Related papers: Classification of Generalized Multiresolution Anal…
Generalized multiresolution analyses are increasing sequences of subspaces of a Hilbert space $\H$ that fail to be multiresolution analyses in the sense of wavelet theory because the core subspace does not have an orthonormal basis…
We discuss the possibility of introducing a multi-resolution in a Hilbert space which is not necessarily a space of functions. We investigate which of the classical properties can be translated to this more general framework and the way in…
We study generalized filters that are associated to multiplicity functions and homomorphisms of the dual of an abelian group. These notions are based on the structure of generalized multiresolution analyses. We investigate when the Ruelle…
We discuss a procedure to construct multi-resolution analyses (MRA) of $\Lc^2(\R)$ starting from a given {\em seed} function $h(s)$ which should satisfy some conditions. Our method, originally related to the quantum mechanical hamiltonian…
Automated sensing instruments on satellites and aircraft have enabled the collection of massive amounts of high-resolution observations of spatial fields over large spatial regions. If these datasets can be efficiently exploited, they can…
We present a notion of frame multiresolution analysis on the Heisenberg group, abbreviated by FMRA, and study its properties. Using the irreducible representations of this group, we shall define a sinc-type function which is our starting…
In modern data analytics, analysts frequently face the challenge of searching for desirable entities by evaluating, for each entity, a collection of its feature relations to derive key analytical properties. This search is challenging…
In this paper we use the equivalence result originally proved by the author which relates a multi-resolution analysis (MRA) of $L^2(R)$ and an orthonormal set of single electron wave-functions in the lowest Landau level, to build up a…
We introduce a new approximate multiresolution analysis (MRA) using a single Gaussian as the scaling function, which we call Gaussian MRA (GMRA). As an initial application, we employ this new tool to accurately and efficiently compute the…
Given the remarkable success that large visual language models (LVLMs) have achieved in image perception tasks, the endeavor to make LVLMs perceive the world like humans is drawing increasing attention. Current multi-modal benchmarks…
Toric ideals to hierarchical models are invariant under the action of a product of symmetric groups. Taking the number of factors, say m, into account, we introduce and study invariant filtrations and their equivariant Hilbert series. We…
Several years ago, O. Bratelli and P. Jorgensen developed the concept of m-systems of filters for dilation by a positive integer N>1 on L^2(R). They constructed a loop group action on m-systems. By work of Mallat and Meyer, these m-systems…
Dimensionality reduction is a main step in the learning process which plays an essential role in many applications. The most popular methods in this field like SVD, PCA, and LDA, only can be applied to data with vector format. This means…
This article surveys results on graded algebras and their Hilbert series. We give simple constructions of finitely generated graded associative algebras $R$ with Hilbert series $H(R,t)$ very close to an arbitrary power series $a(t)$ with…
Let k be a field of characteristic zero. We consider graded subalgebras A of k[x_1,...,x_m]/(x_1^2,...,x_m^2) generated by d linearly independant linear forms. Representations of matroids over k provide a natural description of the…
The purpose of this paper is to present new classes of function systems as part of multiresolution analyses. Our approach is representation theoretic, and it makes use of generalized multiresolution function systems (MRSs). It further…
To a multi-index filtration (say, on the ring of germs of functions on a germ of a complex analytic variety) one associates several invariants: the Hilbert function, the Poincar\'e series, the generalized Poincar\'e series, and the…
This paper introduces a comprehensive framework for the evaluation and validation of generative language models (GLMs), with a focus on Retrieval-Augmented Generation (RAG) systems deployed in high-stakes domains such as banking. GLM…
The necessary and sufficient conditions for existence of a generalized representer theorem are presented for learning Hilbert space-valued functions. Representer theorems involving explicit basis functions and Reproducing Kernels are a…
Composite function minimization captures a wide spectrum of applications in both computer vision and machine learning. It includes bound constrained optimization, $\ell_1$ norm regularized optimization, and $\ell_0$ norm regularized…