Related papers: On periodic distributed representations using Four…

This note gives a summary of ideas concerning Applied Fourier Analysis, mostly formulated for those who have to give such courses to engineers or mathematicians interested in real life applications. It tries to answer recurrent questions…

Covariance representations are developed for the uniform distributions on the Euclidean spheres in terms of spherical gradients and Hessians. They are applied to derive a number of Sobolev type inequalities and to recover and refine the…

Let $\mu$ be a positive measure on the real line with locally finite support $\Lambda$ and integer masses such that its Fourier transform in the sense of distributions is a purely point measure. An explicit form is found for an entire…

Partial Differential Equations are infinite dimensional encoded representations of physical processes. However, imbibing multiple observation data towards a coupled representation presents significant challenges. We present a fully…

The numerical approximation of partial differential equations (PDEs) using neural networks has seen significant advancements through Physics-Informed Neural Networks (PINNs). Despite their straightforward optimization framework and…

We are studying the problem of estimating density in a wide range of metric spaces, including the Euclidean space, the sphere, the ball, and various Riemannian manifolds. Our framework involves a metric space with a doubling measure and a…

Gaussian processes are flexible function approximators, with inductive biases controlled by a covariance kernel. Learning the kernel is the key to representation learning and strong predictive performance. In this paper, we develop…

This paper gives an introduction to the theory of orthogonal projection of functions or signals. Several kinds of decomposition are explored: Fourier, Fourier-Legendre, Fourier-Bessel series for 1D signals, and Spherical Harmonic series for…

Tokens are discrete representations that allow modern deep learning to scale by transforming high-dimensional data into sequences that can be efficiently learned, generated, and generalized to new tasks. These have become foundational for…

Performing machine learning on structured data is complicated by the fact that such data does not have vectorial form. Therefore, multiple approaches have emerged to construct vectorial representations of structured data, from kernel and…

In this work, we investigate Gaussian Processes indexed by multidimensional distributions. While directly constructing radial positive definite kernels based on the Wasserstein distance has been proven to be possible in the unidimensional…

EEG emotion recognition faces significant hurdles due to noise interference, signal nonstationarity, and the inherent complexity of brain activity which make accurately emotion classification. In this study, we present the Fourier Adjacency…

To make progress in science, we often build abstract representations of physical systems that meaningfully encode information about the systems. The representations learnt by most current machine learning techniques reflect statistical…

This article gives a new insight of kernel-based (approximation) methods to solve the high-dimensional stochastic partial differential equations. We will combine the techniques of meshfree approximation and kriging interpolation to extend…

Tensor algebras give rise to one of the most powerful measures of similarity for sequences of arbitrary length called the signature kernel accompanied with attractive theoretical guarantees from stochastic analysis. Previous algorithms to…

Diffusion probabilistic models (DPMs) have achieved remarkable quality in image generation that rivals GANs'. But unlike GANs, DPMs use a set of latent variables that lack semantic meaning and cannot serve as a useful representation for…

The rigorous solution to the grating diffraction problem is a cornerstone step in many scientific fields and industrial applications ranging from the study of the fundamental properties of metasurfaces to the simulation of photolithography…

We present a novel approach to generate a fingerprint for crystalline materials that balances efficiency for machine processing and human interpretability, allowing its application in both machine learning inference and understanding of…

Kernel means are frequently used to represent probability distributions in machine learning problems. In particular, the well known kernel density estimator and the kernel mean embedding both have the form of a kernel mean. Unfortunately,…

Kernels are often developed and used as implicit mapping functions that show impressive predictive power due to their high-dimensional feature space representations. In this study, we gradually construct a series of simple feature maps that…