arXiv subjects

Nataliia Kushnerchuk

Publications and source records attributed to Nataliia Kushnerchuk.

5 recordsLinked to original sources

arXiv2026

Posets of trek polynomials for directed trees

When a variety $V_φ$ equals the image of a polynomial map $φ$ whose coordinate functions are combinatorial generating polynomials (i.e.~polynomials enumerating combinatorial objects), the geometry of $V_φ$ reflects identities satisfied by the generating polynomials. The resulting interplay between combinatorics and algebraic geometry can be used to answer questions about $V_φ$. A recent technique proposes to do so using a partially ordered set (poset) $P_φ$ defined via the coefficient vectors of the polynomials defining $φ$. This paper characterizes the poset $P_φ$ when the generating polynomials defining $φ$ enumerate subgraphs of a directed tree known as treks. The characterization is used to compute the linear span of $V_φ$, prove it is toric and deduce a basis for its vanishing ideal. It is also shown that this poset of trek polynomials for a directed tree is a so-called $π$-system if and only if the tree satisfies a property characterized via Stanley's P-partitions. As an additional consequence, it is shown that the varieties for two distinct directed trees intersect in a strictly lower-dimensional variety. This solves an instance of the structural identifiability problem in the graphical models program from statistics.

math.CO↗

arXiv2026

On unirational varieties with poset parameterizations

We use partially ordered sets (posets) to provide a canonical parameterization for the Zariski closure of the image of a semialgebraic set under a rational map whose coordinate functions are polynomials with nonnegative integral coefficients. The resulting poset parametrization of such a unirational variety allows us to translate several well-studied problems into combinatorics; e.g. reducing the problems to describing the poset associated to the variety. These problems include, the implicitization problem from algebraic geometry, the toric reparameterization problem, the computation of the linear span of the variety, and the problem of distinguishing two semialgebraic subsets of the same ambient space. The technique applies to instances of these problems in several fields, including algebraic geometry, algebraic combinatorics, statistics and applied algebra. We demonstrate the technique on examples from each field, including degenerate subvarieties of secant varieties, matroid flat varieties -- which generalize toric varieties of edge polytopes, as well as varieties arising in multivariate data analysis and evolutionary biology.

math.CO↗

arXiv2026

Discrete-time, discrete-state multistate Markov models from the perspective of algebraic statistics

We study discrete-time, discrete-state multistate Markov models from the perspective of algebraic statistics. These models are widely studied in event history analysis, and are characterized by the state space, the initial distribution and the transition probabilities. A finite path under the multistate Markov model is a particular set of states occupied at finite time instances $\{1, \dots, n\}$. The main goal of this paper is to establish a bridge between event history analysis and algebraic statistics. The joint probabilities of finite paths in these models have a natural monomial parametrization in terms of the initial distribution and the transition probabilities. We study the polynomial relations among joint path probabilities. When the statistical constraints on the parameters are disregarded, nonhomogeneous multistate Markov models of arbitrary order can be viewed as slices of decomposable hierarchical models. This yields a complete description of their vanishing ideals as toric ideals generated by explicit families of binomials. Moreover, the variety of this vanishing ideal equals the nonhomogeneous multistate Markov model on the probability simplex. In contrast, homogeneous multistate Markov models exhibit different algebraic behavior, as time homogeneity imposes additional polynomial relations, leading to vanishing ideals that are strictly larger than in the nonhomogeneous case. We also derive families of binomial relations that vanish on homogeneous multistate Markov models. We investigate maximum likelihood estimation from statistical and algebraic perspectives. For nonhomogeneous models, classical and algebraic formulas agree; in the homogeneous case, the algebraic approach is more complex. Lastly, we provide data applications where we demonstrate the statistical theory to obtain the maximum likelihood estimates of the parameters under specific multistate Markov models.

math.ST↗

arXiv2026

Identifiability in Graphical Discrete Lyapunov Models

In this paper, we study discrete Lyapunov models, which consist of steady-state distributions of first-order vector autoregressive models. The parameter matrix of such a model encodes a directed graph whose vertices correspond to the components of the random vector. This combinatorial framework naturally allows for cycles in the graph structure. We focus on the fundamental problem of identifying the entries of the parameter matrix. In contrast to the classical setting, we assume non-Gaussian error terms, which allows us to use the higher-order cumulants of the model. In this setup, we show generic identifiability for directed acyclic graphs with self-loops at each vertex and show how to express the parameters as a rational function of the cumulants. Furthermore, we establish local identifiability for all directed graphs containing self loops at each vertex and no isolated vertices. Finally, we provide first results on the defining equations of the models, showing model equivalence for certain graphs and paving the way towards structure learning.

math.ST↗

arXiv2024

Matroid Stratification of ML Degrees of Independence Models

We study the maximum likelihood (ML) degree of discrete exponential independence models and models defined by the second hypersimplex. For models with two independent variables, we show that the ML degree is an invariant of a matroid associated to the model. We use this description to explore ML degrees via hyperplane arrangements. For independence models with more variables, we investigate the connection between the vanishing of factors of its principal $A$-determinant and its ML degree. Similarly, for models defined by the second hypersimplex, we determine its principal $A$-determinant and give computational evidence towards a conjectured lower bound of its ML degree.

math.ST↗