Abstract We present an algorithm for computing zero-dimensional tropical varieties using projections. Our main tools are fast monomial transforms of triangular sets. Given a Gröbner basis, we prove that our algorithm requires only a polynomial number of arithmetic operations, and, for ideals in shape position, we show that its timings compare well against univariate factorization and backsubstitution. We conclude that the complexity of computing positive-dimensional tropical varieties via a traversal of the Gröbner complex is dominated by the complexity of the Gröbner walk.

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.

We consider the set of Hermitian matrices corresponding to a given graph, that is, Hermitian matrices whose nonzero entries correspond to the edges of the graph. When a particular vertex is removed from a graph a number of eigenvalues of the resulting principal submatrix may coincide with eigenvalues of the original Hermitian matrix. Here, we count the maximum number of “interlacing equalities” when the graph is a tree and the original matrix has distinct eigenvalues. We provide an upper bound and lower bound for the count and discuss some conditions under which the count is equal to the upper bound.

Various mechanisms exist to secure users' passwords, yet users continue to struggle with the complexity of multiple password management. We explore the effectiveness of a feedback loop to improve users' password management. We introduce YourPassword, a web-based application that uses feedback to inform users about the security of their password behavior. YourPassword has two main components: a password behavior checker that converts password strengths into numerical scores and a dashboard interface that visualizes users' overall password behavior and provides visual feedback in real time. YourPassword not only provides a total score on all passwords, but also visualizes when passwords are too similar to each other. To test the efficacy of YourPassword, we conducted a between-subjects experiment and think-aloud test with 48 participants. Participants either had access to YourPassword, an existing commercial password checker, or no password tool (control condition). YourPassword helped participants improve their password behavior as compared with the commercial tool or no tool.