arXiv subjects

Olga Kuznetsova

Publications and source records attributed to Olga Kuznetsova.

9 recordsLinked to original sources

arXiv2026

A computational approach to maximum likelihood thresholds for colored Gaussian graphical models

Gaussian graphical models (GGMs) are essential tools for interpretable structure learning. However, in high-dimensional, small-sample regimes, the available data is often insufficient for the maximum likelihood estimator to exist. Colored Gaussian graphical models (CGGMs) mitigate this limitation by imposing symmetry constraints through graph coloring, which reduces the required sample size. This minimal number of observations needed to guarantee that the estimator exists almost surely is defined as the maximum likelihood threshold (MLT). Here, we address the computation of the MLT for CGGMs by focusing on its geometric formulation: finding the minimum rank of a sample covariance matrix such that its projection lies almost surely within the interior of the cone of sufficient statistics. We establish a unified theoretical framework, extending results from uncolored to colored models and introducing new symbolic algorithms. Furthermore, we present a computational study integrating sampling with topological data analysis (TDA) to investigate the local geometry of the cone of sufficient statistics. Our results demonstrate the potential of TDA to overcome the computational bottlenecks of traditional symbolic algebraic methods, particularly Groebner basis computations, in analyzing the likelihood geometry of CGGMs.

stat.ML↗

arXiv2023

Exact Solutions in Log-Concave Maximum Likelihood Estimation

We study probability density functions that are log-concave. Despite the space of all such densities being infinite-dimensional, the maximum likelihood estimate is the exponential of a piecewise linear function determined by finitely many quantities, namely the function values, or heights, at the data points. We explore in what sense exact solutions to this problem are possible. First, we show that the heights given by the maximum likelihood estimate are generically transcendental. For a cell in one dimension, the maximum likelihood estimator is expressed in closed form using the generalized W-Lambert function. Even more, we show that finding the log-concave maximum likelihood estimate is equivalent to solving a collection of polynomial-exponential systems of a special form. Even in the case of two equations, very little is known about solutions to these systems. As an alternative, we use Smale's alpha-theory to refine approximate numerical solutions and to certify solutions to log-concave density estimation.

math.ST↗

arXiv2022

Combinatorial Derived Matroids

Let $M$ be an arbitrary matroid with circuits $\mathcal{C}(M)$. We propose a definition of a derived matroid $δM$ that has as its ground set $\mathcal{C}(M)$. Unlike previous attempts of such a definition, our definition applies to arbitrary matroids, and is completely combinatorial. We prove that the rank of $δM$ is bounded from above by $|M|-r(M)$, that it is connected if and only if $M$ is connected. We compute examples including the derived matroids of uniform matroids, the Vámos matroid and the graphical matroid $M(K_4)$. We formulate conjectures relating our construction to previous definitions of derived matroids.

math.CO↗

arXiv2022

Log-concave density estimation in undirected graphical models

We study the problem of maximum likelihood estimation of densities that are log-concave and lie in the graphical model corresponding to a given undirected graph $G$. We show that the maximum likelihood estimate (MLE) is the product of the exponentials of several tent functions, one for each maximal clique of $G$. While the set of log-concave densities in a graphical model is infinite-dimensional, our results imply that the MLE can be found by solving a finite-dimensional convex optimization problem. We provide an implementation and a few examples. Furthermore, we show that the MLE exists and is unique with probability 1 as long as the number of sample points is larger than the size of the largest clique of $G$ when $G$ is chordal. We show that the MLE is consistent when the graph $G$ is a disjoint union of cliques. Finally, we discuss the conditions under which a log-concave density in the graphical model of $G$ has a log-concave factorization according to $G$.

math.ST↗

arXiv2022

Computing Maximum Likelihood Estimates for Gaussian Graphical Models with Macaulay2

We introduce the package "GraphicalModelsMLE" for computing the maximum likelihood estimates (MLEs) of a Gaussian graphical model in the computer algebra system Macaulay2. This package allows the computation of MLEs for the class of loopless mixed graphs. Additional functionality allows the user to explore the underlying algebraic structure of the model, such as its maximum likelihood degree and the ideal of score equations.

stat.CO↗

arXiv2021

Algebraic degree of optimization over a variety with an application to $p$-norm distance degree

We study an optimization problem with the feasible set being a real algebraic variety $X$ and whose parametric objective function $f_u$ is gradient-solvable with respect to the parametric data $u$. This class of problems includes Euclidean distance optimization as well as maximum likelihood optimization. For these particular optimization problems, a prominent role is played by the ED and ML correspondence, respectively. To our generalized optimization problem we attach an optimization correspondence and show that it is equidimensional. This leads to the notion of algebraic degree of optimization on $X$. We apply these results to $p$-norm optimization, and define the $p$-norm distance degree of $X$, which coincides with the ED degree of $X$ for $p=2$. Finally, we derive a formula for the $p$-norm distance degree of $X$ as a weighted sum of the polar classes of $X$ under suitable transversality conditions.

math.AG↗

arXiv2021

Information Hiding Using Matroid Theory

Inspired by problems in Private Information Retrieval, we consider the setting where two users need to establish a communication protocol to transmit a secret without revealing it to external observers. This is a question of how large a linear code can be, when it is required to agree with a prescribed code on a collection of coordinate sets. We show how the efficiency of such a protocol is determined by the derived matroid of the underlying linear communication code. Furthermore, we provide several sufficient combinatorial conditions for when no secret transmission is possible.

math.CO↗

arXiv2019

Charge pump phase-locked loop with phase-frequency detector: closed form mathematical model

Charge pump phase-locked loop with phase-frequency detector (CP-PLL) is an electrical circuit, widely used in digital systems for frequency synthesis and synchronization of the clock signals. In this paper a non-linear second-order model of CP-PLL is rigorously derived. The obtained model obviates the shortcomings of previously known second-order models of CP-PLL. Pull-in time is estimated for the obtained second-order CP-PLL.

eess.SP↗

arXiv2009

Length functions of lemniscates

We study metric and analytic properties of generalized lemniscates E_t(f)={z:ln|f(z)|=t}, where f is an analytic function. Our main result states that the length function |E_t(f)| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln|E_t(f)| is convex on any interval free of critical points of ln|f|. As another application we deduce explicit formulas of the length function in some special cases.

math.CV↗