arXiv subjects

Bartosz Kołodziejek

Publications and source records attributed to Bartosz Kołodziejek.

At least 19 recordsLinked to original sources

arXiv2026

From Graphical Lasso to Atomic Norms: High-Dimensional Pattern Recovery

We study high-dimensional precision-matrix estimation with polyhedral atomic-norm penalties on the off-diagonal entries. This framework covers the graphical lasso, $\ell_\infty$-type penalties, SLOPE/OWL penalties, and penalties motivated by colored graphical models. We define the target pattern through the subdifferential face of the dual ball and derive sufficient conditions for exact pattern recovery from the sample covariance matrix. Our analysis introduces two geometric thresholds, a generalized irrepresentability condition, and extends the primal-dual witness method beyond the $\ell_1$ setting. For the graphical lasso, our results yield weaker covariance-deviation requirements and a less restrictive irrepresentability condition than existing theory, leading to sharper asymptotic guarantees. Numerical experiments for several graph topologies show that the resulting bounds are often close to the empirical recovery threshold.

math.ST↗

arXiv2026

Probability laws associated to the independence preserving quadrirational Yang-Baxter maps -- the ultimate case

A map $F\colon\mathcal X\times\mathcal Y\to \mathcal U\times \mathcal V$ is said to be independence preserving (IP) if there exists a pair of independent random variables $(X,Y)$ valued in $\mathcal X\times\mathcal Y$ such that the two coordinates of $(U,V)=F(X,Y)$ are also independent. Recently, Sasada and Uozumi (2024) observed that a hierarchy of quadrirational Yang-Baxter maps gives rise to independence preserving transformations, and identified corresponding families of probability distributions. In view of the limiting properties of these IP maps, the newly defined generalized second kind beta ($\mathrm{GB}_{II}$) model stands at the top of the hierarchy: for independent random variables $X$ and $Y$ following a $\mathrm{GB}_{II}$ distribution, Sasada and Uozumi (2024) showed that when a special quadrirational Yang-Baxter map $F^{(α,β)}$, parameterized by $(α,β)\in(0,\infty)^2$, is applied to the pair $(X,Y)$, it produces another pair $(U,V)$ of independent $\mathrm{GB}_{II}$-distributed random variables. The aim of this paper is to show that the IP property of $F^{(α,β)}$ uniquely identifies distributions of $X,Y,U$ and $V$ as belonging to the $\mathrm{GB}_{II}$ family. To this end, we introduce specially designed Laplace-type transforms. First, we carefully explain the connection between the results from Sasada and Uozumi (2024) and Koudou and Vallois (2012). Next, we focus on the characterization of the second kind beta and the generalized second kind beta distributions through the IP map $F^{(α,\infty)}$. Finally, extending considerably the methodology developed for the case $(α,\infty)$, we prove the characterization of $\mathrm{GB}_{II}$ distributions in the case $(α,β)\in(0,\infty)^2$ with $α\neqβ$, which implies uniqueness in the ultimate missing case of the quadrirational Yang-Baxter hierarchy of IP models.

math.PR↗

arXiv2026

A new class of colored Gaussian graphical models with explicit normalizing constants

We study Bayesian model selection in colored Gaussian graphical models (CGGMs), which combine sparsity of conditional independencies with symmetry constraints encoded by vertex- and edge-colored graphs. A key computational bottleneck in Bayesian inference for CGGMs is the evaluation of the Diaconis-Ylvisaker normalizing constants, given by gamma-type integrals over cones of precision matrices with prescribed zeros and equality constraints. We introduce a new subclass of RCON models for which these normalizing constants admit closed-form expressions. On the algebraic side, we identify conditions on the space of colored precision matrices that guarantee tractability of the associated integrals, leading to the notions of Block-Cholesky spaces (BC-spaces) and Diagonally Commutative Block-Cholesky spaces (DCBC-spaces). On the combinatorial side, we characterize the colored graphs inducing such spaces via a color perfect elimination ordering and a 2-path regularity condition, and define the resulting Color Elimination-Regular (CER) graphs and their symmetric variants. Our framework extends the theory of uncolored decomposable graphs to the colored setting, and the CER class contains all RCOP models associated with decomposable graphs. For models with a single vertex color, our framework reveals a close connection between DCBC-spaces and Bose-Mesner algebras. For models defined on BC-spaces, we derive finite product formulas for the normalizing constants in terms of gamma functions, matrix determinants, and model-specific structure constants. For RCON models whose matrix spaces are DCBC-spaces, we provide efficient methods for computing all structure constants and determinant factors required to evaluate these formulas. Our results substantially broaden the range of CGGMs amenable to Bayesian structure learning in high-dimensional applications.

math.ST↗

arXiv2026

Graphical Models for Multivariate Count Data

The classical multinomial, negative multinomial, hypergeometric, and negative hypergeometric distributions are naturally organized by two features of the sampling scheme: sampling with or without replacement and stopping after a fixed number of draws or a fixed number of failures. We complete the graphical analogue of this scheme for decomposable graphs by adding graphical hypergeometric and graphical negative hypergeometric distributions to the previously introduced graphical multinomial and graphical negative multinomial models in Danielewska et al. (2025). The resulting four families provide a unified parametric framework for graphical modeling of multivariate count data, in which dependence and admissible configurations are encoded by a graph. They interpolate between products of univariate distributions for the empty graph and the corresponding classical multivariate distributions for the complete graph, while retaining explicit Markov factorizations and tractable sampling representations. We further develop a unified Bayesian hierarchy based on graphical Dirichlet-type distributions, obtaining explicit posterior and predictive laws. The framework is particularly natural for count data arising under exclusion or incompatibility constraints. We discuss several such applications and illustrate its practical potential using Rydberg-atom excitation data.

stat.ME↗

arXiv2026

On the empirical spectral distribution of matrix perpetuities

We study matrix perpetuities, that is, solutions to affine fixed-point equations of the form \[ \mathbf{X} \stackrel{d}{=} \mathbf{A}\,\mathbf{X} \,\mathbf{A}^\top+\mathbf{B},\qquad (\mathbf{A},\mathbf{B})\mbox{ and }\mathbf{X} \mbox{ are independent}, \] with particular emphasis on the empirical spectral distribution of the solution. We first establish existence and uniqueness results by relating the problem to classical vector perpetuities. Under orthogonal invariance, we then prove a compression identity for symmetric multiplicative convolution and show that principal submatrices of a matrix perpetuity are themselves lower-dimensional matrix perpetuities. For positive semidefinite, orthogonally invariant models, we prove a finite-dimensional spectral Kesten theorem: we obtain precise power-law tail asymptotics for the expected empirical spectral distribution, show that its tail is governed by the largest eigenvalue, and relate it explicitly to the tail of any diagonal entry. We also prove that, in the subcritical regime, the expected empirical spectral distribution of matrix perpetuities converges weakly, as the dimension tends to infinity, to the distribution of the corresponding free perpetuity. Our results are illustrated by matrix Beta prime perpetuities, for which explicit limiting spectral distributions are available.

math.PR↗

arXiv2026

Finite free perpetuities

We introduce and study finite free perpetuities, defined as monic polynomial solutions of degree $n$ to the affine fixed-point equation \[ p(z) = \mathbb{E}\!\left[ A^{n}\,p\!\left(\frac{z-B}{A}\right)\mathbf{1}_{\{A\neq0\}} \right] + \mathbb{E}\!\left[ (z-B)^n\mathbf{1}_{\{A=0\}} \right], \] where $A$ and $B$ are complex-valued random variables with finite moments up to order $n$. Equivalently, if $p(z)=\mathbb{E}[(z-X)^n]$, then $p$ encodes a truncated moment version of the classical perpetuity equation $X\stackrel{d}{=}AX+B$ with $X$ and $(A,B)$ independent. This places finite free perpetuities between classical perpetuities and free-probabilistic fixed-point laws. We prove existence and uniqueness under weak conditions, and we identify a broad class of admissible pairs $(A,B)$ for which the resulting polynomial has only real, nonnegative zeros. Our approach uses finite free additive and multiplicative convolutions together with a probabilistic representation via the $U$-transform. As a motivating example, we exhibit an explicit family of finite free perpetuities expressed in terms of Jacobi polynomials and show that their empirical root distributions converge to a free-beta-prime law. More generally, for admissible sequences of parameters, we prove weak convergence of the empirical root distributions of finite free perpetuities to the law of a free perpetuity characterized by the corresponding free fixed-point equation. This yields a finite-degree polynomial model approximating free perpetuities and clarifies the connection between classical affine recursions, finite free convolutions, and free probability.

math.PR↗

arXiv2026

Identifying Network Hubs with the Partial Correlation Graphical LASSO

Graphical LASSO (GLASSO) is a widely used method for estimating sparse precision matrices and learning undirected graphical models in high-dimensional settings. Because GLASSO penalizes entries of the precision matrix directly, however, it is not scale-invariant. Partial Correlation Graphical LASSO (PCGLASSO), introduced by Carter et al. (2024), addresses this limitation by penalizing partial correlations, which directly characterize conditional dependence. In this paper, we study both statistical and computational properties of the PCGLASSO estimator. Our main contribution is the introduction of a scale-invariant irrepresentability condition for PCGLASSO and the proof that this condition is sufficient for consistent model selection. We further show that this condition is weaker than the corresponding irrepresentability condition for GLASSO, helping to explain the improved empirical behavior of PCGLASSO in settings such as hub-structured graphs. In addition, we develop two efficient algorithms for computing the estimator and analyze the nonconvex optimization problem underlying PCGLASSO, deriving conditions for global uniqueness and showing consistency of all minimizers.

math.ST↗

arXiv2025

Pattern recovery by SLOPE

SLOPE is a popular method for dimensionality reduction in the high-dimensional regression. Indeed some regression coefficient estimates of SLOPE can be null (sparsity) or can be equal in absolute value (clustering). Consequently, SLOPE may eliminate irrelevant predictors and may identify groups of predictors having the same influence on the vector of responses. The notion of SLOPE pattern allows to derive theoretical properties on sparsity and clustering by SLOPE. Specifically, the SLOPE pattern of a vector provides: the sign of its components (positive, negative or null), the clusters (indices of components equal in absolute value) and clusters ranking. In this article we give a necessary and sufficient condition for SLOPE pattern recovery of an unknown vector of regression coefficients.

math.ST↗

arXiv2025

Free Perpetuities I: Existence, Subordination and Tail Asymptotics

We study the free analogue of the classical affine fixed-point (or perpetuity) equation \[ \mathbb{X} \stackrel{d}{=} \mathbb{A}^{1/2}\mathbb{X}\,\mathbb{A}^{1/2} + \mathbb{B}, \] where $\mathbb{X}$ is assumed to be $*$-free from the pair $(\mathbb{A},\mathbb{B})$, with $\mathbb{A}\ge 0$ and $\mathbb{B}=\mathbb{B}^*$. Our analysis covers both the subcritical regime, where $τ(\mathbb{A})<1$, and the critical case $τ(\mathbb{A})=1$, in which the solution $\mathbb{X}$ is necessarily unbounded. When $τ(\mathbb{A})=1$, we prove that the series defining $\mathbb{X}$ converges bilaterally almost uniformly (and almost uniformly under additional tail assumptions), while the perpetuity fails to have higher moments even if all moments of $\mathbb{A}$ and $\mathbb{B}$ exist. Our approach relies on a detailed study of the asymptotic behavior of moments under free multiplicative convolution, which reveals a markedly different behavior from the classical setting. By employing subordination techniques for non-commutative random variables, we derive precise asymptotic estimates for the tail of the distributions of $\mathbb{X}$ in both one-sided and symmetric cases. Interestingly, in the critical case, the free perpetuity exhibits a power-law tail behavior that mirrors the phenomenon observed in the celebrated Kesten's theorem.

math.PR↗

arXiv2025

Graphical Negative Multinomial and Multinomial Models with Dirichlet-type priors

Bayesian statistical graphical models are typically classified as either continuous and parametric (Gaussian, parameterized by the graph-dependent precision matrix with Wishart-type priors) or discrete and non-parametric (with graph-dependent structure of probabilities of cells and Dirichlet-type priors). We propose to break this dichotomy by introducing two discrete parametric graphical models on finite decomposable graphs: the graph negative multinomial and the graph multinomial distributions (the former related to the Cartier-Foata theorem for the graph genereted free quotient monoid). These models interpolate between the product of univariate negative binomial laws and the negative multinomial distribution, and between the product of binomial laws and the multinomial distribution, respectively. We derive their Markov decompositions and provide related probabilistic representations. We also introduce graphical versions of the Dirichlet and inverted Dirichlet distributions, which serve as conjugate priors for the two discrete graphical Markov models. We derive explicit normalizing constants for both graphical Dirichlet laws and establish their independence structure (a graphical version of neutrality), which yields a strong hyper Markov property for both Bayesian models. We also provide characterization theorems for graphical Dirichlet laws via respective graphical versions of neutrality, which extends previously known results.

math.PR↗

arXiv2025

Perpetuities with light tails and the local dependence measure

This work investigates the tail behavior of solutions to the affine stochastic fixed-point equation of the form $X\stackrel{d}{=}AX+B$, where $X$ and $(A,B)$ are independent. Focusing on the light-tail regime, following [Burdzy et al. (2022), Ann. Appl. Probab.] we introduce a local dependence measure along with an associated Legendre-type transform. These tools allow us to effectively describe the logarithmic right-tail asymptotics of the solution $X$. Moreover, we extend our analysis to a related recursive sequence $X_n=A_n X_{n-1}+B_n$, where $(A_n,B_n)_{n}$ are i.i.d. copies of $(A,B)$. For this sequence, we construct deterministic scaling $(f_n)_{n}$ such that $\limsup_{n\to\infty} X_n/ f_n$ is a.s. positive and finite, with its non-random explicit value provided.

math.PR↗

arXiv2023

Learning permutation symmetries with gips in R

The study of hidden structures in data presents challenges in modern statistics and machine learning. We introduce the $\mathbf{gips}$ package in R, which identifies permutation subgroup symmetries in Gaussian vectors. $\mathbf{gips}$ serves two main purposes: exploratory analysis in discovering hidden permutation symmetries and estimating the covariance matrix under permutation symmetry. It is competitive to canonical methods in dimensionality reduction while providing a new interpretation of the results. $\mathbf{gips}$ implements a novel Bayesian model selection procedure within Gaussian vectors invariant under the permutation subgroup introduced in Graczyk, Ishi, Kołodziejek, Massam, Annals of Statistics, 50 (3) (2022).

stat.CO↗

arXiv2023

Analysis of Conditional Randomisation and Permutation schemes with application to conditional independence testing

We study properties of two resampling scenarios: Conditional Randomisation and Conditional Permutation schemes, which are relevant for testing conditional independence of discrete random variables $X$ and $Y$ given a random variable $Z$. Namely, we investigate asymptotic behaviour of estimates of a vector of probabilities in such settings, establish their asymptotic normality and ordering between asymptotic covariance matrices. The results are used to derive asymptotic distributions of the empirical Conditional Mutual Information in those set-ups. Somewhat unexpectedly, the distributions coincide for the two scenarios, despite differences in the asymptotic distributions of the estimates of probabilities. We also prove validity of permutation p-values for the Conditional Permutation scheme. The above results justify consideration of conditional independence tests based on resampled p-values and on the asymptotic chi-square distribution with an adjusted number of degrees of freedom. We show in numerical experiments that when the ratio of the sample size to the number of possible values of the triple exceeds 0.5, the test based on the asymptotic distribution with the adjustment made on a limited number of permutations is a viable alternative to the exact test for both the Conditional Permutation and the Conditional Randomisation scenarios. Moreover, there is no significant difference between the performance of exact tests for Conditional Permutation and Randomisation schemes, the latter requiring knowledge of conditional distribution of $X$ given $Z$, and the same conclusion is true for both adaptive tests.

math.ST↗

arXiv2022

Graphical Gaussian models associated to a homogeneous graph with permutation symmetries

We consider multivariate centered Gaussian models for the random vector $(Z^1,\ldots, Z^p)$, whose conditional structure is described by a homogeneous graph and which is invariant under the action of a permutation subgroup. The following paper concerns with model selection within colored graphical Gaussian models, when the underlying conditional dependency graph is known. We derive an analytic expression of the normalizing constant of the Diaconis-Ylvisaker conjugate prior for the precision parameter and perform Bayesian model selection in the class of graphical Gaussian models invariant by the action of a permutation subgroup. We illustrate our results with a toy example of dimension $5$.

math.ST↗

arXiv2022

Pattern recovery and signal denoising by SLOPE when the design matrix is orthogonal

Sorted $\ell_1$ Penalized Estimator (SLOPE) is a relatively new convex regularization method for fitting high-dimensional regression models. SLOPE allows to reduce the model dimension by shrinking some estimates of the regression coefficients completely to zero or by equating the absolute values of some nonzero estimates of these coefficients. This allows to identify situations where some of~true regression coefficients are equal. In this article we will introduce the SLOPE pattern, i.e., the set of relations between the true regression coefficients, which can be identified by SLOPE. We will also present new results on the strong consistency of SLOPE estimators and on~the~strong consistency of pattern recovery by~SLOPE when the design matrix is orthogonal and illustrate advantages of~the~SLOPE clustering in the context of high frequency signal denoising.

math.ST↗

arXiv2022

Model selection in the space of Gaussian models invariant by symmetry

We consider multivariate centered Gaussian models for the random variable $Z=(Z_1,\ldots, Z_p)$, invariant under the action of a subgroup of the group of permutations on $\{1,\ldots, p\}$. Using the representation theory of the symmetric group on the field of reals, we derive the distribution of the maximum likelihood estimate of the covariance parameter $Σ$ and also the analytic expression of the normalizing constant of the Diaconis-Ylvisaker conjugate prior for the precision parameter $K=Σ^{-1}$. We can thus perform Bayesian model selection in the class of complete Gaussian models invariant by the action of a subgroup of the symmetric group, which we could also call complete RCOP models. We illustrate our results with a toy example of dimension $4$ and several examples for selection within cyclic groups, including a high dimensional example with $p=100$.

math.ST↗

arXiv2022

A phase transition for tails of the free multiplicative convolution powers

We study the behavior of the tail of a measure $μ^{\boxtimes t}$, where $\boxtimes t$ is the $t$-fold free multiplicative convolution power for $t\geq 1$. We focus on the case where $μ$ is a probability measure on the positive half-line with a regularly varying tail i.e. of the form $x^{-α} L(x)$, where $L$ is slowly varying. We obtain a phase transition in the behavior of the tail of $μ^{\boxplus t}$ between regimes $α<1$ and $α>1$. Our main tool is a description of the regularly varying tails of $μ$ in terms of the behavior of the corresponding $S$-transform at $0^-$. We also describe the tails of $\boxtimes$ infinitely divisible measures in terms of the tails of corresponding Lévy measure, treat symmetric measures with regularly varying tails and prove the free analog of the Breiman lemma.

math.PR↗

arXiv2020

Stochastic fixed point equation and local dependence measure

We study solutions to the stochastic fixed point equation $X\stackrel{d}{=}AX+B$ where the coefficients $A$ and $B$ are nonnegative random variables. We introduce the ``local dependence measure'' (LDM) and its Legendre-type transform to analyze the left tail behavior of the distribution of $X$. We discuss the relationship of LDM with earlier results on the stochastic fixed point equation and we apply LDM to prove a theorem on a Fleming-Viot-type process.

math.PR↗