arXiv subjects

S. Sivananthan

Publications and source records attributed to S. Sivananthan.

13 recordsLinked to original sources

arXiv2026

Scalable Operator Learning via Nystr\"om Approximation With Denoising Applications

In this paper, we study Nystr\"om subsampling for vector-valued regression in vector-valued reproducing kernel Hilbert spaces. Standard kernel methods often suffer from prohibitive computational costs due to the construction and inversion of large kernel matrices, which limits their scalability to large datasets. To overcome this bottleneck, we propose an efficient operator learning algorithm based on Nystr\"om subsampling that accommodates functional outputs. Under general source conditions characterized by index functions-extending beyond the classical H\"older-type and operator-monotone frameworks-we establish minimax-optimal convergence rates for the proposed estimator. As an application of the proposed framework, we consider function denoising problems. Unlike classical denoising methods, which are typically tailored to specific signal representations or noise models, our approach formulates denoising within a general operator learning framework. Numerical experiments on signal denoising, real-time audio denoising, image denoising, inverse Radon transform reconstruction, and energy-efficiency prediction confirm that the proposed method achieves performance comparable to full kernel methods while substantially reducing computational cost.

math.ST↗

arXiv2026

Towards regularized learning from functional data with covariate shift

This paper investigates a general regularization framework for unsupervised domain adaptation in vector-valued regression under the covariate shift assumption, utilizing vector-valued reproducing kernel Hilbert spaces (vRKHS). Covariate shift occurs when the input distributions of the training and test data differ, introducing significant challenges for reliable learning. By restricting the hypothesis space, we develop a practical operator learning algorithm capable of handling functional outputs. We establish optimal convergence rates for the proposed framework under a general source condition, providing a theoretical foundation for regularized learning in this setting. We also propose an aggregation-based approach that forms a linear combination of estimators corresponding to different regularization parameters and different kernels. The proposed approach addresses the challenge of selecting appropriate tuning parameters, which is crucial for constructing a good estimator, and we provide a theoretical justification for its effectiveness. Furthermore, we illustrate the proposed method on a real-world face image dataset, demonstrating robustness and effectiveness in mitigating distributional discrepancies under covariate shift.

math.ST↗

arXiv2025

Gabor frames generated by Random-Periodic time-frequency shifts

In this article, we consider a variation of the existence of Gabor frames in a probabilistic setting, in which we consider time-frequency shifts taken over random-periodic sets. We demonstrate that the method of selecting random-periodic time-frequency shifts is successful with high probability for specific categories of well-behaved functions, notably including Hermite functions, totally positive functions, and B-spline functions. In particular, we show that if $x_1, x_2, \ldots ,x_m$ are independent and uniformly distributed in $[0,1),$ with $m$ sufficiently large, then the set of time-frequency shifts $Λ\times \ZZ, $ where $Λ=\ZZ + \{x_1, x_2, \ldots, x_m\},$ forms Gabor frame with high probability.

math.FA↗

arXiv2025

Revisiting general source condition in learning over a Hilbert space

In Learning Theory, the smoothness assumption on the target function (known as source condition) is a key factor in establishing theoretical convergence rates for an estimator. The existing general form of the source condition, as discussed in learning theory literature, has traditionally been restricted to a class of functions that can be expressed as a product of an operator monotone function and a Lipschitz continuous function. In this note, we remove these restrictions on the index function and establish optimal convergence rates for least-square regression over a Hilbert space with general regularization under a general source condition, thereby significantly broadening the scope of existing theoretical results.

math.ST↗

arXiv2024

Optimal Rates for Functional Linear Regression with General Regularization

Functional linear regression is one of the fundamental and well-studied methods in functional data analysis. In this work, we investigate the functional linear regression model within the context of reproducing kernel Hilbert space by employing general spectral regularization to approximate the slope function with certain smoothness assumptions. We establish optimal convergence rates for estimation and prediction errors associated with the proposed method under a Hölder type source condition, which generalizes and sharpens all the known results in the literature.

math.ST↗

arXiv2024

Discrete Translates of an Operator in the Schatten $p$-Classes

In this manuscript, we investigate the properties of systems formed by translations of an operator in the Schatten $p$-classes $\mathcal{T}^p$. We establish the existence of Schauder frames of integer translates in $\mathcal{T}^p$ for $p>2$. Later, we provide an instance of a uniformly discrete $Λ\subset \mathbb{R}^{2d}$ such that there exists an operator whose $Λ$-translates are complete in $\mathcal{T}^p$ for all $p>1$.

math.FA↗

arXiv2023

A Note on the Completeness of All Translates of a Function in the Orlicz Spaces

We give a characterization of those functions whose all translates are complete in certain Orlicz space $L^Φ(\mathbb{R})$. As a consequence, we identified those discrete sets $Λ\subseteq \mathbb{R}$ such that there exists a function in $L^Φ(\mathbb{R})$ whose $Λ$-translates are complete. We then prove the completeness of all translates of any simple step function in other Orlicz spaces.

math.FA↗

arXiv2023

Convergence Analysis of Kernel Conjugate Gradient for Functional Linear Regression

In this paper, we discuss the convergence analysis of the conjugate gradient-based algorithm for the functional linear model in the reproducing kernel Hilbert space framework, utilizing early stopping results in regularization against over-fitting. We establish the convergence rates depending on the regularity condition of the slope function and the decay rate of the eigenvalues of the operator composition of covariance and kernel operator. Our convergence rates match the minimax rate available from the literature.

math.ST↗

arXiv2023

Random Sampling of Mellin Band-limited Signals

In this paper, we address the random sampling problem for the class of Mellin band-limited functions BT which is concentrated on a bounded cube. It is established that any function in BT can be approximated by an element in a finite-dimensional subspace of BT. Utilizing the notion of covering number and Bernstein's inequality to the sum of independent random variables, we prove that the random sampling inequality holds with an overwhelming probability provided the sampling size is large enough.

math.FA↗

arXiv2021

Completeness of Discrete Translates in $H^1(\mathbb{R})$

We provide a characterization of discrete sets $Λ\subset \mathbb{R}$ that admit a function whose $Λ$-translates are complete in the Hardy space $H^1(\mathbb{R})$. In particular, we show that such a set cannot be uniformly discrete. We then give a uniformly discrete $Λ\subset \mathbb{R}$ which admits a pair of functions such that their $Λ$-translates are complete in $H^1(\mathbb{R})$.

math.FA↗

arXiv2021

Random sampling of signals concentrated on compact set in localized reproducing kernel subspace of $L^p({\mathbb R}^n)$

The paper is devoted to studying the stability of random sampling in a localized reproducing kernel space. We show that if the sampling set on $Ω$ (compact) discretizes the integral norm of simple functions up to a given error, then the sampling set is stable for the set of functions concentrated on $Ω$. Moreover, we prove with an overwhelming probability that ${\mathcal O}(μ(Ω)(\log μ(Ω))^3)$ many random points uniformly distributed over $Ω$ yield a stable set of sampling for functions concentrated on $Ω$.

math.FA↗

arXiv2005

Growth of Tellurium on As-exposed Si(211)

Electronic structure calculations are performed to obtain the As-exposed Si(211) and the Te adsorbed As-exposed Si(211) surface. Arsenic-exposed Si(211) may be obtained by adsorbing As on Si(211) or by replacing surface Si atoms by As. First, we carry out systematic investigations to obtain stable As-exposed Si(211) due to As adsorption at various coverages. We find that at 1/2 monolayer (ML) coverage of As, the highly terraced Si(211) surface becomes flat decorated with parallel As chains extending along the [$01\bar{1}$] direction. At 1 ML coverage the Si surface essentially retains its ideal structure with an added layer of As. Motivated by the adsorption sequence in the HgCdTe (MCT) growth on Si, Te adsorption on such an As-exposed Si(211) is studied and 1/2 ML of Te coverage is found to be energetically feasible. Next, we explore a stable As-exposed Si(211) upon replacement of surface Si atoms by As. An energetic comparison reveals that the As-exposed Si(211) obtained by replacing surface Si atoms with As is more favorable compared to that obtained by adsorbing As on Si(211). In line with the adsorption sequence in the MCT growth on Si, Te is then adsorbed on the most favorable As-exposed Si(211) and in contrast to earlier situation, Te coverage here is found to be 1/4 of ML which agrees with the experiment.

cond-mat.mtrl-sci↗

arXiv2003

Electronic Structure of Te and As Covered Si(211)

Electronic and atomic structures of the clean, and As and Te covered Si(211) surface are studied using pseudopotential density functional method. The clean surface is found to have (2 X 1) and rebonded (1 X 1) reconstructions as stable surface structures, but no π-bonded chain reconstruction. Binding energies of As and Te adatoms at a number of symmetry sites on the ideal and (2 X 1) reconstructed surfaces have been calculated because of their importance in the epitaxial growth of CdTe and other materials on the Si(211) surface. The special symmetry sites on these surfaces having the highest binding energies for isolated As and Te adatoms are identified. But more significantly, several sites are found to be nearly degenerate in binding energy values. This has important consequences for epitaxial growth processes. Optimal structures calculated for 0.5 ML of As and Te coverage reveal that the As adatoms dimerize on the surface while the Te adatoms do not. However, both As and Te covered surfaces are found to be metallic in nature.

cond-mat.mtrl-sci↗