arXiv subjects

Maciej Wilczyński

Publications and source records attributed to Maciej Wilczyński.

3 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↗

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↗

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↗