arXiv subjects

Anna Seigal

Publications and source records attributed to Anna Seigal.

35 records · Page 2Linked to original sources

arXiv2022

Symmetries in Directed Gaussian Graphical Models

We define Gaussian graphical models on directed acyclic graphs with coloured vertices and edges, calling them RDAG (restricted directed acyclic graph) models. If two vertices or edges have the same colour, their parameters in the model must be the same. We present an algorithm to find the maximum likelihood estimate (MLE) in an RDAG model, and characterise when the MLE exists, via linear independence conditions. We relate properties of a graph, and its colouring, to the number of samples needed for the MLE to exist and to be unique. We also characterise when an RDAG model is equal to an associated undirected graphical model and study connections to groups and invariant theory. We provide examples and simulations to study the benefits of RDAGs over uncoloured DAGs.

math.ST↗

arXiv2021

Principal Components along Quiver Representations

Quiver representations arise naturally in many areas across mathematics. Here we describe an algorithm for calculating the vector space of sections, or compatible assignments of vectors to vertices, of any finite-dimensional representation of a finite quiver. Consequently, we are able to define and compute principal components with respect to quiver representations. These principal components are solutions to constrained optimisation problems defined over the space of sections, and are eigenvectors of an associated matrix pencil.

math.RT↗

arXiv2021

Toric invariant theory for maximum likelihood estimation in log-linear models

We establish connections between invariant theory and maximum likelihood estimation for discrete statistical models. We show that norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We use notions of stability under a torus action to characterize the existence of the maximum likelihood estimate, and discuss connections to scaling algorithms.

math.ST↗

arXiv2021

Invariant theory and scaling algorithms for maximum likelihood estimation

We uncover connections between maximum likelihood estimation in statistics and norm minimization over a group orbit in invariant theory. We focus on Gaussian transformation families, which include matrix normal models and Gaussian graphical models given by transitive directed acyclic graphs. We use stability under group actions to characterize boundedness of the likelihood, and existence and uniqueness of the maximum likelihood estimate. Our approach reveals promising consequences of the interplay between invariant theory and statistics. In particular, existing scaling algorithms from statistics can be used in invariant theory, and vice versa.

math.ST↗

arXiv2021

The Maximum Likelihood Degree of Linear Spaces of Symmetric Matrices

We study multivariate Gaussian models that are described by linear conditions on the concentration matrix. We compute the maximum likelihood (ML) degrees of these models. That is, we count the critical points of the likelihood function over a linear space of symmetric matrices. We obtain new formulae for the ML degree, one via Schubert calculus, and another using Segre classes from intersection theory. We settle the case of codimension one models, and characterize the degenerate case when the ML degree is zero.

math.AG↗

arXiv2020

Ranks and Singularities of Cubic Surfaces

We explore the connection between the rank of a polynomial and the singularities of its vanishing locus. We first describe the singularity of generic polynomials of fixed rank. We then focus on cubic surfaces. Cubic surfaces with isolated singularities are known to fall into 22 singularity types. We compute the rank of a cubic surface of each singularity type. This enables us to find the possible singular loci of a cubic surface of fixed rank. Finally, we study connections to the Hessian discriminant. We show that a cubic surface with singularities that are not ordinary double points lies on the Hessian discriminant, and that the Hessian discriminant is the closure of the rank six cubic surfaces.

math.AG↗

arXiv2019

Tensor clustering with algebraic constraints gives interpretable groups of crosstalk mechanisms in breast cancer

We introduce a tensor-based clustering method to extract sparse, low-dimensional structure from high-dimensional, multi-indexed datasets. This framework is designed to enable detection of clusters of data in the presence of structural requirements which we encode as algebraic constraints in a linear program. Our clustering method is general and can be tailored to a variety of applications in science and industry. We illustrate our method on a collection of experiments measuring the response of genetically diverse breast cancer cell lines to an array of ligands. Each experiment consists of a cell line-ligand combination, and contains time-course measurements of the early-signalling kinases MAPK and AKT at two different ligand dose levels. By imposing appropriate structural constraints and respecting the multi-indexed structure of the data, the analysis of clusters can be optimized for biological interpretation and therapeutic understanding. We then perform a systematic, large-scale exploration of mechanistic models of MAPK-AKT crosstalk for each cluster. This analysis allows us to quantify the heterogeneity of breast cancer cell subtypes, and leads to hypotheses about the signalling mechanisms that mediate the response of the cell lines to ligands.

q-bio.QM↗

arXiv2018

Learning Paths from Signature Tensors

Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry and numerical optimization to this group action. Given a tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. Our primary application is an inverse problem from stochastic analysis: the recovery of paths from their third order signature tensors. We establish identifiability results, both exact and numerical, for piecewise linear paths, polynomial paths, and generic dictionaries. Numerical optimization is applied for recovery from inexact data. We also compute the shortest path with a given signature tensor.

math.NA↗

arXiv2018

Ranks and Symmetric Ranks of Cubic Surfaces

We study cubic surfaces as symmetric tensors of format $4 \times 4 \times 4$. We consider the non-symmetric tensor rank and the symmetric Waring rank of cubic surfaces, and show that the two notions coincide over the complex numbers. The corresponding algebraic problem concerns border ranks. We show that the non-symmetric border rank coincides with the symmetric border rank for cubic surfaces. As part of our analysis, we obtain minimal ideal generators for the symmetric analogue to the secant variety from the salmon conjecture. We also give a test for symmetric rank given by the non-vanishing of certain discriminants. The results extend to order three tensors of all sizes, implying the equality of rank and symmetric rank when the symmetric rank is at most seven, and the equality of border rank and symmetric border rank when the symmetric border rank is at most five. We also study real ranks via the real substitution method.

math.AG↗

arXiv2017

Duality of Graphical Models and Tensor Networks

In this article we show the duality between tensor networks and undirected graphical models with discrete variables. We study tensor networks on hypergraphs, which we call tensor hypernetworks. We show that the tensor hypernetwork on a hypergraph exactly corresponds to the graphical model given by the dual hypergraph. We translate various notions under duality. For example, marginalization in a graphical model is dual to contraction in the tensor network. Algorithms also translate under duality. We show that belief propagation corresponds to a known algorithm for tensor network contraction. This article is a reminder that the research areas of graphical models and tensor networks can benefit from interaction.

math.ST↗

arXiv2017

Mixtures and products in two graphical models

We compare two statistical models of three binary random variables. One is a mixture model and the other is a product of mixtures model called a restricted Boltzmann machine. Although the two models we study look different from their parametrizations, we show that they represent the same set of distributions on the interior of the probability simplex, and are equal up to closure. We give a semi-algebraic description of the model in terms of six binomial inequalities and obtain closed form expressions for the maximum likelihood estimates. We briefly discuss extensions to larger models.

stat.ML↗

arXiv2017

Real Rank Two Geometry

The real rank two locus of an algebraic variety is the closure of the union of all secant lines spanned by real points. We seek a semi-algebraic description of this set. Its algebraic boundary consists of the tangential variety and the edge variety. Our study of Segre and Veronese varieties yields a characterization of tensors of real rank two.

math.AG↗

arXiv2016

Gram Determinants of Real Binary Tensors

A binary tensor consists of $2^n$ entries arranged into hypercube format $2 \times 2 \times \cdots \times 2$. There are $n$ ways to flatten such a tensor into a matrix of size $2 \times 2^{n-1}$. For each flattening, $M$, we take the determinant of its Gram matrix, ${\rm det }(M M^T)$. We consider the map that sends a tensor to its $n$-tuple of Gram determinants. We propose a semi-algebraic characterization of the image of this map. This offers an answer to a question raised by Hackbusch and Uschmajew concerning the higher-order singular values of tensors.

math.SP↗

arXiv2016

Does Antibiotic Resistance Evolve in Hospitals?

Nosocomial outbreaks of bacteria are well-documented. Based on these incidents, and the heavy usage of antibiotics in hospitals, it has been assumed that antibiotic resistance evolves in hospital environments. To test this assumption, we studied resistance phenotypes of bacteria collected from patient isolates at a community hospital over a 2.5-year period. A graphical model analysis shows no association between resistance and patient information other than time of arrival. This allows us to focus on time course data. We introduce a Hospital Transmission Model, based on negative binomial delay. Our main contribution is a statistical hypothesis test called the Nosocomial Evolution of Resistance Detector (NERD). It calculates the significance of resistance trends occurring in a hospital. It can inform hospital staff about the effects of various practices and interventions, can help detect clonal outbreaks, and is available as an R-package. We applied the NERD method to each of the 16 antibiotics in the study via 16 hypothesis tests. For 13 of the antibiotics, we found that the hospital environment had no significant effect upon the evolution of resistance; the hospital is merely a piece of the larger picture. The p-values obtained for the other three antibiotics (Cefepime, Ceftazidime and Gentamicin) indicate that particular care should be taken in hospital practices with these antibiotics. One of the three, Ceftazidime, was significant after accounting for multiple hypotheses, indicating a trend of decreased resistance for this drug.

q-bio.PE↗

arXiv2016

Singular Vectors of Orthogonally Decomposable Tensors

Orthogonal decomposition of tensors is a generalization of the singular value decomposition of matrices. In this paper, we study the spectral theory of orthogonally decomposable tensors. For such a tensor, we give a description of its singular vector tuples as a variety in a product of projective spaces.

math.SP↗

arXiv2016

Applied Algebra and Geometry: A SIAGA of Seven Pictures

The cover of the SIAM Journal on Applied Algebra and Geometry shows seven pictures. We describe these pictures and discuss the topics they represent. About the Author: Anna Seigal is a graduate student at UC Berkeley working in applied algebra. Her interests lie in tensors and applications to biological systems. She has a particular penchant for writing about mathematics in terms of pictures and also blogs on this subject at https://picturethismaths.wordpress.com/.

math.HO↗

arXiv2015

Eigenconfigurations of Tensors

Square matrices represent linear self-maps of vector spaces, and their eigenpoints are the fixed points of the induced map on projective space. Likewise, polynomial self-maps of a projective space are represented by tensors. We study the configuration of fixed points of a tensor or symmetric tensor.

math.AG↗