arXiv subjects

Francesca Bartolucci

Publications and source records attributed to Francesca Bartolucci.

17 recordsLinked to original sources

arXiv2025

Neural reproducing kernel Banach spaces and representer theorems for deep networks

Characterizing the function spaces defined by neural networks helps understanding the corresponding learning models and their inductive bias. While in some limits neural networks correspond to function spaces that are Hilbert spaces, these regimes do not capture the properties of the networks used in practice. Indeed, several results have shown that shallow networks can be better characterized in terms of suitable Banach spaces. However, analogous results for deep networks are limited. In this paper we show that deep neural networks define suitable reproducing kernel Banach spaces. These spaces are equipped with norms that enforce a form of sparsity, enabling them to adapt to potential latent structures within the input data and their representations. In particular, by leveraging the theory of reproducing kernel Banach spaces, combined with variational results, we derive representer theorems that justify the finite architectures commonly employed in applications. Our study extends analogous results for shallow networks and represents a step towards understanding the function spaces induced by neural architectures used in practice.

stat.ML↗

arXiv2024

A Lipschitz spaces view of infinitely wide shallow neural networks

We revisit the mean field parametrization of shallow neural networks, using signed measures on unbounded parameter spaces and duality pairings that take into account the regularity and growth of activation functions. This setting directly leads to the use of unbalanced Kantorovich-Rubinstein norms defined by duality with Lipschitz functions, and of spaces of measures dual to those of continuous functions with controlled growth. These allow to make transparent the need for total variation and moment bounds or penalization to obtain existence of minimizers of variational formulations, under which we prove a compactness result in strong Kantorovich-Rubinstein norm, and in the absence of which we show several examples demonstrating undesirable behavior. Further, the Kantorovich-Rubinstein setting enables us to combine the advantages of a completely linear parametrization and ensuing reproducing kernel Banach space framework with optimal transport insights. We showcase this synergy with representer theorems and uniform large data limits for empirical risk minimization, and in proposed formulations for distillation and fusion applications.

math.FA↗

arXiv2024

Unique wavelet sign retrieval from samples without bandlimiting

We study the problem of recovering a signal from magnitudes of its wavelet frame coefficients when the analyzing wavelet is real-valued. We show that every real-valued signal can be uniquely recovered, up to global sign, from its multi-wavelet frame coefficients \[ \{\lvert \mathcal{W}_{ϕ_i} f(α^{m}βn,α^{m}) \rvert: i\in\{1,2,3\}, m,n\in\mathbb{Z}\} \] for every $α>1,β>0$ with $β\ln(α)\leq 4π/(1+4p)$, $p>0$, when the three wavelets $ϕ_i$ are suitable linear combinations of the Poisson wavelet $P_p$ of order $p$ and its Hilbert transform $\mathscr{H}P_p$. For complex-valued signals we find that this is not possible for any choice of the parameters $α>1,β>0$, and for any window. In contrast to the existing literature on wavelet sign retrieval, our uniqueness results do not require any bandlimiting constraints or other a priori knowledge on the real-valued signals to guarantee their unique recovery from the absolute values of their wavelet coefficients.

math.FA↗

arXiv2023

Convolutional Neural Operators for robust and accurate learning of PDEs

Although very successfully used in conventional machine learning, convolution based neural network architectures -- believed to be inconsistent in function space -- have been largely ignored in the context of learning solution operators of PDEs. Here, we present novel adaptations for convolutional neural networks to demonstrate that they are indeed able to process functions as inputs and outputs. The resulting architecture, termed as convolutional neural operators (CNOs), is designed specifically to preserve its underlying continuous nature, even when implemented in a discretized form on a computer. We prove a universality theorem to show that CNOs can approximate operators arising in PDEs to desired accuracy. CNOs are tested on a novel suite of benchmarks, encompassing a diverse set of PDEs with possibly multi-scale solutions and are observed to significantly outperform baselines, paving the way for an alternative framework for robust and accurate operator learning. Our code is publicly available at https://github.com/bogdanraonic3/ConvolutionalNeuralOperator

cs.LG↗

arXiv2023

On the connection between uniqueness from samples and stability in Gabor phase retrieval

Gabor phase retrieval is the problem of reconstructing a signal from only the magnitudes of its Gabor transform. Previous findings suggest a possible link between unique solvability of the discrete problem (recovery from measurements on a lattice) and stability of the continuous problem (recovery from measurements on an open subset of $\mathbb{R}^2$). In this paper, we close this gap by proving that such a link cannot be made. More precisely, we establish the existence of functions which break uniqueness from samples without affecting stability of the continuous problem. Furthermore, we prove the novel result that counterexamples to unique recovery from samples are dense in $L^2(\mathbb{R})$. Finally, we develop an intuitive argument on the connection between directions of instability in phase retrieval and certain Laplacian eigenfunctions associated to small eigenvalues.

math.FA↗

arXiv2023

Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learning

Recently, operator learning, or learning mappings between infinite-dimensional function spaces, has garnered significant attention, notably in relation to learning partial differential equations from data. Conceptually clear when outlined on paper, neural operators necessitate discretization in the transition to computer implementations. This step can compromise their integrity, often causing them to deviate from the underlying operators. This research offers a fresh take on neural operators with a framework Representation equivalent Neural Operators (ReNO) designed to address these issues. At its core is the concept of operator aliasing, which measures inconsistency between neural operators and their discrete representations. We explore this for widely-used operator learning techniques. Our findings detail how aliasing introduces errors when handling different discretizations and grids and loss of crucial continuous structures. More generally, this framework not only sheds light on existing challenges but, given its constructive and broad nature, also potentially offers tools for developing new neural operators.

cs.LG↗

arXiv2023

Uncovering the limits of uniqueness in sampled Gabor phase retrieval: A dense set of counterexamples in $L^2(\mathbb{R})$

Sampled Gabor phase retrieval - the problem of recovering a square-integrable signal from the magnitude of its Gabor transform sampled on a lattice - is a fundamental problem in signal processing, with important applications in areas such as imaging and audio processing. Recently, a classification of square-integrable signals which are not phase retrievable from Gabor measurements on parallel lines has been presented. This classification was used to exhibit a family of counterexamples to uniqueness in sampled Gabor phase retrieval. Here, we show that the set of counterexamples to uniqueness in sampled Gabor phase retrieval is dense in $L^2(\mathbb{R})$, but is not equal to the whole of $L^2(\mathbb{R})$ in general. Overall, our work contributes to a better understanding of the fundamental limits of sampled Gabor phase retrieval.

math.FA↗

arXiv2021

Understanding neural networks with reproducing kernel Banach spaces

Characterizing the function spaces corresponding to neural networks can provide a way to understand their properties. In this paper we discuss how the theory of reproducing kernel Banach spaces can be used to tackle this challenge. In particular, we prove a representer theorem for a wide class of reproducing kernel Banach spaces that admit a suitable integral representation and include one hidden layer neural networks of possibly infinite width. Further, we show that, for a suitable class of ReLU activation functions, the norm in the corresponding reproducing kernel Banach space can be characterized in terms of the inverse Radon transform of a bounded real measure, with norm given by the total variation norm of the measure. Our analysis simplifies and extends recent results in [34,29,30].

stat.ML↗

arXiv2021

Unitarization of the Horocyclic Radon Transform on Symmetric Spaces

We consider the Radon transform for a dual pair $(X,Ξ)$, where $X=G/K$ is a noncompact symmetric space and $Ξ$ is the space of horocycles of $X$. We address the unitarization problem that was considered (and solved in some cases) by Helgason, namely the determination of a pseudo-differential operator such that the pre-composition with the Radon transform extends to a unitary operator $\mathcal{Q}\colon L^2(X)\to L_\flat^2(Ξ)$, where $L_\flat^2(Ξ)$ is a closed subspace of $L^2(Ξ)$ which accounts for the Weyl symmetries. Furthermore, we show that the unitary extension intertwines the quasi-regular representations of $G$ on $L^2(X)$ and $L_\flat^2(Ξ)$.

math.RT↗

arXiv2021

Unitarization of the Horocyclic Radon Transform on Homogeneous Trees

Following previous work in the continuous setup, we construct the unitarization of the horocyclic Radon transform on a homogeneous tree X and we show that it intertwines the quasi regular representations of the group of isometries of X on the tree itself and on the space of horocycles.

math.RT↗

arXiv2020

Phase retrieval of bandlimited functions for the wavelet transform

We study the recovery of square-integrable signals from the absolute values of their wavelet transforms, also called wavelet phase retrieval. We present a new uniqueness result for wavelet phase retrieval. To be precise, we show that any wavelet with finitely many vanishing moments allows for the unique recovery of real-valued bandlimited signals up to global sign. Additionally, we present the first uniqueness result for sampled wavelet phase retrieval in which the underlying wavelets are allowed to be complex-valued and we present a uniqueness result for phase retrieval from sampled Cauchy wavelet transform measurements.

math.FA↗

arXiv2020

The Shearlet Transform and Lizorkin Spaces

We prove a continuity result for the shearlet transform when restricted to the space of smooth and rapidly decreasing functions with all vanishing moments. We define the dual shearlet transform, called here the shearlet synthesis operator, and we prove its continuity on the space of smooth and rapidly decreasing functions over $\mathbb{R}^2\times\mathbb{R}\times\mathbb{R}^\times$. Then, we use these continuity results to extend the shearlet transform to the space of Lizorkin distributions, and we prove its consistency with the classical definition for test functions.

math.FA↗

arXiv2020

Continuity properties of the shearlet transform and the shearlet synthesis operator on the Lizorkin type spaces

We develop a distributional framework for the shearlet transform $\mathcal{S}_ψ\colon\mathcal{S}_0(\mathbb{R}^2)\to\mathcal{S}(\mathbb{S})$ and the shearlet synthesis operator $\mathcal{S}^t_ψ\colon\mathcal{S}(\mathbb{S})\to\mathcal{S}_0(\mathbb{R}^2)$, where $\mathcal{S}_0(\mathbb{R}^2)$ is the Lizorkin test function space and $\mathcal{S}(\mathbb{S})$ is the space of highly localized test functions on the standard shearlet group $\mathbb{S}$. These spaces and their duals $\mathcal{S}_0^\prime (\mathbb R^2),\, \mathcal{S}^\prime (\mathbb{S})$ are called Lizorkin type spaces of test functions and distributions. We analyze the continuity properties of these transforms when the admissible vector $ψ$ belongs to $\mathcal{S}_0(\mathbb{R}^2)$. Then, we define the shearlet transform and the shearlet synthesis operator of Lizorkin type distributions as transpose mappings of the shearlet synthesis operator and the shearlet transform, respectively. They yield continuous mappings from $\mathcal{S}_0^\prime (\mathbb R^2)$ to $\mathcal{S}^\prime (\mathbb{S})$ and from $\mathcal{S}^\prime (\mathbb S)$ to $\mathcal{S}_0^\prime (\mathbb{R}^2)$. Furthermore, we show the consistency of our definition with the shearlet transform defined by direct evaluation of a distribution on the shearlets. The same can be done for the shearlet synthesis operator. Finally, we give a reconstruction formula for Lizorkin type distributions, from which follows that the action of such generalized functions can be written as an absolutely convergent integral over the standard shearlet group.

math.FA↗

arXiv2020

Radon Transform: Dual Pairs and Irreducible Representations

We illustrate the general point of view developed in [SIAM J. Math. Anal., 51(6), 4356-4381] that can be described as a variation of Helgason's theory of dual $G$-homogeneous pairs $(X,Ξ)$ and which allows us to prove intertwining properties and inversion formulae of many existing Radon transforms. Here we analyze in detail one of the important aspects in the theory of dual pairs, namely the injectivity of the map label-to-manifold $ξ\to\hatξ$ and we prove that it is a necessary condition for the irreducibility of the quasi-regular representation of $G$ on $L^2(Ξ)$. We further explain how the theory in [SIAM J. Math. Anal., 51(6), 4356-4381] applies to the classical Radon and X-ray transforms in $\mathbb R^3$.

math.FA↗

arXiv2019

Cone-Adapted Shearlets and Radon Transforms

We show that the cone-adapted shearlet coefficients can be computed by means of the limited angle horizontal and vertical (affine) Radon transforms and the one-dimensional wavelet transform. This yields formulas that open new perspectives for the inversion of the Radon transform.

math.FA↗

arXiv2018

Unitarization and Inversion Formulae for the Radon Transform between Dual Pairs

We consider the Radon transform associated to dual pairs $(X,Ξ)$ in the sense of Helgason, with $X=G/K$ and $Ξ=G/H$, where $G=\mathbb{R}^d\rtimes K$, $K$ is a closed subgroup of ${\rm GL}(d,\mathbb{R})$ and $H$ is a closed subgroup of $G$. Under some technical assumptions, we prove that if the quasi regular representations of $G$ acting on $L^2(X)$ and $L^2(Ξ)$ are irreducible, then the Radon transform admits a unitarization intertwining the two representations. If, in addition, the representations are square integrable, we provide an inversion formula for the Radon transform based on the voice transform associated to these representations.

math.RT↗

arXiv2017

Radon transform intertwines shearlets and wavelets

We prove that the unitary affine Radon transform intertwines the quasi-regular representation of a class of semidirect products, built by shearlet dilation groups and translations, and the tensor product of a standard wavelet representation with a wavelet-like representation. This yields a formula for shearlet coefficients that involves only integral transforms applied to the affine Radon transform of the signal, thereby opening new perspectives in the inversion of the Radon transform.

math.FA↗