Searcharxiv⌕ Search

arXiv subjects

Piotr Graczyk

Publications and source records attributed to Piotr Graczyk.

At least 19 recordsLinked to original sources
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 computational bottleneck in Bayesian inference for CGGMs is the evaluation of Diaconis-Ylvisaker normalizing constants, given by gamma-type integrals over cones of precision matrices with prescribed zeros and equality constraints. While explicit formulas are known for standard Gaussian graphical models only in special cases (e.g. decomposable graphs) and for a limited class of RCOP models, no general tractable framework has been available for broader families of CGGMs. 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 spaces of colored precision matrices that guarantee tractability of the associated integrals, leading to 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. This class strictly extends decomposable graphs in the uncolored setting and contains all RCOP models associated with decomposable graphs. In the one-color case, our framework reveals a close connection between DCBC-spaces and Bose-Mesner algebras. For models defined on BC- and DCBC-spaces, we derive explicit closed-form formulas for the normalizing constants in terms of a finite collection of structure constants and propose an efficient method for computing them in the commutative case. Our results broaden the range of CGGMs amenable to principled Bayesian structure learning in high-dimensional applications.

math.ST↗
arXiv2025

Hybrid structure with a ferromagnetic film and an array of magnetic molecules for deep-nanoscale reprogrammable magnonics

Miniaturization is an essential element in the development of information processing technologies and is also one of the main determinants of the usability of the tested artificial neural networks. It is also a key element and one of the main challenges in the development of magnonic neuromorphic systems. In this work, we propose a new platform for the development of these new spin-wave-based technologies. Using micromagnetic simulations, we demonstrate that magnetic molecules regularly arranged on the surface of a thin ferromagnetic layer enable resonant coupling of propagating spin waves with the dynamics of the molecules' magnetic moments, opening a gap in the transmission spectrum up to 150 MHz. The gap, its width, and frequency can be controlled by an external magnetic field or the arrangement of molecules on the ferromagnetic surface. Furthermore, the antiferromagnetic arrangement of the magnetic moments of molecules or clusters of molecules allows for control of the gap's position and width. Thus, the proposed hybrid structure offers reprogrammability and miniaturization down to the deep nanoscale, operating frequencies in the range of several GHz, key properties for the implementation of artificial neural networks.

cond-mat.mes-hall↗
arXiv2025

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

Estimating high-dimensional precision matrices is a fundamental problem in modern statistics, with the graphical lasso and its $\ell_1$-penalty being a standard approach for recovering sparsity patterns. However, many statistical models, e.g. colored graphical models, exhibit richer structures like symmetry or equality constraints, which the $\ell_1$-norm cannot adequately capture. This paper addresses the gap by extending the high-dimensional analysis of pattern recovery to a general class of atomic norm penalties, particularly those whose unit balls are polytopes, where patterns correspond to the polytope's facial structure. We establish theoretical guarantees for recovering the true pattern induced by these general atomic norms in precision matrix estimation. Our framework builds upon and refines the primal-dual witness methodology of Ravikumar et al. (2011). Our analysis provides conditions on the deviation between sample and true covariance matrices for successful pattern recovery, given a novel, generalized irrepresentability condition applicable to any atomic norm. When specialized to the $\ell_1$-penalty, our results offer improved conditions -- including weaker deviation requirements and a less restrictive irrepresentability condition -- leading to tighter bounds and better asymptotic performance than prior work. The proposed general irrepresentability condition, based on a new thresholding concept, provides a unified perspective on model selection consistency. Numerical examples demonstrate the tightness of the derived theoretical bounds.

math.ST↗
arXiv2025

Optical excitation and detection of high-frequency Sezawa modes in Si/SiO2 system decorated with Ni80Fe20 nanodot arrays

Surface acoustic waves have emerged as one of the potential candidates for the development of next-generation wave-based information and computing technologies. For practical devices, it is essential to develop the excitation techniques for different types of surface acoustic waves, especially at higher microwave frequencies, and to tailor their frequency versus wave vector characteristics. We show that this can be done by using ultrashort laser pulses incident on the surface of a multilayer decorated with a periodic array of metallic nanodots. Specifically, we study surface acoustic waves in the dielectric substrate Si/SiO2 decorated with a square lattice of thin Ni80Fe20 (Py) dots. Using a femtosecond laser-based optical pump-probe measurement, we detect a number of high-frequency phononic modes. By performing finite element simulations, we identify them as Sezawa modes from the second and third Brillouin zone in addition to the modes confined within the Py dots. The frequency of the Sezawa modes strongly depends on the period of the Py dots and varies in the range between 5 to 15 GHz. Both types of waves cover the same frequency range for Py dots with period less than 400 nm, providing a promising system for magnetoelastic studies.

cond-mat.mtrl-sci↗
arXiv2023

Generation of femtosecond spin-polarized current pulses at Fe/MgO interface by quasi-static voltage

The generation of short spin-current pulses is essential for fast spintronic devices. So far, spin current pulses are generated by femtosecond laser pulses which drive spins from a ferromagnetic metal layer. However, the need for miniaturization, simplicity and energy efficiency favour electric-field control of spintronic devices over optic or thermal control. Here, we combine ab initio calculations of electronic density of states at MgO/Fe interface with continuous model for charge transport to investigate the dynamics of the spin-dependent potential. We demonstrate that the voltage-driven instability of the electronic band structure due to the electronic resonant states at the Fe/MgO interface results in the generation of the femtosecond spin-polarized current pulse with the spin polarization up to P=700% that propagates from the interface to the bulk. The dynamics of the current pulses driven by the Stoner instability depends neither on the dielectric relaxation time nor on the details of how the instability is achieved by changing the voltage, i.e. as long as the voltage changes are slow (quasi-static) with respect to the time determined by the spin diffusion constant, being of the order of fs.The presence of the instability may be detected by transport or magnetic measurements.

cond-mat.mes-hall↗
arXiv2023

A Unified Framework for Pattern Recovery in Penalized and Thresholded Estimation and its Geometry

We consider the framework of penalized estimation where the penalty term is given by a real-valued polyhedral gauge, which encompasses methods such as LASSO, generalized LASSO, SLOPE, OSCAR, PACS and others. Each of these estimators is defined through an optimization problem and can uncover a different structure or ``pattern'' of the unknown parameter vector. We define a novel and general notion of patterns based on subdifferentials and formalize an approach to measure pattern complexity. For pattern recovery, we provide a minimal condition for a particular pattern to be detected by the procedure with positive probability, the so-called accessibility condition. Using our approach, we also introduce the stronger noiseless recovery condition. For the LASSO, it is well known that the irrepresentability condition is necessary for pattern recovery with probability larger than $1/2$ and we show that the noiseless recovery plays exactly the same role in our general framework, thereby unifying and extending the irrepresentability condition to a broad class of penalized estimators. We also show that the noiseless recovery condition can be relaxed when turning to so-called thresholded penalized estimators: we prove that the necessary condition of accessibility is already sufficient for sure pattern recovery by thresholded penalized estimation provided that the noise is small enough. Throughout the article, we demonstrate how our findings can be interpreted through a geometrical lens.

math.ST↗
arXiv2023

Advanced magnon-optic effects with spin-wave leaky modes

We numerically demonstrate the excitation of leaky spin waves (SWs) guided along a ferromagnetic stripe by an obliquely incident SW beam on the thin film edge placed below the stripe. During propagation, leaky waves emit energy back to the layer in the form of plane waves and several laterally shifted parallel SW beams. This resonance excitation, combined with interference effects of the reflected and re-emitted waves, results in the magnonic Woods anomaly and significant increase of the Goos-Hanchen shift magnitude. Hence, we provide a unique platform to control SW reflection and to transfer SWs from a 2D platform into the 1D guiding mode that can be used to form a transdimensional magnonic router.

cond-mat.mes-hall↗
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

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↗
arXiv2021

Sharp estimates for $W$-invariant Dunkl and heat kernels in the $A_n$ case

In this article, we prove exact estimates for the $W$-invariant Dunkl kernel and heat kernel, for the root system of type $A$ with arbitrary positive multiplicities. We apply the estimates of the $W$-invariant Dunkl heat kernel to compute sharp estimates for the Newton kernel and for the $s$-stable semigroups generated by a fractional power of the $W$-invariant Dunkl Laplacian.

math.RT↗
arXiv2020

VARCLUST: clustering variables using dimensionality reduction

VARCLUST algorithm is proposed for clustering variables under the assumption that variables in a given cluster are linear combinations of a small number of hidden latent variables, corrupted by the random noise. The entire clustering task is viewed as the problem of selection of the statistical model, which is defined by the number of clusters, the partition of variables into these clusters and the 'cluster dimensions', i.e. the vector of dimensions of linear subspaces spanning each of the clusters. The optimal model is selected using the approximate Bayesian criterion based on the Laplace approximations and using a non-informative uniform prior on the number of clusters. To solve the problem of the search over a huge space of possible models we propose an extension of the ClustOfVar algorithm which was dedicated to subspaces of dimension only 1, and which is similar in structure to the $K$-centroid algorithm. We provide a complete methodology with theoretical guarantees, extensive numerical experimentations, complete data analyses and implementation. Our algorithm assigns variables to appropriate clusterse based on the consistent Bayesian Information Criterion (BIC), and estimates the dimensionality of each cluster by the PEnalized SEmi-integrated Likelihood Criterion (PESEL), whose consistency we prove. Additionally, we prove that each iteration of our algorithm leads to an increase of the Laplace approximation to the model posterior probability and provide the criterion for the estimation of the number of clusters. Numerical comparisons with other algorithms show that VARCLUST may outperform some popular machine learning tools for sparse subspace clustering. We also report the results of real data analysis including TCGA breast cancer data and meteorological data. The proposed method is implemented in the publicly available R package varclust.

stat.CO↗
arXiv2020

Analysis on Lambert-Tsallis functions

In this paper, we study the Lambert-Tsallis function, which is a generalization of the Lambert function with two real parameters. We give a condition on the parameters such that there exists a complex domain touching zero on boundary which is mapped bijectively to the upper half plane by the Lambert-Tsallis function.

math.CV↗
arXiv2020

Wigner and Wishart Ensembles for graphical models

Vinberg cones and the ambient vector spaces are important in modern statistics of sparse models and of graphical models. The aim of this paper is to study eigenvalue distributions of Gaussian, Wigner and covariance matrices related to growing Vinberg matrices, corresponding to growing daisy graphs. For Gaussian or Wigner ensembles, we give an explicit formula for the limiting distribution. For Wishart ensembles defined naturally on Vinberg cones, their limiting Stieltjes transforms, support and atom at 0 are described explicitly in terms of the Lambert-Tsallis functions, which are defined by using the Tsallis $q$-exponential functions.

math.ST↗
arXiv2020

Spin-wave diode and circulator based on unidirectional coupling

In magnonics, an emerging branch of wave physics characterized by low-energy consumption, it is highly desirable to realize circuit elements within the scope of spin-wave computing. Here, based on numerical simulations, we demonstrate the functionality of the spin-wave diode and the circulator to steer and manipulate spin waves over a wide range of frequency in the GHz regime. They take advantage of the unidirectional coupling induced by the interfacial Dzyaloshinskii-Moriya interaction to transfer the spin wave between thin ferromagnetic layers in only one direction of propagation. Using the multilayered structure consisting of Py and Co in direct contact with Pt, we obtain sub-micrometer-size devices of high efficiency. In the diode, the power loss ratio between forward and reverse direction reaches 22 dB, while in the four-port circulator, the efficiency exceeds 13 dB. Thus, our work contributes to the emerging branch of energy-efficient magnonic logic devices, where, thanks to short wavelength of spin waves, it is possible to realize nanoscale devices.

cond-mat.mes-hall↗