arXiv subjects

Anna Veselovska

Publications and source records attributed to Anna Veselovska.

9 recordsLinked to original sources

arXiv2026

Reliable one-bit quantization of bandlimited graph data via single-shot noise shaping

Graph data are ubiquitous in natural sciences and machine learning. In this paper, we consider the problem of quantizing graph structured, bandlimited data to few bits per entry while preserving its information under low-pass filtering. We propose an efficient single-shot noise shaping method that achieves state-of-the-art performance and comes with rigorous error bounds. In contrast to existing methods it allows reliable quantization to arbitrary bit-levels including the extreme case of using a single bit per data coefficient.

cs.IT↗

arXiv2026

Fast training of accurate physics-informed neural networks without gradient descent

Solving time-dependent Partial Differential Equations (PDEs) is one of the most critical problems in computational science. While Physics-Informed Neural Networks (PINNs) offer a promising framework for approximating PDE solutions, their accuracy and training speed are limited by two core barriers: gradient-descent-based iterative optimization over complex loss landscapes and non-causal treatment of time as an extra spatial dimension. We present Frozen-PINN, a novel PINN based on the principle of space-time separation that leverages random features instead of training with gradient descent, and incorporates temporal causality by construction. On eight PDE benchmarks, including challenges such as extreme advection speeds, shocks, and high dimensionality, Frozen-PINNs achieve superior training efficiency and accuracy over state-of-the-art PINNs, often by several orders of magnitude. Our work addresses longstanding training and accuracy bottlenecks of PINNs, delivering quickly trainable, highly accurate, and inherently causal PDE solvers, a combination that prior methods could not realize. Our approach challenges the reliance of PINNs on stochastic gradient-descent-based methods and specialized hardware, leading to a paradigm shift in PINN training and providing a challenging benchmark for the community.

math.NA↗

arXiv2026

Low-Bit Quantization of Bandlimited Graph Signals via Iterative Methods

We study the quantization of real-valued bandlimited signals on graphs, focusing on low-bit representations. We propose iterative noise-shaping algorithms for quantization, including sampling approaches with and without vertex replacement. The methods leverage the spectral properties of the graph Laplacian and exploit graph incoherence to achieve high-fidelity approximations. Theoretical guarantees are provided for the random sampling method, and extensive numerical experiments on synthetic and real-world graphs illustrate the efficiency and robustness of the proposed schemes.

eess.SP↗

arXiv2024

Implicit Regularization for Tubal Tensor Factorizations via Gradient Descent

We provide a rigorous analysis of implicit regularization in an overparametrized tensor factorization problem beyond the lazy training regime. For matrix factorization problems, this phenomenon has been studied in a number of works. A particular challenge has been to design universal initialization strategies which provably lead to implicit regularization in gradient-descent methods. At the same time, it has been argued by Cohen et. al. 2016 that more general classes of neural networks can be captured by considering tensor factorizations. However, in the tensor case, implicit regularization has only been rigorously established for gradient flow or in the lazy training regime. In this paper, we prove the first tensor result of its kind for gradient descent rather than gradient flow. We focus on the tubal tensor product and the associated notion of low tubal rank, encouraged by the relevance of this model for image data. We establish that gradient descent in an overparametrized tensor factorization model with a small random initialization exhibits an implicit bias towards solutions of low tubal rank. Our theoretical findings are illustrated in an extensive set of numerical simulations show-casing the dynamics predicted by our theory as well as the crucial role of using a small random initialization.

cs.LG↗

arXiv2024

The Mathematics of Dots and Pixels: On the Theoretical Foundations of Image Halftoning

The evolution of image halftoning, from its analog roots to contemporary digital methodologies, encapsulates a fascinating journey marked by technological advancements and creative innovations. Yet the theoretical understanding of halftoning is much more recent. In this article, we explore various approaches towards shedding light on the design of halftoning approaches and why they work. We discuss both halftoning in a continuous domain and on a pixel grid. We start by reviewing the mathematical foundation of the so-called electrostatic halftoning method, which departed from the heuristic of considering the back dots of the halftoned image as charged particles attracted by the grey values of the image in combination with mutual repulsion. Such an attraction-repulsion model can be mathematically represented via an energy functional in a reproducing kernel Hilbert space allowing for a rigorous analysis of the resulting optimization problem as well as a convergence analysis in a suitable topology. A second class of methods that we discuss in detail is the class of error diffusion schemes, arguably among the most popular halftoning techniques due to their ability to work directly on a pixel grid and their ease of application. The main idea of these schemes is to choose the locations of the black pixels via a recurrence relation designed to agree with the image in terms of the local averages. We discuss some recent mathematical understanding of these methods that is based on a connection to Sigma-Delta quantizers, a popular class of algorithms for analog-to-digital conversion.

eess.IV↗

arXiv2023

Regularized Shannon sampling formulas related to the special affine Fourier transform

In this paper, we present new regularized Shannon sampling formulas related to the special affine Fourier transform (SAFT). These sampling formulas use localized sampling with special compactly supported window functions, namely B-spline, sinh-type, and continuous Kaiser-Bessel window functions. In contrast to the Shannon sampling series for SAFT, the regularized Shannon sampling formulas for SAFT possesses an exponential decay of the approximation error and are numerically robust in the presence of noise, if certain oversampling condition is fulfilled. Several numerical experiments illustrate the theoretical results.

math.NA↗

arXiv2022

Non-intrusive surrogate modelling using sparse random features with applications in crashworthiness analysis

Efficient surrogate modelling is a key requirement for uncertainty quantification in data-driven scenarios. In this work, a novel approach of using Sparse Random Features for surrogate modelling in combination with self-supervised dimensionality reduction is described. The method is compared to other methods on synthetic and real data obtained from crashworthiness analyses. The results show a superiority of the here described approach over state of the art surrogate modelling techniques, Polynomial Chaos Expansions and Neural Networks.

cs.LG↗

arXiv2022

Enhanced Digital Halftoning via Weighted Sigma-Delta Modulation

In this paper, we study error diffusion techniques for digital halftoning from the perspective of 1-bit Sigma-Delta quantization. We introduce a method to generate Sigma-Delta schemes for two-dimensional signals as a weighted combination of its one-dimensional counterparts and show that various error diffusion schemes proposed in the literature can be represented in this framework via Sigma-Delta schemes of first order. Under the model of two-dimensional bandlimited signals, which is motivated by a mathematical model of human visual perception, we derive quantitative error bounds for such weighted Sigma-Delta schemes. We see these bounds as a step towards a mathematical understanding of the good empirical performance of error diffusion, even though they are formulated in the supremum norm, which is known to not fully capture the visual similarity of images. Motivated by the correspondence between existing error diffusion algorithms and first-order Sigma-Delta schemes, we study the performance of the analogous weighted combinations of second-order Sigma-Delta schemes and show that they exhibit a superior performance in terms of guaranteed error decay for two-dimensional bandlimited signals. In extensive numerical simulations for real world images, we demonstrate that with some modifications to enhance stability this superior performance also translates to the problem of digital halftoning. More concretely, we find that certain second-order weighted Sigma-Delta schemes exhibit competitive performance for digital halftoning of real world images in terms of the Feature Similarity Index (FSIM), a state-of-the-art measure for image quality assessment.

math.NA↗

arXiv2021

Super-Resolution on the Two-Dimensional Unit Sphere

We study the problem of recovering an atomic measure on the unit 2-sphere $\mathbb{S}^2$ given finitely many moments with respect to spherical harmonics. The analysis relies on the formulation of this problem as an optimization problem on the space of bounded Borel measures on $\mathbb{S}^2$ as it was considered by Y. de Castro & F. Gamboa and E. Candés & C. Fernandez-Granda. We construct a dual certificate using a kernel given in an explicit form and make a concrete analysis of the interpolation problem. Numerical examples are provided and analyzed.

math.FA↗