SVMs). Like classical SVMs, a tropical SVM is a discriminative classifier defined by the tropical hyperplane which maximizes the minimum tropical distance from data points to itself in order to separate these data points into sectors (half-spaces) in the tropical projective torus. Both hard margin tropical SVMs and soft margin tropical SVMs can be formulated as linear programming problems. We focus on classifying two categories of data, and we study a simpler case by assuming the data points from the same category ideally stay in the same sector of a tropical separating hyperplane. For hard margin tropical SVMs, we prove the necessary and sufficient conditions for two categories of data points to be separated, and we show an explicit formula for the optimal value of the feasible linear programming problem. For soft margin tropical SVMs, we develop novel methods to compute an optimal tropical separating hyperplane. Computational experiments show our methods work well. We end this paper with open problems.

math.CO↗

arXiv2019

Tropical principal component analysis on the space of ultrametrics

In 2019, Yoshida et al. introduced a notion of tropical principal component analysis (PCA). The output is a tropical polytope with a fixed number of vertices that best fits the data. We here apply tropical PCA to dimension reduction and visualization of data sampled from the space of phylogenetic trees. Our main results are twofold: the existence of a tropical cell decomposition into regions of fixed tree topology and the development of a stochastic optimization method to estimate the tropical PCA using a Markov Chain Monte Carlo (MCMC) approach. This method performs well with simulation studies, and it is applied to three empirical datasets: Apicomplexa and African coelacanth genomes as well as sequences of hemagglutinin for influenza from New York.

math.CO↗

arXiv2018

Tropical Fermat-Weber points

In a metric space, the Fermat-Weber points of a sample are statistics to measure the central tendency of the sample and it is well-known that the Fermat-Weber point of a sample is not necessarily unique in the metric space. We investigate the computation of Fermat-Weber points under the tropical metric on the quotient space $\mathbb{R}^{n} \!/ \mathbb{R} {\bf 1}$ with a fixed $n \in \mathbb{N}$, motivated by its application to the space of equidistant phylogenetic trees with $N$ leaves (in this case $n=\binom{N}{2}$) realized as the tropical linear space of all ultrametrics. We show that the set of all tropical Fermat-Weber points of a finite sample is always a classical convex polytope, and we present a combinatorial formula for a key value associated to this set. We identify conditions under which this set is a singleton. We apply numerical experiments to analyze the set of the tropical Fermat-Weber points within a space of phylogenetic trees. We discuss the issues in the computation of the tropical Fermat-Weber points.

math.CO↗

arXiv2018

Semigroups and sequential importance sampling for multiway tables

When an interval of integers between the lower bound $l_i$ and the upper bound $u_i$ is the support of the marginal distribution $n_i|(n_{i-1}, ...,n_1)$, Chen et al, 2005 noticed that sampling from the interval at each step, for $n_i$ during a sequential importance sampling (SIS) procedure, always produces a table which satisfies the marginal constraints. However, in general, the interval may not be equal to the support of the marginal distribution. In this case, the SIS procedure may produce tables which do not satisfy the marginal constraints, leading to rejection Chen et al 2006. In this paper we consider the uniform distribution as the target distribution. First we show that if we fix the number of rows and columns of the design matrix of the model for contingency tables then there exists a polynomial time algorithm in terms of the input size to sample a table from the set of all tables satisfying all marginals defined by the given model via the SIS procedure without rejection. We then show experimentally that in general the SIS procedure may have large rejection rates even with small tables. Further we show that in general the classical SIS procedure in Chen et al, 2005 can have a large rejection rate whose limit is one. When estimating the number of tables in our simulation study, we used the univariate and bivariate logistic regression models since under this model the SIS procedure seems to have higher rate of rejections even with small tables.

math.CO↗

arXiv2017

Tropical Principal Component Analysis and its Application to Phylogenetics

Principal component analysis is a widely-used method for the dimensionality reduction of a given data set in a high-dimensional Euclidean space. Here we define and analyze two analogues of principal component analysis in the setting of tropical geometry. In one approach, we study the Stiefel tropical linear space of fixed dimension closest to the data points in the tropical projective torus; in the other approach, we consider the tropical polytope with a fixed number of vertices closest to the data points. We then give approximative algorithms for both approaches and apply them to phylogenetics, testing the methods on simulated phylogenetic data and on an empirical dataset of Apicomplexa genomes.

math.CO↗

arXiv2017

Semigroups --- A Computational Approach

The question whether there exists an integral solution to the system of linear equations with non-negative constraints, $A\x = \b, \, \x \ge 0$, where $A \in \Z^{m\times n}$ and ${\mathbf b} \in \Z^m$, finds its applications in many areas, such as operation research, number theory and statistics. In order to solve this problem, we have to understand the semigroup generated by the columns of the matrix $A$ and the structure of the "holes" which are the difference between the semigroup generated by the columns of the matrix $A$ and its saturation. In this paper, we discuss the implementation of an algorithm by Hemmecke, Takemura, and Yoshida that computes the set of holes of a semigroup, % generated by the columns of $A$ and we discuss applications to problems in combinatorics. Moreover, we compute the set of holes for the common diagonal effect model, and we show that the $n$th linear ordering polytope has the integer-decomposition property for $n\leq 7$. The software is available at \url{http://ehrhart.math.fu-berlin.de/People/fkohl/HASE/}.

math.CO↗