Related papers: Tropical $k$-means clustering for phylogenetic tre…
We define and study the cyclic Bergman fan of a matroid M, which is a simplicial polyhedral fan supported on the tropical linear space T(M) of M and is amenable to computational purposes. It slightly refines the nested set structure on…
Phylogenetic trees summarize evolutionary relationships between organisms, and tools to analyze collections of phylogenetic trees enable contrasts between different genes' ancestry. The BHV metric space has enabled the analysis of…
There are several tools available to infer phylogenetic trees, which depict the evolutionary relationships among biological entities such as viral and bacterial strains in infectious outbreaks, or cancerous cells in tumor progression trees.…
Given a matroid M on the ground set E, the Bergman fan B(M), or space of M-ultrametrics, is a polyhedral complex in R^E which arises in several different areas, such as tropical algebraic geometry, dynamical systems, and phylogenetics.…
In this work, we introduce a Tropical Axial Attention neural reasoning architecture that replaces vanilla softmax dot-product attention with max-plus operators, inducing a piecewise-linear structure aligned with dynamic programming…
Phylogenetic networks are generalizations of trees that allow for the modeling of non-tree like evolutionary processes. Split networks give a useful way to construct networks with intuitive distance structures induced from the associated…
The search for similarity and dissimilarity measures on phylogenetic trees has been motivated by the computation of consensus trees, the search by similarity in phylogenetic databases, and the assessment of clustering results in…
Given a dissimilarity map $\delta$ on finite set $X$, the set of ultrametrics (equidistant tree metrics) which are $l^\infty$-nearest to $\delta$ is a tropical polytope. We give an internal description of this tropical polytope which we use…
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…
We show that the cophylogenetic distance, k-interval cospeciation, is distinct from other metrics and accounts for global congruence between locally incongruent trees. The growth of the neighborhood of trees which satisfy the largest…
We investigate uniqueness issues that arise in $l^\infty$-optimization to linear spaces and Bergman fans of matroids. For linear spaces, we give a polyhedral decomposition of $\mathbb{R}^n$ based on the dimension of the set of…
Motivated by applications to low-rank matrix completion, we give a combinatorial characterization of the independent sets in the algebraic matroid associated to the collection of $m\times n$ rank-2 matrices and $n\times n$ skew-symmetric…
We explicitly describe the tropicalization of a cluster variety of finite type C, realizing it as the space of axially symmetric phylogenetic trees. We also find all occurring sign patterns of coordinates, for both the cluster variety and…
A metric phylogenetic tree relating a collection of taxa induces weighted rooted triples and weighted quartets for all subsets of three and four taxa, respectively. New intertaxon distances are defined that can be calculated from these…
The Neighbor-Joining algorithm is a popular distance-based phylogenetic method that computes a tree metric from a dissimilarity map arising from biological data. Realizing dissimilarity maps as points in Euclidean space, the algorithm…
Phylogenetic trees summarize evolutionary relationships. The Billera-Holmes-Vogtmann (BHV) space for comparing phylogenetic trees has many elegant mathematical properties, but it does not encompass trees with differing leaf sets. To…
We consider the numerical taxonomy problem of fitting a positive distance function ${D:{S\choose 2}\rightarrow \mathbb R_{>0}}$ by a tree metric. We want a tree $T$ with positive edge weights and including $S$ among the vertices so that…
Each gene has its own evolutionary history which can substantially differ from the evolutionary histories of other genes. For example, some individual genes or operons can be affected by specific horizontal gene transfer and recombination…
The k-means clustering algorithm is a popular algorithm that partitions data into k clusters. There are many improvements to accelerate the standard algorithm. Most current research employs upper and lower bounds on point-to-cluster…
The presence of reticulate evolutionary events in phylogenies turn phylogenetic trees into phylogenetic networks. These events imply in particular that there may exist multiple evolutionary paths from a non-extant species to an extant one,…