arXiv subjects

Aida Maraj

Publications and source records attributed to Aida Maraj.

14 recordsLinked to original sources

arXiv2026

Computing the continuous symmetries of a parametrized variety

We prove that the symmetry Lie algebra of a parametrized variety can be determined directly from the parametrization, without computing the vanishing ideal of the variety. We derive a practical polynomial-time Monte Carlo algorithm for computing the symmetry Lie algebra of a parametrized variety. We discuss applications to testing the binomiality of the ideal of a parametrized variety after changing coordinates, and test this property on varieties arising from staged tree models and colored Gaussian graphical models. Finally, we discuss symmetries and binomiality after changing coordinates for rational curves and give a characterization of the symmetries of many secant varieties.

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arXiv2025

Halfspace Representations of Path Polytopes of Trees

Given a tree $T$, its path polytope is the convex hull of the edge indicator vectors for the paths between any two distinct leaves in $T$. These polytopes arise naturally in polyhedral geometry and applications, such as phylogenetics, tropical geometry, and algebraic statistics. We provide a minimal halfspace representation of these polytopes. The construction is made inductively using toric fiber products.

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arXiv2024

Toric Multivariate Gaussian Models from Symmetries in a Tree

Given a rooted tree $T$ on $n$ non-root leaves with colored and zeroed nodes, we construct a linear space $L_T$ of $n\times n$ symmetric matrices with constraints determined by the combinatorics of the tree. When $L_T$ represents the covariance matrices of a Gaussian model, it provides natural generalizations of Brownian motion tree (BMT) models in phylogenetics. When $L_T$ represents a space of concentration matrices of a Gaussian model, it gives certain colored Gaussian graphical models, which we refer to as BMT derived models. We investigate conditions under which the reciprocal variety $L_T^{-1}$ is toric. Relying on the birational isomorphism of the inverse matrix map, we show that if the BMT derived graph of $T$ is vertex-regular and a block graph, under the derived Laplacian transformation, $L_T^{-1}$ is the vanishing locus of a toric ideal. This ideal is given by the sum of the toric ideal of the Gaussian graphical model on the block graph, the toric ideal of the original BMT model, and binomial linear conditions coming from vertex-regularity. To this end, we provide monomial parametrizations for these toric models realized through paths among leaves in $T$.

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arXiv2024

Contrastive independent component analysis

In recent years, there has been growing interest in jointly analyzing a foreground dataset, representing an experimental group, and a background dataset, representing a control group. The goal of such contrastive investigations is to identify salient features in the experimental group relative to the control. Independent component analysis (ICA) is a powerful tool for learning independent patterns in a dataset. We generalize it to contrastive ICA (cICA). For this purpose, we devise a new linear algebra based tensor decomposition algorithm, which is more expressive but just as efficient and identifiable as other linear algebra based algorithms. We establish the identifiability of cICA and demonstrate its performance in finding patterns and visualizing data, using synthetic, semi-synthetic, and real-world datasets, comparing the approach to existing methods.

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arXiv2024

Maximum Likelihood Degrees of Brownian Motion Tree Models: Star Trees and Root Invariance

A Brownian motion tree (BMT) model is a Gaussian model whose associated set of covariance matrices is linearly constrained according to common ancestry in a phylogenetic tree. We study the complexity of inferring the maximum likelihood (ML) estimator for a BMT model by computing its ML-degree. Our main result is that the ML-degree of the BMT model on a star tree with $n + 1$ leaves is $2^{n+1}-2n-3$, which was previously conjectured by Am\'endola and Zwiernik. We also prove that the ML-degree of a BMT model is independent of the choice of the root. The proofs rely on the toric geometry of concentration matrices in a BMT model. Toward this end, we produce a combinatorial formula for the determinant of the concentration matrix of a BMT model, which generalizes the Cayley-Pr\"ufer theorem to complete graphs with weights given by a tree.

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arXiv2023

Lawrence Lifts, Matroids, and Maximum Likelihood Degrees

We express the maximum likelihood (ML) degrees of a family toric varieties in terms of Mobius invariants of matroids. The family of interest are those parametrized by monomial maps given by Lawrence lifts of totally unimodular matrices with even circuits. Specifying these matrices to be vertex-edge incidence matrices of bipartite graphs gives the ML degrees of some hierarchical models and three dimensional quasi-independence models. Included in this list are the no-three-way interaction models with one binary random variable, for which, we give closed formulae.

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arXiv2023

Symmetry Lie Algebras of Varieties with Applications to Algebraic Statistics

The motivation for this paper is to detect when an irreducible projective variety V is not toric. We do this by analyzing a Lie group and a Lie algebra associated to V. If the dimension of V is strictly less than the dimension of the above mentioned objects, then V is not a toric variety. We provide an algorithm to compute the Lie algebra of an irreducible variety and use it to provide examples of non-toric statistical models in algebraic statistics.

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arXiv2022

Symmetrically colored Gaussian graphical models with toric vanishing ideal

A colored Gaussian graphical model is a linear concentration model in which equalities among the concentrations are specified by a coloring of an underlying graph. The model is called RCOP if this coloring is given by the edge and vertex orbits of a subgroup of the automorphism group of the graph. We show that RCOP Gaussian graphical models on block graphs are toric in the space of covariance matrices and we describe Markov bases for them. To this end, we learn more about the combinatorial structure of these models and their connection with Jordan algebras.

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arXiv2022

Shift Invariant Algebras, Segre Products and Regular Languages

Motivated by results on the rationality of equivariant Hilbert series of some hierarchical models in algebraic statistics we introduce the Segre product of formal languages and apply it to establish rationality of equivariant Hilbert series in new cases. To this end we show that the Segre product of two regular languages is again regular. We also prove that every filtration of algebras given as a tensor product of families of algebras with rational equivariant Hilbert series has a rational equivariant Hilbert series. The term equivariant is used broadly to include the action of the monoid of nonnegative integers by shifting variables. Furthermore, we exhibit a filtration of shift invariant monomial algebras that has a rational equivariant Hilbert series, but whose presentation ideals do not stabilize.

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arXiv2021

Reciprocal Maximum Likelihood Degrees of Brownian Motion Tree Models

We give an explicit formula for the reciprocal maximum likelihood degree of Brownian motion tree models. To achieve this, we connect them to certain toric (or log-linear) models, and express the Brownian motion tree model of an arbitrary tree as a toric fiber product of star tree models.

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arXiv2021

Staged tree models with toric structure

A staged tree model is a discrete statistical model encoding relationships between events. These models are realised by directed trees with coloured vertices. In algebro-geometric terms, the model consists of points inside a toric variety. For certain trees, called balanced, the model is in fact the intersection of the toric variety and the probability simplex. This gives the model a straightforward description, and has computational advantages. In this paper we show that the class of staged tree models with a toric structure extends far outside of the balanced case, if we allow a change of coordinates. It is an open problem whether all staged tree models have toric structure.

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arXiv2021

Nonlinear Algebra and Applications

We showcase applications of nonlinear algebra in the sciences and engineering. Our review is organized into eight themes: polynomial optimization, partial differential equations, algebraic statistics, integrable systems, configuration spaces of frameworks, biochemical reaction networks, algebraic vision, and tensor decompositions. Conversely, developments on these topics inspire new questions and algorithms for algebraic geometry.

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arXiv2020

Generalized Cut Polytopes for Binary Hierarchical Models

Marginal polytopes are important geometric objects that arise in statistics as the polytopes underlying hierarchical log-linear models. These polytopes can be used to answer geometric questions about these models, such as determining the existence of maximum likelihood estimates or the normality of the associated semigroup. Cut polytopes of graphs have been useful in analyzing binary marginal polytopes in the case where the simplicial complex underlying the hierarchical model is a graph. We introduce a generalized cut polytope that is isomorphic to the binary marginal polytope of an arbitrary simplicial complex via a generalized covariance map. This polytope is full dimensional in its ambient space and has a natural switching operation among its facets that can be used to deduce symmetries between the facets of the correlation and binary marginal polytopes. We find complete H-representations of the generalized cut polytope for some important families of simplicial complexes. We also compute the volume of these polytopes in some instances.

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arXiv2019

Equivariant Hilbert series for hierarchical models

Toric ideals to hierarchical models are invariant under the action of a product of symmetric groups. Taking the number of factors, say m, into account, we introduce and study invariant filtrations and their equivariant Hilbert series. We present a condition that guarantees that the equivariant Hilbert series is a rational function in m+1 variables with rational coefficients. Furthermore we give explicit formulas for the rational functions with coefficients in a number field and an algorithm for determining the rational functions with rational coefficients. A key is to construct finite automata that recognize languages corresponding to invariant filtrations.

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