arXiv subjects
Publications and source records attributed to Bartosz Trojan.
This article deals with harmonic analysis on affine buildings. Its main goal is to construct suitable kernels associated to a discrete multitemporal wave equations on the latter spaces, the long-standing motivation being to contribute to progress in arithmetic quantum unique ergodicity (AQUE) on certain Riemannian manifolds.
Modern Transformer-based models frequently suffer from miscalibration, producing overconfident predictions that do not reflect true empirical frequencies. This work investigates the calibration dynamics of LoRA: Low-Rank Adaptation and a novel hyper-network-based adaptation framework as parameter-efficient alternatives to full fine-tuning for RoBERTa. Evaluating across the GLUE benchmark, we demonstrate that LoRA-based adaptation consistently achieves calibration parity with (and in specific tasks exceeds) full fine-tuning, while maintaining significantly higher parameter efficiency. We further explore a dynamic approach where a shared hyper-network generates LoRA factors (A and B matrices) to induce structural coupling across layers. This approach produced results similar to standard LoRA fine-tuning, even achieving better MCC on CoLA dataset. Our study also reveal a critical trade-off: constraining the adaptation space (e.g., freezing matrices A) acts as a powerful regularizer that enhances Expected Calibration Error (ECE), but necessitates a carefully balanced sacrifice in downstream task accuracy. To support future research, we provide a unified and reproducible implementation of contemporary calibration metrics, including ECE, MCE, and ACE. Our findings clarify the relationship between parameter efficiency and probabilistic reliability, positioning structured low-rank updates as a viable foundation for uncertainty-aware Transformer architectures. Code available at: https://github.com/btrojan-official/HypeLoRA
We introduce a new class of Sturm-Liouville operators with periodically modulated parameters. Their spectral properties depend on the monodromy matrix of the underlying periodic problem computed for the spectral parameter equal to $0$. Under certain assumptions, by studying the asymptotic behavior of Christoffel functions and density of states, we prove that the spectral density is a continuous positive everywhere function on the real line.
We carry out an in-depth study of Martin compactifications of affine buildings, from the viewpoint of potential theory and random walks. This work does not use any group action on buildings, although all the results are also stated within the framework of the Bruhat--Tits theory of semisimple groups over non-Archimedean local fields. This choice should allow the use of these building compactifications in intriguing geometric group theory situations, where only lattice actions are available. The resulting compactified spaces use and, at the same time, make it possible to understand geometrically the descriptions of asymptotic behavior of kernels resulting from the non-Archimedean harmonic analysis on affine buildings. Along the paper, we make explicit the most substantial differences with the case of symmetric spaces, namely absence of a group action but existence of precise asymptotics of Green kernels and, of course, no possibility to stand by standard techniques from PDEs.
We study the large-time asymptotic behavior of solutions to the discrete-time heat equation, i.e., caloric functions, on affine buildings, including those without transitive group actions. For each $p \in [1, \infty]$, we introduce a notion of a $p$-mass function and prove that caloric functions with initial data belonging to certain weighted-$\ell^1$ spaces or to the radial $\ell^1$ class, asymptotically decouple as the product of this mass function and the heat kernel. These results extend classical analogues from Euclidean spaces and symmetric spaces of non-compact type to the non-Archimedean setting, and remain valid even for exotic buildings beyond the Bruhat--Tits framework. We characterize the spatial concentration of heat kernels in $p$-norms and describe the geometry of associated critical regions. Our results highlight substantial differences in the asymptotic regimes depending on the value of $p$, and clarify the interplay between volume growth and heat diffusion.
The ability of deep learning models to learn continuously is essential for adapting to new data categories and evolving data distributions. In recent years, approaches leveraging frozen feature extractors after an initial learning phase have been extensively studied. Many of these methods estimate per-class covariance matrices and prototypes based on backbone-derived feature representations. Within this paradigm, we introduce FeNeC (Feature Neighborhood Classifier) and FeNeC-Log, its variant based on the log-likelihood function. Our approach generalizes the existing concept by incorporating data clustering to capture greater intra-class variability. Utilizing the Mahalanobis distance, our models classify samples either through a nearest neighbor approach or trainable logit values assigned to consecutive classes. Our proposition may be reduced to the existing approaches in a special case while extending them with the ability of more flexible adaptation to data. We demonstrate that two FeNeC variants achieve competitive performance in scenarios where task identities are unknown and establish state-of-the-art results on several benchmarks.
While stochastic resetting (or total resetting) is less young and more established concept in stochastic processes, partial stochastic resetting (PSR) is a relatively new field. PSR means that, at random moments in time, a stochastic process gets multiplied by a factor between 0 and 1, thus approaching but not reaching the resetting position. In this paper, we present new results on PSR highlighting the main similarities and discrepancies with total resetting. Specifically, we consider both symmetric $α$-stable Lévy processes (Lévy flights) and Brownian motion with PSR in arbitrary d dimensions. We derive explicit expressions for the propagator and its stationary measure, and discuss in detail their asymptotic behavior. Interestingly, while approaching to stationarity, a dynamical phase transition occurs for the Brownian motion, but not for Lévy flights. We also analyze the behavior of the process around the resetting position and find significant differences between PSR and total resetting.
We study a $d$-dimensional stochastic process $\mathbf{X}$ which arises from a Lévy process $\mathbf{Y}$ by partial resetting, that is the position of the process $\mathbf{X}$ at a Poisson moment equals $c$ times its position right before the moment, and it develops as $\mathbf{Y}$ between these two consecutive moments, $c \in (0, 1)$. We focus on $\mathbf{Y}$ being a strictly $α$-stable process with $α\in (0,2]$ having a transition density: We analyze properties of the transition density $p$ of the process $\mathbf{X}$. We establish a series representation of $p$. We prove its convergence as time goes to infinity (ergodicity), and we show that the limit $ρ_{\mathbf{Y}}$ (density of the ergodic measure) can be expressed by means of the transition density of the process $\mathbf{Y}$ starting from zero, which results in closed concise formulae for its moments. We show that the process $\mathbf{X}$ reaches a non-equilibrium stationary state. Furthermore, we check that $p$ satisfies the Fokker--Planck equation, and we confirm the harmonicity of $ρ_{\mathbf{Y}}$ with respect to the adjoint generator. In detail, we discuss the following cases: Brownian motion, isotropic and $d$-cylindrical $α$-stable processes for $α\in (0,2)$, and $α$-stable subordinator for $α\in (0,1)$. We find the asymptotic behavior of $p(t;x,y)$ as $t\to +\infty$ while $(t,y)$ stays in a certain space-time region. For Brownian motion, we discover a phase transition, that is a change of the asymptotic behavior of $p(t;0,y)$ with respect to $ρ_{\mathbf{Y}}(y)$.
We obtain the optimal global upper and lower bounds for the transition density $p_n(x,y)$ of a finite range isotropic random walk on affine buildings. We present also sharp estimates for the corresponding Green function.
We study a distinguished random walk on affine buildings of type Ar , which was already considered by Cartwright, Saloff-Coste and Woess. In rank r=2, it is the simple random walk and we obtain optimal global bounds for its transition density (same upper and lower bound, up to multiplicative constants). In the higher rank case, we obtain sharp uniform bounds in fairly large space-time regions which are sufficient for most applications.
We study spectrum of finite truncations of unbounded Jacobi matrices with periodically modulated entries. In particular, we show that under some hypotheses a sequence of properly normalized eigenvalue counting measures converge vaguely to an explicit infinite Radon measure. To do so we link the asymptotic behavior of the Christoffel-Darboux kernel on the diagonal with the limiting measure. Finally, we derive strong asymptotics of the associated orthogonal polynomials in the complex plane, which allows us to prove that Cauchy transforms of the normalized eigenvalue counting measures converge pointwise and which leads to a stronger notion of convergence.
We establish sharp upper and lower estimates of the Dunkl kernel in the case of dihedral groups.
We study Jacobi matrices with $N$-periodically modulated recurrence coefficients when the sequence of $N$-step transfer matrices is convergent to a non-trivial Jordan block. In particular, we describe asymptotic behavior of their generalized eigenvectors, we prove convergence of $N$-shifted Turán determinants as well as of the Christoffel--Darboux kernel on the diagonal. Finally, by means of subordinacy theory, we identify their absolutely continuous spectrum as well as their essential spectrum. By quantifying the speed of convergence of transfer matrices we were able to cover a large class of Jacobi matrices. In particular, those related to generators of birth-death processes.
We study orthogonal polynomials with periodically modulated recurrence coefficients when $0$ lies on the hard edge of the spectrum of the corresponding periodic Jacobi matrix. In particular, we show that their orthogonality measure is purely absolutely continuous on a real half-line and purely discrete on its complement. Additionally, we provide the constructive formula for the density in terms of Turán determinants. Moreover, we determine the exact asymptotic behavior of the orthogonal polynomials. Finally, we study scaling limits of the Christoffel-Darboux kernel.
We prove sharp estimates on heat kernels and Green functions for subordinate Markov processes with both discrete an continuous time, under relatively weak assumptions about original processes as well as Laplace exponents of subordinators. We also show stability of heat kernel estimates in the discrete settings.
We show the pointwise convergence of the averages \[ \mathcal{A}_N f(x) = \frac{1}{\# \mathbf{B}_N} \sum_{n \in \mathbf{B}_N} f(x + n) \] for $f \in \ell^1(\mathbb{Z})$ where $\mathbf{B}_N = \mathbf{B} \cap [1, N]$, and $\mathbf{B}$ is a $1$-regular sequence of integers, for example $\mathbf{B} = \{\lfloor n \log n \rfloor : n \in \mathbb{N}\}$.
We prove existence and asymptotic behavior of the transition density for a large class of subordinators whose Laplace exponents satisfy lower scaling condition at infinity. Furthermore, we present lower and upper bounds for the density. Sharp estimates are provided if additional upper scaling condition on the Laplace exponent is imposed. In particular, we cover the case when the (minus) second derivative of the Laplace exponent is a function regularly varying at infinity with regularity index bigger than -2.
We prove that for any martingale with respect to a biparameter atomic filtration satisfying $(F_4)$ condition there is a martingale having the same joint distribution but with respect to the canonical $(F_4)$ filtration. Even in one parameter case our result is an improvement of the theorem due to Montgomery-Smith, since the construction gives a morphism of filtrations and does not depend on underlying sequence.