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The Hilbert function of standard graded algebras are well understood by Macaulay's theorem and very little is known in the local case, even if we assume that the local ring is a complete intersection. An extension to the power series ring…
Score-based diffusion models in infinite-dimensional function spaces provide a mathematically principled framework for modelling function-valued data, offering key advantages such as resolution invariance and the ability to handle irregular…
Radial basis functions (RBFs) are prominent examples for reproducing kernels with associated reproducing kernel Hilbert spaces (RKHSs). The convergence theory for the kernel-based interpolation in that space is well understood and optimal…
Marginalization -- summing a function over all assignments to a subset of its inputs -- is a fundamental computational problem with applications from probabilistic inference to formal verification. Despite its computational hardness in…
We consider iterated functions systems (IFS) on compact metric spaces and introduce the concept of target sets. Such sets have very rich dynamical properties and play a similar role as semifractals introduced by Lasota and Myjak do for…
In most adaptive signal processing applications, system linearity is assumed and adaptive linear filters are thus used. The traditional class of supervised adaptive filters rely on error-correction learning for their adaptive capability.…
We present decompositions of various positive kernels as integrals or sums of positive kernels. Within this framework we study the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions. As a…
We present an analysis of a novel spherical Radon transform, $R$, which defines the integrals of a function, $f$, in $\mathbb{R}^n$ over spheres with arbitrary center ($\mathbf{y}$) and radii, $r(\mathbf{y})$, which vary smoothly with…
Our community has a deep and sophisticated understanding of phase transitions and their universal scaling functions. We outline and advocate an ambitious program to use this understanding as an anchor for describing the surrounding phases.…
Implicit Neural Representations (INRs) have recently shown impressive results, but their fundamental capacity, implicit biases, and scaling behavior remain poorly understood. We investigate the performance of diverse INRs across a suite of…
Grokking occurs when a model achieves high training accuracy but generalization to unseen test points happens long after that. This phenomenon was initially observed on a class of algebraic problems, such as learning modular arithmetic…
In this paper, we present the general theory of embedding independence tests on Hilbert spaces that generalizes the concepts of distance covariance, distance multivariance and HSIC. This is done by defining new types of kernel on an $n$…
To investigate neural network parameters, it is easier to study the distribution of parameters than to study the parameters in each neuron. The ridgelet transform is a pseudo-inverse operator that maps a given function $f$ to the parameter…
Kernel interpolation is a fundamental technique for approximating functions from scattered data, with a well-understood convergence theory when interpolating elements of a reproducing kernel Hilbert space. Beyond this classical setting,…
We obtain invariance principles for a wide class of fractionally integrated nonlinear processes. The limiting distributions are shown to be fractional Brownian motions. Under very mild conditions, we extend earlier ones on long memory…
This paper studies a general class of Iterated Function Systems (IFS). No contractivity assumptions are made, other than the existence of some compact attractor. The possibility of escape to infinity is considered. Our present approach is…
We propose a simple architecture for deep reinforcement learning by embedding inputs into a learned Fourier basis and show that it improves the sample efficiency of both state-based and image-based RL. We perform infinite-width analysis of…
This paper studies a machine learning regression problem as a multivariate approximation problem using the framework of the theory of random functions. An ab initio derivation of a regression method is proposed, starting from postulates of…
We study the regularity properties of random wavelet series constructed by multiplying the coefficients of a deterministic wavelet series with unbounded I.I.D. random variables. In particular, we show that, at the opposite to what happens…
We study the minimality of almost every orbital branch of minimal iterated function systems (IFSs). We prove that this kind of minimality holds for forward and backward minimal IFSs generated by orientation-preserving homeomorphisms of the…