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Piotr Zwiernik

Piotr Zwiernik contributes to research discovery and scholarly infrastructure.

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math.ST Machine Learning Statistics Theory Methodology math.AG math.OC

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preprint2026arXiv

Causal discovery under mean independence and linearity

Causal discovery methods such as LiNGAM identify causal structure from observational data by assuming mutually independent disturbances. This assumption is fragile: shared volatility, common scale effects, or other forms of dependence can cause the methods to recover the wrong causal order, even with infinite data. We introduce the Linear Mean-Independent Acyclic Model (LiMIAM), which replaces full independence with weaker one-sided mean-independence restrictions on the disturbances. Under finite-order consequences of these restrictions, source nodes are generically identifiable, and hence a compatible causal order can be recovered recursively. Our proof is constructive and leads to DirectLiMIAM, a sequential residual-based algorithm for causal discovery under dependent noise. In simulations with mean-independent but dependent disturbances, DirectLiMIAM outperforms LiNGAM methods. A large-scale empirical application to the oil market highlights the implausibility of the independence assumption and the ability of DirectLiMIAM to recover a realistic causal ordering, from policy to production and from prices to inflation.

preprint2026arXiv

Testing properties of trees in graphical models with covariance queries

We consider the problem of testing properties of graphs underlying high-dimensional graphical models. We adopt the model of covariance queries introduced by Lugosi, Truszkowski, Velona, and Zwiernik (2021). We study the case when the underlying graph is a tree. The main results of the paper show that, while reconstructing the entire tree may be costly, certain global structural properties can be tested efficiently. In particular, we design randomized tests for global structural properties that use a sub-quadratic number of queries. We develop testing procedures for several fundamental properties, including the number of leaves, the maximum degree, the typical distance, and the diameter of the tree. For each property, we obtain explicit query complexity bounds that depend on the target threshold and tolerance parameters.

preprint2022arXiv

Locally associated graphical models and mixed convex exponential families

The notion of multivariate total positivity has proved to be useful in finance and psychology but may be too restrictive in other applications. In this paper we propose a concept of local association, where highly connected components in a graphical model are positively associated and study its properties. Our main motivation comes from gene expression data, where graphical models have become a popular exploratory tool. The models are instances of what we term mixed convex exponential families and we show that a mixed dual likelihood estimator has simple exact properties for such families as well as asymptotic properties similar to the maximum likelihood estimator. We further relax the positivity assumption by penalizing negative partial correlations in what we term the positive graphical lasso. Finally, we develop a GOLAZO algorithm based on block-coordinate descent that applies to a number of optimization procedures that arise in the context of graphical models, including the estimation problems described above. We derive results on existence of the optimum for such problems.

preprint2021arXiv

Likelihood Geometry of Correlation Models

Correlation matrices are standardized covariance matrices. They form an affine space of symmetric matrices defined by setting the diagonal entries to one. We study the geometry of maximum likelihood estimation for this model and linear submodels that encode additional symmetries. We also consider the problem of minimizing two closely related functions of the covariance matrix: the Stein's loss and the symmetrized Stein's loss. Unlike the Gaussian log-likelihood these two functions are convex and hence admit a unique positive definite optimum. Some of our results hold for general affine covariance models.

preprint2021arXiv

Robust estimation of tree structured models

Consider the problem of learning undirected graphical models on trees from corrupted data. Recently Katiyar et al. showed that it is possible to recover trees from noisy binary data up to a small equivalence class of possible trees. Their other paper on the Gaussian case follows a similar pattern. By framing this as a special phylogenetic recovery problem we largely generalize these two settings. Using the framework of linear latent tree models we discuss tree identifiability for binary data under a continuous corruption model. For the Ising and the Gaussian tree model we also provide a characterisation of when the Chow-Liu algorithm consistently learns the underlying tree from the noisy data.

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