arXiv subjects

Sonja Petrović

Publications and source records attributed to Sonja Petrović.

At least 19 recordsLinked to original sources

arXiv2026

Analytical solutions for some quadratic ODEs found via linear rational eigenfunctions and the rational eigenfunction variety

Many important systems across biology, engineering, physics, and economics are characterized by polynomial ordinary differential equations (ODEs), yet analytical solutions are rare. We develop a framework for identifying and solving a broad class of two-dimensional quadratic ODEs using linear rational Koopman eigenfunctions. By imposing a linear rational form on the eigenfunctions, we convert the Koopman eigenfunction PDE into a large algebraic system of polynomials. We then study the solutions of this polynomial system that satisfy the ODE restrictions; we call the solution set the rational eigenfunction variety of an ODE system. The nonlinear algebra method uses formal algebraic geometry theory to analyze and solve systems otherwise intractable and to discover relationships between ODE and eigenfunction parameters that must hold to extract eigenfunctions. We identify families of quadratic ODEs that can be solved analytically, characterize their eigenfunction parameters, and use the resulting eigenfunctions to produce closed-form analytical solutions.

math.DS↗

arXiv2026

Asymptotic properties of random monomial ideals

This paper focuses on asymptotic properties of random monomial ideals through a statistical viewpoint. It extends the study of redundancy in monomial ideals by analyzing the poset density of the LCM-lattice. We explore how this density behaves across random algebraic models and structured networks. Experimental data reveal that the LCM-lattice exhibits sharp threshold behavior rather than changing smoothly. We observe a strong negative correlation between the number of generators and LCM-lattice density, abruptly separating three distinct regimes: a low-density Taylor-like regime, a high-density redundant regime, and a narrow transition window. We show that increasing the generator degree causes this density drop to occur at lower probability thresholds. We conclude by conjecturing that for equigenerated squarefree ideals, the LCM-lattice density undergoes a sharp phase transition, analogous to the emergence of giant components in hypergraphs. This suggests that the classical, ideal-by-ideal role of the LCM-lattice as a combinatorial invariant also admits a statistical/asymptotic counterpart: in natural random families, redundancy and resolution-complexity indicators concentrate into distinct typical regimes separated by a narrow transition window.

cs.SC↗

arXiv2026

Irreducible Markov Chains on spaces of graphs with fixed degree-color sequences

We study a colored generalization of the famous simple-switch Markov chain for sampling the set of graphs with a fixed degree sequence. Here we consider the space of graphs with colored vertices, in which we fix the degree sequence and another statistic arising from the vertex coloring, and prove that the set can be connected with simple color-preserving switches or moves. These moves form a basis for defining an irreducible Markov chain necessary for testing statistical model fit to block-partitioned network data. Our methods further generalize well-known algebraic results from the 1990s: namely, that the corresponding moves can be used to construct a regular triangulation for a generalization of the second hypersimplex. On the other hand, in contrast to the monochromatic case, we show that for \emph{simple} graphs, the 1-norm of the moves necessary to connect the space increases with the number of colors.

cs.DM↗

arXiv2025

Multilevel Sampling in Algebraic Statistics

This paper proposes a multilevel sampling algorithm for fiber sampling problems in algebraic statistics, inspired by Henry Wynn's suggestion to adapt multilevel Monte Carlo (MLMC) ideas to discrete models. Focusing on log-linear models, we sample from high-dimensional lattice fibers defined by algebraic constraints. Building on Markov basis methods and results from Diaconis and Sturmfels, our algorithm uses variable step sizes to accelerate exploration and reduce the need for long burn-in. We introduce a novel Fiber Coverage Score (FCS) based on Voronoi partitioning to assess sample quality, and highlight the utility of the Maximum Mean Discrepancy (MMD) quality metric. Simulations on benchmark fibers show that multilevel sampling outperforms naive MCMC approaches. Our results demonstrate that multilevel methods, when properly applied, provide practical benefits for discrete sampling in algebraic statistics.

stat.CO↗

arXiv2025

Non-asymptotic goodness-of-fit tests and model selection in valued stochastic blockmodels

A valued stochastic blockmodel (SBM) is a general way to view networked data in which nodes are grouped into blocks and links between them are measured by counts or labels. This family allows for varying dyad sampling schemes, thereby including the classical, Poisson, and labeled SBMs, as well as those in which some edge observations are censored. This paper addresses the question of testing goodness-of-fit of such non-Bernoulli SBMs, focusing in particular on finite-sample tests. We derive explicit Markov bases moves necessary to generate samples from reference distributions and define goodness-of-fit statistics for determining model fit, comparable to those in the literature for related model families. For the labeled SBM, which includes in particular the censored-edge model, we study the asymptotic behavior of said statistics. One of the main purposes of testing goodness-of-fit of an SBM is to determine whether block membership of the nodes influences network formation. Power and Type 1 error rates are verified on simulated data. Additionally, we discuss the use of asymptotic results in selecting the number of blocks under the latent-block modeling assumption. The method derived for Poisson SBM is applied to ecological networks of host-parasite interactions. Our data analysis conclusions differ in selecting the number of blocks for the species from previous results in the literature.

stat.ME↗

arXiv2025

Learning to sample fibers for goodness-of-fit testing

We consider the problem of constructing exact goodness-of-fit tests for discrete exponential family models. This classical problem remains practically unsolved for many types of structured or sparse data, as it rests on a computationally difficult core task: to produce a reliable sample from lattice points in a high-dimensional polytope. We translate the problem into a Markov decision process and demonstrate a reinforcement learning approach for learning `good moves' for sampling. We illustrate the approach on data sets and models for which traditional MCMC samplers converge too slowly due to problem size, sparsity structure, and the requirement to use prohibitive non-linear algebra computations in the process. The differentiating factor is the use of scalable tools from \emph{linear} algebra in the context of theoretical guarantees provided by \emph{non-linear} algebra. Our algorithm is based on an actor-critic sampling scheme, with provable convergence. The discovered moves can be used to efficiently obtain an exchangeable sample, significantly cutting computational times with regards to statistical testing.

stat.ML↗

arXiv2025

Predicting the cardinality and maximum degree of a reduced Gröbner basis

We construct neural network regression models to predict key metrics of complexity for Gröbner bases of binomial ideals. This work illustrates why predictions with neural networks from Gröbner computations are not a straightforward process. Using two probabilistic models for random binomial ideals, we generate and make available a large data set that is able to capture sufficient variability in Gröbner complexity. We use this data to train neural networks and predict the cardinality of a reduced Gröbner basis and the maximum total degree of its elements. While the cardinality prediction problem is unlike classical problems tackled by machine learning, our simulations show that neural networks, providing performance statistics such as $r^2 = 0.401$, outperform naive guess or multiple regression models with $r^2 = 0.180$.

math.AC↗

arXiv2025

Computation of dominant ideals

We consider the problem of determining whether a monomial ideal is dominant. This property is critical for determining for which monomial ideals the Taylor resolution is minimal. We first analyze dominant ideals with a fixed least common multiple of generators using combinatorial methods. Then, we adopt a probabilistic approach via the \er\ type model, examining both homogeneous and non-homogeneous cases. This model offers an efficient alternative to exhaustive enumeration, allowing the study of dominance through small random samples, even in high-dimensional settings.

math.AC↗

arXiv2024

Longitudinal Network Models and Permutation-Uniform Markov Chains

Consider longitudinal networks whose edges turn on and off according to a discrete-time Markov chain with exponential-family transition probabilities. We characterize when their joint distributions are also exponential families with the same parameter, improving data reduction. Further we show that the permutation-uniform subclass of these chains permit interpretation as an independent, identically distributed sequence on the same state space. We then apply these ideas to temporal exponential random graph models, for which permutation uniformity is well suited, and discuss mean-parameter convergence, dyadic independence, and exchangeability. Our framework facilitates our introducing a new network model; simplifies analysis of some network and autoregressive models from the literature, including by permitting closed-form expressions for maximum likelihood estimates for some models; and facilitates applying standard tools to longitudinal-network Markov chains from either asymptotics or single-observation exponential random graph models.

stat.ME↗

arXiv2024

Monte Carlo goodness-of-fit tests for degree corrected and related stochastic blockmodels

We construct Bayesian and frequentist finite-sample goodness-of-fit tests for three different variants of the stochastic blockmodel for network data. Since all of the stochastic blockmodel variants are log-linear in form when block assignments are known, the tests for the \emph{latent} block model versions combine a block membership estimator with the algebraic statistics machinery for testing goodness-of-fit in log-linear models. We describe Markov bases and marginal polytopes of the variants of the stochastic blockmodel, and discuss how both facilitate the development of goodness-of-fit tests and understanding of model behavior. The general testing methodology developed here extends to any finite mixture of log-linear models on discrete data, and as such is the first application of the algebraic statistics machinery for latent-variable models.

stat.ME↗

arXiv2024

Goodness of fit for log-linear ERGMs

Many popular models from the networks literature can be viewed through a common lens of contingency tables on network dyads, resulting in \emph{log-linear ERGMs}: exponential family models for random graphs whose sufficient statistics are linear on the dyads. We propose a new model in this family, the \emph{$p_1$-SBM}, which combines node and group effects common in network formation mechanisms. In particular, it is a generalization of several well-known ERGMs including the stochastic blockmodel for undirected graphs with known block assignment, the degree-corrected version of it, and the directed $p_1$ model without group structure. We frame the problem of testing model fit for the log-linear ERGM class through an exact conditional test whose $p$-value can be approximated efficiently in networks of both small and moderately large sizes. The sampling methods we build rely on a dynamic adaptation of Markov bases. We use quick estimation algorithms adapted from the contingency table literature and effective sampling methods rooted in graph theory and algebraic statistics. The performance and scalability of the method is demonstrated on two data sets from biology: the connectome of \emph{C. elegans} and the interactome of \emph{Arabidopsis thaliana}. These two networks -- a network and a protein-protein interaction network -- have been popular examples in the network science literature. Our work provides a model-based approach to studying them.

stat.ME↗

arXiv2024

The Weak Lefschetz property and unimodality of Hilbert functions of random monomial algebras

In this work, we investigate the presence of the weak Lefschetz property (WLP) and Hilbert functions for various types of random standard graded Artinian algebras. If an algebra has the WLP then its Hilbert function is unimodal. Using probabilistic models for random monomial algebras, our results and simulations suggest that in each considered regime the Hilbert functions of the produced algebras are unimodal with high probability. The WLP appears to be present with high probability most of the time. However, we propose that there is one scenario where the generated algebras fail to have the WLP with high probability.

math.AC↗

arXiv2024

New directions in algebraic statistics: Three challenges from 2023

In the last quarter of a century, algebraic statistics has established itself as an expanding field which uses multilinear algebra, commutative algebra, computational algebra, geometry, and combinatorics to tackle problems in mathematical statistics. These developments have found applications in a growing number of areas, including biology, neuroscience, economics, and social sciences. Naturally, new connections continue to be made with other areas of mathematics and statistics. This paper outlines three such connections: to statistical models used in educational testing, to a classification problem for a family of nonparametric regression models, and to phase transition phenomena under uniform sampling of contingency tables. We illustrate the motivating problems, each of which is for algebraic statistics a new direction, and demonstrate an enhancement of related methodologies.

math.ST↗

arXiv2024

Markov bases: a 25 year update

In this paper, we evaluate the challenges and best practices associated with the Markov bases approach to sampling from conditional distributions. We provide insights and clarifications after 25 years of the publication of the fundamental theorem for Markov bases by Diaconis and Sturmfels. In addition to a literature review we prove three new results on the complexity of Markov bases in hierarchical models, relaxations of the fibers in log-linear models, and limitations of partial sets of moves in providing an irreducible Markov chain.

stat.ME↗

arXiv2023

Sampling lattice points in a polytope: a Bayesian biased algorithm with random updates

The set of nonnegative integer lattice points in a polytope, also known as the fiber of a linear map, makes an appearance in several applications including optimization and statistics. We address the problem of sampling from this set using three ingredients: an easy-to-compute lattice basis of the constraint matrix, a biased sampling algorithm with a Bayesian framework, and a step-wise selection method. The bias embedded in our algorithm updates sampler parameters to improve fiber discovery rate at each step chosen from previously discovered elements. We showcase the performance of the algorithm on several examples, including fibers that are out of reach for the state-of-the-art Markov bases samplers.

stat.CO↗

arXiv2023

The Spark Randomizer: a learned randomized framework for computing Gröbner bases

We define a violator operator which captures the definition of a minimal Gröbner basis of an ideal. This construction places the problem of computing a Gröbner basis within the framework of violator spaces, introduced in 2008 by G{ä}rtner, Matou{š}ek, R{ü}st, and {Š}kovro{ň} in a different context. The key aspect which we use is their successful utilization of a Clarkson-style fast sampling algorithm from geometric optimization. Using the output of a machine learning algorithm, we combine the prediction of the size of a minimal Gröbner basis of an ideal with the Clarkson-style biased random sampling method to compute a Gröbner basis in expected runtime linear in the size of the violator space.

math.AC↗

arXiv2022

Learning a performance metric of Buchberger's algorithm

What can be (machine) learned about the complexity of Buchberger's algorithm? Given a system of polynomials, Buchberger's algorithm computes a Gröbner basis of the ideal these polynomials generate using an iterative procedure based on multivariate long division. The runtime of each step of the algorithm is typically dominated by a series of polynomial additions, and the total number of these additions is a hardware independent performance metric that is often used to evaluate and optimize various implementation choices. In this work we attempt to predict, using just the starting input, the number of polynomial additions that take place during one run of Buchberger's algorithm. Good predictions are useful for quickly estimating difficulty and understanding what features make Gröbner basis computation hard. Our features and methods could also be used for value models in the reinforcement learning approach to optimize Buchberger's algorithm introduced in [Peifer, Stillman, and Halpern-Leistner, 2020]. We show that a multiple linear regression model built from a set of easy-to-compute ideal generator statistics can predict the number of polynomial additions somewhat well, better than an uninformed model, and better than regression models built on some intuitive commutative algebra invariants that are more difficult to compute. We also train a simple recursive neural network that outperforms these linear models. Our work serves as a proof of concept, demonstrating that predicting the number of polynomial additions in Buchberger's algorithm is a feasible problem from the point of view of machine learning.

math.AC↗

arXiv2022

Marginal Independence Models

We impose rank one constraints on marginalizations of a tensor, given by a simplicial complex. Following work of Kirkup and Sullivant, such marginal independence models can be made toric by a linear change of coordinates. We study their toric ideals, with emphasis on random graph models and independent set polytopes of matroids. We develop the numerical algebra of parameter estimation, using both Euclidean distance and maximum likelihood, and we present a comprehensive database of small models.

math.ST↗