arXiv subjects

Eugenio Hernández

Publications and source records attributed to Eugenio Hernández.

At least 19 recordsLinked to original sources

arXiv2025

Scattering Networks on Noncommutative Finite Groups

Scattering Networks were initially designed to elucidate the behavior of early layers in Convolutional Neural Networks (CNNs) over Euclidean spaces and are grounded in wavelets. In this work, we introduce a scattering transform on an arbitrary finite group (not necessarily abelian) within the context of group-equivariant convolutional neural networks (G-CNNs). We present wavelets on finite groups and analyze their similarity to classical wavelets. We demonstrate that, under certain conditions in the wavelet coefficients, the scattering transform is non-expansive, stable under deformations, preserves energy, equivariant with respect to left and right group translations, and, as depth increases, the scattering coefficients are less sensitive to group translations of the signal, all desirable properties of convolutional neural networks. Furthermore, we provide examples illustrating the application of the scattering transform to classify data with domains involving abelian and nonabelian groups.

math.NA↗

arXiv2023

Learning optimal smooth invariant subspaces for data approximation

In this article, we consider the problem of approximating a finite set of data (usually huge in applications) by invariant subspaces generated through a small set of smooth functions. The invariance is either by translations under a full-rank lattice or through the action of crystallographic groups. Smoothness is ensured by stipulating that the generators belong to a Paley-Wiener space, that is selected in an optimal way based on the characteristics of the given data. To complete our investigation, we analyze the fundamental role played by the lattice in the process of approximation.

math.OC↗

arXiv2023

On Approximation Spaces and Greedy-type Bases

The purpose of this paper is to introduce $ω$-Chebyshev-greedy and $ω$-partially greedy approximation classes and to study their relation with $ω$-approximation spaces, where the latter are a generalization of the classical approximation spaces. The relation gives us sufficient conditions of when certain continuous embeddings imply different greedy-type properties. Along the way, we generalize a result by P. Wojtaszczyk as well as characterize semi-greedy Schauder bases in quasi-Banach spaces, generalizing a previous result by the first author.

math.FA↗

arXiv2021

The structure of group preserving operators

In this paper, we prove the existence of a particular diagonalization for normal bounded operators defined on subspaces of $L^2(\mathfrak{S})$ where $\mathfrak{S}$ is a second countable LCA group. The subspaces where the operators act are invariant under the action of a group $Γ$ which is a semi-direct product of a uniform lattice of $\mathfrak{S}$ with a discrete group of automorphisms. This class includes the crystal groups which are important in applications as models for images. The operators are assumed to be $Γ$ preserving. i.e. they commute with the action of $Γ$. In particular we obtain a spectral decomposition for these operators. This generalizes recent results on shift-preserving operators acting on lattice invariant subspaces where $\mathfrak{S}$ is the Euclidean space.

math.FA↗

arXiv2020

Spaces invariant under unitary representations of discrete groups

We investigate the structure of subspaces of a Hilbert space that are invariant under unitary representations of a discrete group. We work with square integrable representations, and we show that they are those for which we can construct an isometry intertwining the representation with the right regular representation, that we call a Helson map. We then characterize invariant subspaces using the Helson map, and provide general characterizations of Riesz and frame sequences of orbits. These results extend to the nonabelian setting several known results for abelian groups. They also extend to countable families of generators previous results obtained for principal subspaces.

math.FA↗

arXiv2020

Approximation by group invariant subspaces

In this article we study the structure of $Γ$-invariant spaces of $L^2(\bf R)$. Here $\bf R$ is a second countable LCA group. The invariance is with respect to the action of $Γ$, a non commutative group in the form of a semidirect product of a discrete cocompact subgroup of $\bf R$ and a group of automorphisms. This class includes in particular most of the crystallographic groups. We obtain a complete characterization of $Γ$-invariant subspaces in terms of range functions associated to shift-invariant spaces. We also define a new notion of range function adapted to the $Γ$-invariance and construct Parseval frames of orbits of some elements in the subspace, under the group action. These results are then applied to prove the existence and construction of a $Γ$-invariant subspace that best approximates a set of functional data in $L^2(\bf R)$. This is very relevant in applications since in the euclidean case, $Γ$-invariant subspaces are invariant under rigid movements, a very sought feature in models for signal processing.

math.FA↗

arXiv2019

Optimal translational-rotational invariant dictionaries for images

We provide the construction of a set of square matrices whose translates and rotates provide a Parseval frame that is optimal for approximating a given dataset of images. Our approach is based on abstract harmonic analysis techniques. Optimality is considered with respect to the quadratic error of approximation of the images in the dataset with their projection onto a linear subspace that is invariant under translations and rotations. In addition, we provide an elementary and fully self-contained proof of optimality, and the numerical results from datasets of natural images.

eess.IV↗

arXiv2019

Calderón-type inequalities for affine frames

We prove sharp upper and lower bounds for generalized Calderón's sums associated to frames on LCA groups generated by affine actions of cocompact subgroup translations and general measurable families of automorphisms. The proof makes use of techniques of analysis on metric spaces, and relies on a counting estimate of lattice points inside metric balls. We will deduce as special cases Calderón-type inequalities for families of expanding automorphisms as well as for LCA-Gabor systems.

math.FA↗

arXiv2018

Lebesgue inequalities for Chebyshev Thresholding Greedy Algorithms

We establish estimates for the Lebesgue parameters of the Chebyshev Weak Thresholding Greedy Algorithm in the case of general bases in Banach spaces. These generalize and slightly improve earlier results in [9], and are complemented with examples showing the optimality of the bounds. Our results also correct certain bounds recently announced in [18], and answer some questions left open in that paper.

math.FA↗

arXiv2017

Frames of exponentials and sub-multitiles in LCA groups

In this note we investigate the existence of frames of exponentials for $L^2(Ω)$ in the setting of LCA groups. Our main result shows that sub-multitiling properties of $Ω\subset \widehat{G}$ with respect to a uniform lattice $Γ$ of $\widehat{G}$ guarantee the existence of a frame of exponentials with frequencies in a finite number of translates of the annihilator of $Γ$. We also prove the converse of this result and provide conditions for the existence of these frames. These conditions extend recent results on Riesz bases of exponentials and multitilings to frames.

math.CA↗

arXiv2016

Tiling by lattices for locally compact abelian groups

For a locally compact abelian group $G$ a simple proof is given for the known fact that a bounded domain $Ω$ tiles $G$ with translations by a lattice $Λ$ if and only if the set of characters of $G$ indexed by the dual lattice of $Λ$ is an orthogonal basis of $L^2(Ω).$ The proof uses simple techniques from Harmonic Analysis.

math.FA↗

arXiv2015

Noncommutative Shift-Invariant Spaces

The structure of subspaces of a Hilbert space that are invariant under unitary representations of a discrete group is related to a notion of Hilbert modules endowed with inner products taking values in spaces of unbounded operators. A theory of reproducing systems in such modular structures is developed, providing a general framework that includes fundamental results of shift-invariant spaces. In particular, general characterizations of Riesz and frame sequences associated to group representations are provided, extending previous results for abelian groups and for cyclic subspaces of unitary representations of noncommutative discrete groups.

math.FA↗

arXiv2015

The Zak transform and the structure of spaces invariant by the action of an LCA group

We study closed subspaces of $L^2(X)$, where $(X, μ)$ is a $σ$-finite measure space, that are invariant under the unitary representation associated to a measurable action of a discrete countable LCA group $Γ$ on $X$. We provide a complete description for these spaces in terms of range functions and a suitable generalized Zak transform. As an application of our main result, we prove a characterization of frames and Riesz sequences in $L^2(X)$ generated by the action of the unitary representation under consideration on a countable set of functions in $L^2(X)$. Finally, closed subspaces of $L^2(G)$, for $G$ being an LCA group, that are invariant under translations by elements on a closed subgroup $Γ$ of $G$ are studied and characterized.

math.FA↗

arXiv2014

Greedy bases in variable Lebesgue spaces

We compute the right and left democracy functions of admissible wavelet bases in variable Lebesgue spaces defined on $R^n$. As an application we give Lebesgue type inequalities for these wavelet bases. We also show that our techniques can be easily modified to prove analogous results for weighted variable Lebesgue spaces and variable exponent Triebel-Lizorkin spaces.

math.FA↗

arXiv2014

Riesz and frame systems generated by unitary actions of discrete groups

We characterize orthonormal bases, Riesz bases and frames which arise from the action of a countable discrete group $Γ$ on a single element $ψ$ of a given Hilbert space $\mathcal{H}$. As $Γ$ might not be abelian, this is done in terms of a bracket map taking values in the $L^1$-space associated to the group von Neumann algebra of $Γ$. Our result generalizes recent work for LCA groups. In many cases, the bracket map can be computed in terms of a noncommutative form of the Zak transform.

math.FA↗

arXiv2013

Greedy type algorithms for RIP matrices. A study of two selection rules

Some consequences of the Restricted Isometry Property (RIP) of matrices have been applied to develop a greedy algorithm called "ROMP" (Regularized Orthogonal Matching Pursuit) to recover sparse signals and to approximate non-sparse ones. These consequences were subsequently applied to other greedy and thresholding algorithms like "SThresh", "CoSaMP", "StOMP" and "SWCGP". In this paper, we find another consequence of the RIP property and use it to analyze the approximation to k-sparse signals with Stagewise Weak versions of Gradient Pursuit (SWGP), Matching Pursuit (SWMP) and Orthogonal Matching Pursuit (SWOMP). We combine the above mentioned algorithms with another selection rule similar to the ones that have appeared in the literature showing that results are obtained with less restrictions in the RIP constant, but we need a smaller threshold parameter for the coefficients. The results of some experiments are shown.

cs.IT↗

arXiv2012

Lebesgue type inequalities for quasi-greedy bases

We show that for quasi-greedy bases in real or complex Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best N- term error of approximation times a function of N which depends on the democracy functions and the quasi-greedy constant of the basis. If the basis is democratic this function is bounded by C logN. We show with two examples that this bound is attained for quasi-greedy democratic bases.

math.FA↗

arXiv2011

Lebesgue-Type inequalities for quasi-greedy bases

We show that for quasi-greedy bases in real Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best N-term error of approximation times a constant which depends on the democracy functions and the quasi-greedy constant of the basis.

math.FA↗