arXiv subjects

Jacek Grela

Publications and source records attributed to Jacek Grela.

18 recordsLinked to original sources

arXiv2026

Multifractal Signatures of Ageing and Dementia Development: A Multifractal Space-Filling Curve Analysis

Multifractality is an effective formalism for quantifying the nonlinear, scale-free properties of complex data. In this study, we propose a novel and efficient methodology, termed Multifractal Space-filling Curve Analysis (MFSCA), for quantifying the correlation structure of multidimensional data. Within this framework, the original multidimensional data - while preserving both local and long-range organisational properties - are projected onto a one-dimensional representation using a fractal space-filling curve. The resulting one-dimensional signal is then analysed using multifractal algorithms. We demonstrate the utility of the method using both artificially generated multifractal structures and real data. In particular, we apply MFSCA to analyse magnetic resonance imaging (MRI) data from Alzheimer patients at different stages of dementia. Based on the results, we estimate the multifractal profiles of the brain for healthy subjects of different ages as well as for dementia patients. The analysis reveals that the spatial organization of brain structures, as measured by the degree of multifractality, progressively weakens with age and the development of dementia. A transition from multifractality to monofractality is observed both in control groups, when comparing the Young Control and Elderly Control groups, and among dementia subjects of similar age but at different stages of the disease, namely early dementia and mild cognitive impairment. Thus, from the perspective of multiscaling properties, the heterogeneous characteristics of spatial brain organization deteriorate under worsening conditions, leading to a homogeneous and weakly correlated structure. These findings not only effectively capture key aspects of brain organisation, but also demonstrate that the multifractality of MRI data can serve as a marker of structural brain changes.

q-bio.NC↗

arXiv2025

Using Space-Filling Curves and Fractals to Reveal Spatial and Temporal Patterns in Neuroimaging Data

We present a novel method, Fractal Space-Curve Analysis (FSCA), which combines Space-Filling Curve (SFC) mapping for dimensionality reduction with fractal Detrended Fluctuation Analysis (DFA). The method is suitable for multidimensional geometrically embedded data, especially for neuroimaging data which is highly correlated temporally and spatially. We conduct extensive feasibility studies on diverse, artificially generated data with known fractal characteristics: the fractional Brownian motion, Cantor sets, and Gaussian processes. We compare the suitability of dimensionality reduction via Hilbert SFC and a data-driven alternative. FSCA is then successfully applied to real-world magnetic resonance imaging (MRI) and functional MRI (fMRI) scans. The method utilizing Hilbert curves is optimized for computational efficiency, proven robust against boundary effects typical in experimental data analysis, and resistant to data sub-sampling. It is able to correctly quantify and discern correlations in both stationary and dynamic two-dimensional images. In MRI Alzheimer's dataset, patients reveal a progression of the disease associated with a systematic decrease of the Hurst exponent. In fMRI recording of breath-holding task, the change in the exponent allows distinguishing different experimental phases. This study introduces a robust method for fractal characterization of spatial and temporal correlations in many types of multidimensional neuroimaging data. Very few assumptions allow it to be generalized to more dimensions than typical for neuroimaging and utilized in other scientific fields. The method can be particularly useful in analyzing fMRI experiments to compute markers of pathological conditions resulting from neurodegeneration. We also showcase its potential for providing insights into brain dynamics in task-related experiments.

q-bio.NC↗

arXiv2023

Investigating structural and functional aspects of the brain's criticality in stroke

This paper addresses the question of the brain's critical dynamics after an injury such as a stroke. It is hypothesized that the healthy brain operates near a phase transition (critical point), which provides optimal conditions for information transmission and responses to inputs. If structural damage could cause the critical point to disappear and thus make self-organized criticality unachievable, it would offer the theoretical explanation for the post-stroke impairment of brain function. In our contribution, however, we demonstrate using network models of the brain, that the dynamics remain critical even after a stroke. In cases where the average size of the second-largest cluster of active nodes, which is one of the commonly used indicators of criticality, shows an anomalous behavior, it results from the loss of integrity of the network, quantifiable within graph theory, and not from genuine non-critical dynamics. We propose a new simple model of an artificial stroke that explains this anomaly. The proposed interpretation of the results is confirmed by an analysis of real connectomes acquired from post-stroke patients and a control group. The results presented refer to neurobiological data; however, the conclusions reached apply to a broad class of complex systems that admit a critical state.

q-bio.NC↗

arXiv2022

SLOVA: Uncertainty Estimation Using Single Label One-Vs-All Classifier

Deep neural networks present impressive performance, yet they cannot reliably estimate their predictive confidence, limiting their applicability in high-risk domains. We show that applying a multi-label one-vs-all loss reveals classification ambiguity and reduces model overconfidence. The introduced SLOVA (Single Label One-Vs-All) model redefines typical one-vs-all predictive probabilities to a single label situation, where only one class is the correct answer. The proposed classifier is confident only if a single class has a high probability and other probabilities are negligible. Unlike the typical softmax function, SLOVA naturally detects out-of-distribution samples if the probabilities of all other classes are small. The model is additionally fine-tuned with exponential calibration, which allows us to precisely align the confidence score with model accuracy. We verify our approach on three tasks. First, we demonstrate that SLOVA is competitive with the state-of-the-art on in-distribution calibration. Second, the performance of SLOVA is robust under dataset shifts. Finally, our approach performs extremely well in the detection of out-of-distribution samples. Consequently, SLOVA is a tool that can be used in various applications where uncertainty modeling is required.

cs.LG↗

arXiv2022

Glass--like transition described by toppling of stability hierarchy

Building on the work of Fyodorov (2004) and Fyodorov and Nadal (2012) we examine the critical behaviour of population of saddles with fixed instability index $k$ in high dimensional random energy landscapes. Such landscapes consist of a parabolic confining potential and a random part in $N\gg 1$ dimensions. When the relative strength $m$ of the parabolic part is decreasing below a critical value $m_c$, the random energy landscapes exhibit a glass-like transition from a simple phase with very few critical points to a complex phase with the energy surface having exponentially many critical points. We obtain the annealed probability distribution of the instability index $k$ by working out the mean size of the population of saddles with index $k$ relative to the mean size of the entire population of critical points and observe toppling of stability hierarchy which accompanies the underlying glass-like transition. In the transition region $m=m_c + δN^{-1/2}$ the typical instability index scales as $k = κN^{1/4}$ and the toppling mechanism affects whole instability index distribution, in particular the most probable value of $κ$ changes from $κ= 0$ in the simple phase ($δ> 0 $) to a non-zero value $ κ_{\max} \propto (-δ)^{3/2}$ in the complex phase ($δ< 0$). We also show that a similar phenomenon is observed in random landscapes with an additional fixed energy constraint and in the $p$-spin spherical model.

math-ph↗

arXiv2022

Eikonal formulation of large dynamical random matrix models

Standard approach to dynamical random matrix models relies on the description of trajectories of eigenvalues. Using the analogy from optics, based on the duality between the Fermat principle(trajectories) and the Huygens principle (wavefronts), we formulate the Hamilton-Jacobi dynamics for large random matrix models. The resulting equations describe a broad class of random matrix models in a unified way, including normal (Hermitian or unitary) as well as strictly non-normal dynamics. HJ formalism applied to Brownian bridge dynamics allows one for calculations of the asymptotics of the Harish-Chandra-Itzykson-Zuber integrals.

math-ph↗

arXiv2021

Non-intersecting Brownian bridges in the flat-to-flat geometry

We study $N$ vicious Brownian bridges propagating from an initial configuration $\{a_1 < a_2 < \ldots< a_N \}$ at time $t=0$ to a final configuration $\{b_1 < b_2 < \ldots< b_N \}$ at time $t=t_f$, while staying non-intersecting for all $0\leq t \leq t_f$. We first show that this problem can be mapped to a non-intersecting Dyson's Brownian bridges with Dyson index $β=2$. For the latter we derive an exact effective Langevin equation that allows to generate very efficiently the vicious bridge configurations. In particular, for the flat-to-flat configuration in the large $N$ limit, where $a_i = b_i = (i-1)/N$, for $i = 1, \cdots, N$, we use this effective Langevin equation to derive an exact Burgers' equation (in the inviscid limit) for the Green's function and solve this Burgers' equation for arbitrary time $0 \leq t\leq t_f$. At certain specific values of intermediate times $t$, such as $t=t_f/2$, $t=t_f/3$ and $t=t_f/4$ we obtain the average density of the flat-to-flat bridge explicitly. We also derive explicitly how the two edges of the average density evolve from time $t=0$ to time $t=t_f$. Finally, we discuss connections to some well known problems, such as the Chern-Simons model, the related Stieltjes-Wigert orthogonal polynomials and the Borodin-Muttalib ensemble of determinantal point processes.

cond-mat.stat-mech↗

arXiv2020

Extreme matrices or how an exponential map links classical and free extreme laws

Using the proposed by us thinning approach to describe extreme matrices, we find an explicit exponentiation formula linking classical extreme laws of Fréchet, Gumbel and Weibull given by Fisher-Tippet-Gnedenko classification and free extreme laws of free Fréchet, free Gumbel and free Weibull by Ben Arous and Voiculescu [1]. We also develop an extreme random matrix formalism, in which refined questions about extreme matrices can be answered. In particular, we demonstrate explicit calculations for several more or less known random matrix ensembles, providing examples of all three free extreme laws. Finally, we present an exact mapping, showing the equivalence of free extreme laws to the Peak-Over-Threshold method in classical probability.

math-ph↗

arXiv2018

Kinetic energy of a trapped Fermi gas at finite temperature

We study the statistics of the kinetic (or equivalently potential) energy for $N$ non-interacting fermions in a $1d$ harmonic trap of frequency $ω$, at finite temperature $T$. Remarkably, we find an exact solution for the full distribution of the kinetic energy, at any temperature $T$ and for any $N$, using a non-trivial mapping to an integrable Calogero-Moser-Sutherland model. As a function of temperature $T$, and for large $N$, we identify: (i) a quantum regime, for $T \sim \hbar ω$, where quantum fluctuations dominate and (ii) a thermal regime, for $T \sim N \hbar ω$, governed by thermal fluctuations. We show how the mean, the variance as well as the large deviation function associated with the distribution of the kinetic energy cross over from the quantum to the thermal regime as temperature increases.

cond-mat.stat-mech↗

arXiv2018

Full Dysonian dynamics of the complex Ginibre ensemble

We find stochastic equations governing eigenvalues and eigenvectors of a dynamical complex Ginibre ensemble reaffirming the intertwined role played between both sets of matrix degrees of freedom. We solve the accompanying Smoluchowski-Fokker-Planck equation valid for any initial matrix. We derive evolution equations for the averaged extended characteristic polynomial and for a class of $k$-point eigenvalue correlation functions. From the latter we obtain a novel formula for the eigenvector correlation function which we inspect for Ginibre and spiric initial conditions and obtain macro- and microscopic limiting laws.

math-ph↗

arXiv2018

On characteristic polynomials for a generalized chiral random matrix ensemble with a source

We evaluate averages involving characteristic polynomials, inverse characteristic polynomials and ratios of characteristic polynomials for a $N\times N$ random matrix taken from a $L$-deformed Chiral Gaussian Unitary Ensemble with an external source $Ω$. Relation to a recently studied statistics of bi-orthogonal eigenvectors in the complex Ginibre ensemble, see Y.V. Fyodorov arXiv:1710.04699, is briefly discussed as a motivation to study asymptotics of these objects in the case of external source proportional to the identity matrix. In particular, for an associated 'complex bulk/chiral edge' scaling regime we retrieve the kernel related to Bessel/Macdonald functions.

math-ph↗

arXiv2017

What drives transient behaviour in complex systems?

We study transient behaviour in the dynamics of complex systems described by a set of non-linear ODE's. Destabilizing nature of transient trajectories is discussed and its connection with the eigenvalue-based linearization procedure. The complexity is realized as a random matrix drawn from a modified May-Wigner model. Based on the initial response of the system, we identify a novel stable-transient regime. We calculate exact abundances of typical and extreme transient trajectories finding both Gaussian and Tracy-Widom distributions known in extreme value statistics. We identify degrees of freedom driving transient behaviour as connected to the eigenvectors and encoded in a non-orthogonality matrix $T_0$. We accordingly extend the May-Wigner model to contain a phase with typical transient trajectories present. An exact norm of the trajectory is obtained in the vanishing $T_0$ limit where it describes a normal matrix.

nlin.CD↗

arXiv2016

Exact spectral densities of complex noise-plus-structure random matrices

We use supersymmetry to calculate exact spectral densities for a class of complex random matrix models having the form $M=S+LXR$, where $X$ is a random noise part $X$ and $S,L,R$ are fixed structure parts. This is a certain version of the "external field" random matrix models. We found two-fold integral formulas for arbitrary structural matrices. We investigate some special cases in detail and carry out numerical simulations. The presence or absence of a normality condition on $S$ leads to a qualitatively different behavior of the eigenvalue densities.

math-ph↗

arXiv2016

Ornstein-Uhlenbeck diffusion of hermitian and non-hermitian matrices - unexpected links

We compare the Ornstein-Uhlenbeck process for the Gaussian Unitary Ensemble to its non-hermitian counterpart - for the complex Ginibre ensemble. We exploit the mathematical framework based on the generalized Green's functions, which involves a new, hidden complex variable, in comparison to the standard treatment of the resolvents. This new variable turns out to be crucial to understand the pattern of the evolution of non-hermitian systems. The new feature is the emergence of the coupling between the flow of eigenvalues and that of left/right eigenvectors. We analyze local and global equilibria for both systems. Finally, we highlight some unexpected links between both ensembles.

math-ph↗

arXiv2015

Diffusion method in Random Matrix Theory

We introduce a simple yet powerful calculational tool useful in calculating averages of ratios and products of characteristic polynomials. The method is based on Dyson Brownian motion and Grassmann integration formula for determinants. It is intended as an alternative to other RMT techniques applicable to general gaussian measures. Resulting formulas are exact for finite matrix size N and form integral representations convenient for large N asymptotics. Quantities obtained by the method can be interpreted as averages over matrix models with an external source. We provide several explicit and novel calculations showing a range of applications.

math-ph↗

arXiv2015

Unveiling the significance of eigenvectors in diffusing non-hermitian matrices by identifying the underlying Burgers dynamics

Following our recent letter, we study in detail an entry-wise diffusion of non-hermitian complex matrices. We obtain an exact partial differential equation (valid for any matrix size $N$ and arbitrary initial conditions) for evolution of the averaged extended characteristic polynomial. The logarithm of this polynomial has an interpretation of a potential which generates a Burgers dynamics in quaternionic space. The dynamics of the ensemble in the large $N$ is completely determined by the coevolution of the spectral density and a certain eigenvector correlation function. This coevolution is best visible in an electrostatic potential of a quaternionic argument built of two complex variables, the first of which governs standard spectral properties while the second unravels the hidden dynamics of eigenvector correlation function. We obtain general large $N$ formulas for both spectral density and 1-point eigenvector correlation function valid for any initial conditions. We exemplify our studies by solving three examples, and we verify the analytic form of our solutions with numerical simulations.

math-ph↗

arXiv2014

Diffusion in the space of complex Hermitian matrices - microscopic properties of the averaged characteristic polynomial and the averaged inverse characteristic polynomial

We show that the averaged characteristic polynomial and the averaged inverse characteristic polynomial, associated with Hermitian matrices whose elements perform a random walk in the space of complex numbers, satisfy certain partial differential, diffusion-like, equations. These equations are valid for matrices of arbitrary size. Their solutions can be given an integral representation that allows for a simple study of their asymptotic behaviors for a broad range of initial conditions.

math-ph↗

arXiv2014

Dysonian dynamics of the Ginibre ensemble

We study the time evolution of Ginibre matrices whose elements undergo Brownian motion. The non-Hermitian character of the Ginibre ensemble binds the dynamics of eigenvalues to the evolution of eigenvectors in a non-trivial way, leading to a system of coupled nonlinear equations resembling those for turbulent systems. We formulate a mathematical framework allowing simultaneous description of the flow of eigenvalues and eigenvectors, and we unravel a hidden dynamics as a function of new complex variable, which in the standard description is treated as a regulator only. We solve the evolution equations for large matrices and demonstrate that the non-analytic behavior of the Green's functions is associated with a shock wave stemming from a Burgers-like equation describing correlations of eigenvectors. We conjecture that the hidden dynamics, that we observe for the Ginibre ensemble, is a general feature of non-Hermitian random matrix models and is relevant to related physical applications.

math-ph↗