The Shape and Dimensionality of Phylogenetic Tree-Space Based on Mitochondrial Genomes

Phylogenetic analyses of large and diverse data sets generally result in large sets of competing phylogenetic trees. Consensus tree methods used to summarize sets of competing trees discard important information regarding the similarity and distribution of competing trees. A more fine grain approach is to use a dimensionality reduction method to project tree-to-tree distances in 2D or 3D space. In this study, we systematically evaluate the performance of several nonlinear dimensionality reduction (NLDR) methods on tree-to-tree distances obtained from independent nonparametric bootstrap analyses of genes from three mid- to large-sized mitochondrial genome alignments.
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First-order optimization on stratified sets

We consider the problem of minimizing a differentiable function with locally Lipschitz continuous gradient on a stratified set and present a first-order algorithm designed to find a stationary point of that problem. Our assumptions on the stratified set are satisfied notably by the determinantal variety (i.e., matrices of bounded rank), its intersection with the cone of positive-semidefinite matrices, and the set of nonnegative sparse vectors. The iteration map of the proposed algorithm applies a step of projected-projected gradient descent with backtracking line search, as proposed by Schneider and Uschmajew (2015), to its input but also to a projection of the input onto each of the lower strata to which it is considered close, and outputs a point among those thereby produced that maximally reduces the cost function. Under our assumptions on the stratified set, we prove that this algorithm produces a sequence whose accumulation points are stationary, and therefore does not follow the so-called apocalypses described by Levin, Kileel, and Boumal (2022). We illustrate the apocalypse-free property of our method through a numerical experiment on the determinantal variety

A Riemannian Optimization Approach to Clustering Problems

This paper considers the optimization problem in the form of $\min_{X \in \mathcal{F}_v} f(x) + \lambda \|X\|_1,$ where $f$ is smooth, $\mathcal{F}_v = \{X \in \mathbb{R}^{n \times q} : X^T X = I_q, v \in \mathrm{span}(X)\}$, and $v$ is a given positive vector. The clustering models including but not limited to the models used by $k$-means, community detection, and normalized cut can be reformulated as such optimization problems. It is proven that the domain $\mathcal{F}_v$ forms a compact embedded submanifold of $\mathbb{R}^{n \times q}$ and optimization-related tools including a family of computationally efficient retractions and an orthonormal basis of any normal space of $\mathcal{F}_v$ are derived. An inexact accelerated Riemannian proximal gradient method that allows adaptive step size is proposed and its global convergence is established. Numerical experiments on community detection in networks and normalized cut for image segmentation are used to demonstrate the performance of the proposed method

Parallel algorithms for matrix computations

ix, 197 p. : ill. ; 26 cm

An Algorithm for the Detection and Integration of Highly Oscillatory Ordinary Differential Equations Using a Generalized Unified Modified Divided Difference Representation

117 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1983.This thesis describes an algorithm which automatically integrates systems of ordinary differential equations which have highly oscillatory solutions. Natural variable step derivations of the Generalized Adams and Generalized BDF methods are presented. An efficient numerical algorithm for the evaluation of the local period of an oscillation is presented along with a corresponding algorithm which detects behavior that indicates the system may be amenable to solution by the generalized methods. A code, which implements the algorithm and exploits the overwhelming similarity between the generalized methods and conventional integration methods, is discussed along with some numerical results.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD

Appearance-Based Classification And Recognition Using Spectral Histogram Representations And Hierarchical Learning For Oca

ix 1. APPEARANCE-BASED CLASSIFICATION AND RECOGNITION USING SPECTRAL HISTOGRAM REPRESENTATIONS . . . . . . . . . . . 1 1.1