The Fréchet mean is a fundamental notion of central tendency defined as a minimizer of a sum of squared distances in a general metric space. In this paper, we study Fréchet means in tropical geometry—a piecewise linear, combinatorial, and polyhedral variant of algebraic geometry—by formulating and solving the associated tropical quadratic optimization problem. We give a geometric characterization of the collection of all tropical Fréchet means as a bounded set that is simultaneously tropically and classically convex, hence a polytrope. We establish the existence of positivity certificates for maxima of finitely many quadratic polynomials in R [ x 1 , … , x n ] whose homogeneous quadratic components are sums of squares, which provides a symbolic framework for exact optimization. Using this structure, we develop algorithms for computing tropical Fréchet means and the associated Fréchet mean polytrope. We further describe a combinatorial type decomposition of the objective function induced by braid arrangements, yielding a piecewise quadratic representation and a fully symbolic method for exact computation.

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. This can be accomplished with a single additional hidden layer called a tropical embedding layer, and can in principle be added to any neural network architecture. We study the geometry of the resulting decision boundary, and find that like adversarial training and various regularization techniques that have been proposed, adding the tropical embedding layer tends to increase the number of linear regions associated with the decision boundaries. Our numerical experiments show that our approach achieves state-of-the-art levels of adversarial robustness, while requiring much less computational time than adversarial training.

The Fréchet mean is a key measure of central tendency as a barycenter for a given set of points in a general metric space. It is computed by solving an optimization problem and is a fundamental quantity in statistics. In this paper, we study Fréchet means in tropical geometry—a piecewise linear, combinatorial, and polyhedral variant of algebraic geometry that has gained prominence in applications. A key property of Fréchet means is that uniqueness is generally not guaranteed, which is true in tropical settings. In solving the tropical Fréchet mean optimization problem, we obtain a geometric characterization of the collection of all Fréchet means in a general tropical space as a tropically and classically convex polytope. Furthermore, we prove that a certificate of positivity for finitely many quadratic polynomials in \(\mathbb {R}[x_1,\ldots ,x_n]\) always exists, given that their quadratic homogeneous components are sums of squares. We propose an algorithm to symbolically compute the Fréchet mean polytope based on our exact quadratic optimization result and study its complexity.

*Can algebraic geometry enhance the sharpness, robustness, and interpretability of modern neural reasoning models by equipping them with a mathematically grounded inductive bias?* To answer this, we introduce Tropical Attention, an attention mechanism grounded in tropical geometry that lifts the attention kernel into tropical projective space, where reasoning is piecewise-linear and 1-Lipschitz, thus preserving the polyhedral decision structure inherent to combinatorial reasoning. We prove that multi-head Tropical Attention (MHTA) stacks universally approximate tropical circuits and realize tropical transitive closure through composition, achieving polynomial resource bounds without invoking recurrent mechanisms. These guarantees explain why the induced polyhedral decision boundaries remain sharp and scale-invariant, rather than smoothed by Softmax. Empirically, we show that Tropical Attention delivers stronger out-of-distribution generalization in both length and value, with high robustness against perturbative noise, and substantially faster inference with fewer parameters compared to Softmax-based and recurrent attention baselines, respectively. For the first time, we push the domain of neural algorithmic reasoning beyond **PTIME** problems to **NP-hard/complete** problems, paving the way toward sharper and more expressive Large Reasoning Models (LRMs) capable of tackling complex combinatorial challenges in Phylogenetics, Cryptography, Particle Physics, and Mathematical Discovery. The code is available at https://github.com/Baran-phys/Tropical-Attention/.

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 a tropical neural network is a universal approximator and we derive a backpropagation rule for deep tropical 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.

Much evidence from biological theory and empirical data indicates that, gene trees, 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 the gene trees are constrained by the tree topology of the underlying species tree, that is, the phylogenetic tree depicting the evolutionary history of the set of species under consideration. 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 the 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.

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.

Abstract 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.

MOTIVATION: Due to new technology for efficiently generating genome data, machine learning methods are urgently needed to analyze large sets of gene trees over the space of phylogenetic trees. However, the space of phylogenetic trees is not Euclidean, so ordinary machine learning methods cannot be directly applied. In 2019, Yoshida et al. introduced the notion of tropical principal component analysis (PCA), a statistical method for visualization and dimensionality reduction using a tropical polytope with a fixed number of vertices that minimizes the sum of tropical distances between each data point and its tropical projection. However, their work focused on the tropical projective space rather than the space of phylogenetic trees. We focus here on tropical PCA for dimension reduction and visualization over the space of phylogenetic trees. RESULTS: Our main results are 2-fold: (i) theoretical interpretations of the tropical principal components over the space of phylogenetic trees, namely, the existence of a tropical cell decomposition into regions of fixed tree topology; and (ii) the development of a stochastic optimization method to estimate tropical PCs over the space of phylogenetic trees using a Markov Chain Monte Carlo 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. AVAILABILITY AND IMPLEMENTATION: Dataset: http://polytopes.net/Data.tar.gz. Code: http://polytopes.net/tropica_MCMC_codes.tar.gz. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Evolutionary hypotheses provide important underpinnings of biological and medical sciences, and comprehensive, genome-wide understanding of evolutionary relationships among organisms are needed to test and refine such hypotheses. Theory and empirical evidence clearly indicate that phylogenies (trees) of different genes (loci) should not display precisely matching topologies. The main reason for such phylogenetic incongruence is reticulated evolutionary history of most species due to meiotic sexual recombination in eukaryotes, or horizontal transfers of genetic material in prokaryotes. Nevertheless, many genes should display topologically related phylogenies, and should group into one or more (for genetic hybrids) clusters in poly-dimensional "tree space". Unusual evolutionary histories or effects of selection may result in "outlier" genes with phylogenies that fall outside the main distribution(s) of trees in tree space. We present a new phylogenomic method, CURatio, which uses ratios of total branch lengths in gene trees to help identify phylogenetic outliers in a given set of ortholog groups from multiple genomes. An advantage of CURatio over other methods is that genes absent from and/or duplicated in some genomes can be included in the analysis. We conducted a simulation study under the coalescent model, and showed that, given sufficient species depth and topological difference, these ratios are significantly higher for the "outlier" gene phylogenies. Also, we applied CURatio to a set of annotated genomes of the fungal family, Clavicipitaceae, and identified alkaloid biosynthesis genes as outliers, probably due to a history of duplication and loss. The source code is available at https://github.com/QiwenKang/CURatio, and the empirical data set for Clavicipitaceae and simulated data set are available at Mendeley https://data.mendeley.com/datasets/mrxts7wjrr/1.

Advances in modern genomics have allowed researchers to apply phylogenetic analyses on a genome-wide scale. While large volumes of genomic data can be generated cheaply and quickly, data missingness is a non-trivial and somewhat expected problem. Since the available information is often incomplete for a given set of genetic loci and individual organisms, a large proportion of trees that depict the evolutionary history of a single genetic locus, called gene trees, fail to contain all individuals. Data incompleteness causes difficulties in data collection, information extraction, and gene tree inference. Furthermore, identifying outlying gene trees, which can represent horizontal gene transfers, gene duplications, or hybridizations, is difficult when data is missing from the gene trees. The typical approach is to remove all individuals with missing data from the gene trees, and focus the analysis on individuals whose information is fully available - a huge loss of information. In this work, we propose and design an optimization-based imputation approach to infer the missing distances between leaves in a set of gene trees via a mixed integer non-linear programming model. We also present a new research pipeline, imPhy, that can (i) simulate a set of gene trees with leaves randomly missing in each tree, (ii) impute the missing pairwise distances in each gene tree, (iii) reconstruct the gene trees using the Neighbor Joining (NJ) and Unweighted Pair Group Method with Arithmetic Mean (UPGMA) methods, and (iv) analyze and report the efficiency of the reconstruction. To impute the missing leaves, we employ our newly proposed non-linear programming framework, and demonstrate its capability in reconstructing gene trees with incomplete information in both simulated and empirical datasets. In the empirical datasets apicomplexa and lungfish, our imputation has very small normalized mean square errors, even in the extreme case where 50 percent of the individuals in each gene tree are missing. Data, software, and user manuals can be found at https://github.com/yasuiniko/imPhy.

Decision makers use partial information of networks to guide their decision, yet when they act, they act in the real network or the ground truth. Therefore, a way of comparing the partial information to ground truth is required. We introduce a statistical measure that analyzes the network obtained from the partially observed information and ground truth, which of course can be applied to the comparison of any networks. As a first step, in the current research, we restrict ourselves to networks of the same size to introduce such a method, which can be generalized to different size networks. We perform mathematical analysis on the random graph, and then apply our methodology to synthetic networks generated using five different generating models. We conclude with a statistical hypothesis test to decide whether two graphs are correlated or not correlated.

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 $\mathbb{R}^{n} \!/ \mathbb{R} {1}$ 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 with 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.

We study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all ultrametrics. The ${CAT}(0)$ metric of Billera--Holmes--Vogtman arises from the theory of orthant spaces. While its geodesics can be computed by the Owen--Provan algorithm, geodesic triangles are complicated. We show that the dimension of such a triangle can be arbitrarily high. Tropical convexity and the tropical metric exhibit properties that are desirable for geometric statistics, such as geodesics of small depth.

As costs of genome sequencing have dropped precipitously, development of efficient bioinformatic methods to analyze genome structure and evolution have become ever more urgent. For example, most published phylogenomic studies involve either massive concatenation of sequences, or informal comparisons of phylogenies inferred on a small subset of orthologous genes, neither of which provides a comprehensive overview of evolution or systematic identification of genes with unusual and interesting evolution (e.g., horizontal gene transfers, gene duplication, and subsequent neofunctionalization). We are interested in identifying such "outlying" gene trees from the set of gene trees and estimating the distribution of trees over the "tree space". This paper describes an improvement to the kdetrees algorithm, an adaptation of classical kernel density estimation to the metric space of phylogenetic trees (Billera-Holmes-Vogtman treespace), whereby the kernel normalizing constants, are estimated through the use of the novel holonomic gradient methods. As in the original kdetrees paper, we have applied kdetrees to a set of Apicomplexa genes. The analysis identified several unreliable sequence alignments that had escaped previous detection, as well as a gene independently reported as a possible case of horizontal gene transfer. The updated version of the kdetrees software package is available both from CRAN (the official R package system), as well as from the official development repository on Github. ( github.com/grady/kdetrees).

In 2010, M. Studený, R. Hemmecke, and S. Lindner explored a new algebraic description of graphical models, called characteristic imsets. Compared with standard imsets, characteristic imsets have several advantages: they are still unique vector representatives of conditional independence structures, 0-1 vectors, and more intuitive in terms of graphs than standard imsets. After defining a characteristic imset polytope (cim-polytope) as the convex hull of all characteristic imsets with a given set of nodes, they also showed that a model selection in graphical models, which maximizes a quality criterion, can be converted into a linear programming problem over the cim-polytope. However, in general, for a fixed set of nodes, the cim-polytope can have exponentially many vertices over an exponentially high dimension. Therefore, in this paper, we focus on the family of directed acyclic graphs whose nodes have a fixed order. This family includes diagnosis models described by bipartite graphs with a set of $m$ nodes and a set of $n$ nodes for any $m, n \in \mathbb{Z}_+$. We first consider cim-polytopes for all diagnosis models and show that these polytopes are direct products of simplices. Then we give a combinatorial description of all edges and all facets of these polytopes. Finally, we generalize these results to the cim-polytopes for all Bayesian networks with a fixed underlying ordering of nodes with or without fixed (or forbidden) edges.

MOTIVATION: Although the majority of gene histories found in a clade of organisms are expected to be generated by a common process (e.g. the coalescent process), it is well known that numerous other coexisting processes (e.g. horizontal gene transfers, gene duplication and subsequent neofunctionalization) will cause some genes to exhibit a history distinct from those of the majority of genes. Such 'outlying' gene trees are considered to be biologically interesting, and identifying these genes has become an important problem in phylogenetics. RESULTS: We propose and implement kdetrees, a non-parametric method for estimating distributions of phylogenetic trees, with the goal of identifying trees that are significantly different from the rest of the trees in the sample. Our method compares favorably with a similar recently published method, featuring an improvement of one polynomial order of computational complexity (to quadratic in the number of trees analyzed), with simulation studies suggesting only a small penalty to classification accuracy. Application of kdetrees to a set of Apicomplexa genes identified several unreliable sequence alignments that had escaped previous detection, as well as a gene independently reported as a possible case of horizontal gene transfer. We also analyze a set of Epichloë genes, fungi symbiotic with grasses, successfully identifying a contrived instance of paralogy. AVAILABILITY AND IMPLEMENTATION: Our method for estimating tree distributions and identifying outlying trees is implemented as the R package kdetrees and is available for download from CRAN.

Background We present our work on using the XSEDE high-performance computing (HPC) network to support and facilitate hands-on bioinformatics tasks for participants of our Essentials of Next Generation Sequencing (NGS) workshop, as well as for students and other learners. In the summer of 2012, the University of Kentucky hosted the NGS workshop, attended by faculty and students from across the Commonwealth who were introduced to the laboratory and bioinformatic components of next-generation sequencing and sequence analysis. Participants used next-generation technology to sequence real genetic material, then used a variety of bioinformatics software tools to assemble those sequences, compare and align them to other sequences, predict genes, and visualize the genome. Due to the success of the 2012 workshop, the second workshop, planned for this summer, is expected to be larger in scale and to include even more participants. It will furthermore include several additional bioinformatics tools and tasks. Since participants will be simultaneously running intensive bioinformatics computing tasks, the resources required will exceed the capacity of the single twelve-core server used to support the workshop last year. One particular resource that appears promising to meet our intensive computational needs is the XSEDE grid computing network, a follow-on to the TeraGrid project designed specifically for “e-Science” and scientific computing. Many of the systems in the XSEDE network already support some of the software used within our workshop; however, many of the programs we will demonstrate have not previously been installed on tested on the XSEDE network. We will describe our experiences porting these applications to, and deploying them on, XSEDE. We will also discuss the challenges that the HPC approach presents for teaching and learning, particularly the complexities of navigating between time-sharing systems and remote job scheduling.

BACKGROUND: The increased use of multi-locus data sets for phylogenetic reconstruction has increased the need to determine whether a set of gene trees significantly deviate from the phylogenetic patterns of other genes. Such unusual gene trees may have been influenced by other evolutionary processes such as selection, gene duplication, or horizontal gene transfer. RESULTS: Motivated by this problem we propose a nonparametric goodness-of-fit test for two empirical distributions of gene trees, and we developed the software GeneOut to estimate a p-value for the test. Our approach maps trees into a multi-dimensional vector space and then applies support vector machines (SVMs) to measure the separation between two sets of pre-defined trees. We use a permutation test to assess the significance of the SVM separation. To demonstrate the performance of GeneOut, we applied it to the comparison of gene trees simulated within different species trees across a range of species tree depths. Applied directly to sets of simulated gene trees with large sample sizes, GeneOut was able to detect very small differences between two set of gene trees generated under different species trees. Our statistical test can also include tree reconstruction into its test framework through a variety of phylogenetic optimality criteria. When applied to DNA sequence data simulated from different sets of gene trees, results in the form of receiver operating characteristic (ROC) curves indicated that GeneOut performed well in the detection of differences between sets of trees with different distributions in a multi-dimensional space. Furthermore, it controlled false positive and false negative rates very well, indicating a high degree of accuracy. CONCLUSIONS: The non-parametric nature of our statistical test provides fast and efficient analyses, and makes it an applicable test for any scenario where evolutionary or other factors can lead to trees with different multi-dimensional distributions. The software GeneOut is freely available under the GNU public license.

We have developed software called Phylotree as a toolkit for running experiments to study gene cophylogenies for genome evolution using distance-based methods. In particular, the toolkit has been instrumental in conducting processing-heavy experiments with the new “difference of means” statistical method. Phylotree was used to run experiments using simulated data as well as biological sequences of well known host and parasite species, and is distributed with data and configuration files allowing these experiments to be reproduced.