Ernesto De Vito

Antoine Chatalic, Nicolas Schreuder, Ernesto De Vito et al.

In this work we consider the problem of numerical integration, i.e., approximating integrals with respect to a target probability measure using only pointwise evaluations of the integrand. We focus on the setting in which the target distribution is only accessible through a set of $n$ i.i.d. observations, and the integrand belongs to a reproducing kernel Hilbert space. We propose an efficient procedure which exploits a small i.i.d. random subset of $m<n$ samples drawn either uniformly or using approximate leverage scores from the initial observations. Our main result is an upper bound on the approximation error of this procedure for both sampling strategies. It yields sufficient conditions on the subsample size to recover the standard (optimal) $n^{-1/2}$ rate while reducing drastically the number of functions evaluations, and thus the overall computational cost. Moreover, we obtain rates with respect to the number $m$ of evaluations of the integrand which adapt to its smoothness, and match known optimal rates for instance for Sobolev spaces. We illustrate our theoretical findings with numerical experiments on real datasets, which highlight the attractive efficiency-accuracy tradeoff of our method compared to existing randomized and greedy quadrature methods. We note that, the problem of numerical integration in RKHS amounts to designing a discrete approximation of the kernel mean embedding of the target distribution. As a consequence, direct applications of our results also include the efficient computation of maximum mean discrepancies between distributions and the design of efficient kernel-based tests.

Ernesto De Vito, Massimo Fornasier, Valeriya Naumova

Despite a variety of available techniques the issue of the proper regularization parameter choice for inverse problems still remains one of the biggest challenges. The main difficulty lies in constructing a rule, allowing to compute the parameter from given noisy data without relying either on a priori knowledge of the solution or on the noise level. In this paper we propose a novel method based on supervised machine learning to approximate the high-dimensional function, mapping noisy data into a good approximation to the optimal Tikhonov regularization parameter. Our assumptions are that solutions of the inverse problem are statistically distributed in a concentrated manner on (lower-dimensional) linear subspaces and the noise is sub-gaussian. One of the surprising facts is that the number of previously observed examples for the supervised learning of the optimal parameter mapping scales at most linearly with the dimension of the solution subspace. We also provide explicit error bounds on the accuracy of the approximated parameter and the corresponding regularization solution. Even though the results are more of theoretical nature, we present a recipe for the practical implementation of the approach and provide numerical experiments confirming the theoretical results. We also outline interesting directions for future research with some preliminary results, confirming their feasibility.

Francesca Bartolucci, Ernesto De Vito, Lorenzo Rosasco et al.

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.

Shuo Huang, Hippolyte Labarrière, Ernesto De Vito et al.

Deep neural networks excel in high-dimensional problems, outperforming models such as kernel methods, which suffer from the curse of dimensionality. However, the theoretical foundations of this success remain poorly understood. We follow the idea that the compositional structure of the learning task is the key factor determining when deep networks outperform other approaches. Taking a step towards formalizing this idea, we consider a simple compositional model, namely the multi-index model (MIM). In this context, we introduce and study hyper-kernel ridge regression (HKRR), an approach blending neural networks and kernel methods. Our main contribution is a sample complexity result demonstrating that HKRR can adaptively learn MIM, overcoming the curse of dimensionality. Further, we exploit the kernel nature of the estimator to develop ad hoc optimization approaches. Indeed, we contrast alternating minimization and alternating gradient methods both theoretically and numerically. These numerical results complement and reinforce our theoretical findings.

Andrea Della Vecchia, Arnaud Mavakala Watusadisi, Ernesto De Vito et al.

This paper addresses the covariate shift problem in the context of nonparametric regression within reproducing kernel Hilbert spaces (RKHSs). Covariate shift arises in supervised learning when the input distributions of the training and test data differ, presenting additional challenges for learning. Although kernel methods have optimal statistical properties, their high computational demands in terms of time and, particularly, memory, limit their scalability to large datasets. To address this limitation, the main focus of this paper is to explore the trade-off between computational efficiency and statistical accuracy under covariate shift. We investigate the use of random projections where the hypothesis space consists of a random subspace within a given RKHS. Our results show that, even in the presence of covariate shift, significant computational savings can be achieved without compromising learning performance.

Emilia Magnani, Ernesto De Vito, Philipp Hennig et al.

We consider the problem of learning convolution operators associated to compact Abelian groups. We study a regularization-based approach and provide corresponding learning guarantees under natural regularity conditions on the convolution kernel. More precisely, we assume the convolution kernel is a function in a translation invariant Hilbert space and analyze a natural ridge regression (RR) estimator. Building on existing results for RR, we characterize the accuracy of the estimator in terms of finite sample bounds. Interestingly, regularity assumptions which are classical in the analysis of RR, have a novel and natural interpretation in terms of space/frequency localization. Theoretical results are illustrated by numerical simulations.

Giovanni S. Alberti, Ernesto De Vito, Tapio Helin et al.

This paper introduces a novel approach to learning sparsity-promoting regularizers for solving linear inverse problems. We develop a bilevel optimization framework to select an optimal synthesis operator, denoted as $B$, which regularizes the inverse problem while promoting sparsity in the solution. The method leverages statistical properties of the underlying data and incorporates prior knowledge through the choice of $B$. We establish the well-posedness of the optimization problem, provide theoretical guarantees for the learning process, and present sample complexity bounds. The approach is demonstrated through examples, including compact perturbations of a known operator and the problem of learning the mother wavelet, showcasing its flexibility in incorporating prior knowledge into the regularization framework. This work extends previous efforts in Tikhonov regularization by addressing non-differentiable norms and proposing a data-driven approach for sparse regularization in infinite dimensions.

Andrea Della Vecchia, Ernesto De Vito, Lorenzo Rosasco

We study a natural extension of classical empirical risk minimization, where the hypothesis space is a random subspace of a given space. In particular, we consider possibly data dependent subspaces spanned by a random subset of the data, recovering as a special case Nystrom approaches for kernel methods. Considering random subspaces naturally leads to computational savings, but the question is whether the corresponding learning accuracy is degraded. These statistical-computational tradeoffs have been recently explored for the least squares loss and self-concordant loss functions, such as the logistic loss. Here, we work to extend these results to convex Lipschitz loss functions, that might not be smooth, such as the hinge loss used in support vector machines. This unified analysis requires developing new proofs, that use different technical tools, such as sub-gaussian inputs, to achieve fast rates. Our main results show the existence of different settings, depending on how hard the learning problem is, for which computational efficiency can be improved with no loss in performance.

Stefano Vigogna, Giacomo Meanti, Ernesto De Vito et al.

We study the behavior of error bounds for multiclass classification under suitable margin conditions. For a wide variety of methods we prove that the classification error under a hard-margin condition decreases exponentially fast without any bias-variance trade-off. Different convergence rates can be obtained in correspondence of different margin assumptions. With a self-contained and instructive analysis we are able to generalize known results from the binary to the multiclass setting.

Giacomo Meanti, Luigi Carratino, Ernesto De Vito et al.

Kernel methods provide a principled approach to nonparametric learning. While their basic implementations scale poorly to large problems, recent advances showed that approximate solvers can efficiently handle massive datasets. A shortcoming of these solutions is that hyperparameter tuning is not taken care of, and left for the user to perform. Hyperparameters are crucial in practice and the lack of automated tuning greatly hinders efficiency and usability. In this paper, we work to fill in this gap focusing on kernel ridge regression based on the Nyström approximation. After reviewing and contrasting a number of hyperparameter tuning strategies, we propose a complexity regularization criterion based on a data dependent penalty, and discuss its efficient optimization. Then, we proceed to a careful and extensive empirical evaluation highlighting strengths and weaknesses of the different tuning strategies. Our analysis shows the benefit of the proposed approach, that we hence incorporate in a library for large scale kernel methods to derive adaptively tuned solutions.

Antoine Chatalic, Luigi Carratino, Ernesto De Vito et al.

Compressive learning is an approach to efficient large scale learning based on sketching an entire dataset to a single mean embedding (the sketch), i.e. a vector of generalized moments. The learning task is then approximately solved as an inverse problem using an adapted parametric model. Previous works in this context have focused on sketches obtained by averaging random features, that while universal can be poorly adapted to the problem at hand. In this paper, we propose and study the idea of performing sketching based on data-dependent Nyström approximation. From a theoretical perspective we prove that the excess risk can be controlled under a geometric assumption relating the parametric model used to learn from the sketch and the covariance operator associated to the task at hand. Empirically, we show for k-means clustering and Gaussian modeling that for a fixed sketch size, Nyström sketches indeed outperform those built with random features.

Francesca Bartolucci, Ernesto De Vito, Lorenzo Rosasco et al.

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].

Giovanni S. Alberti, Ernesto De Vito, Matti Lassas et al.

In this work, we consider the linear inverse problem $y=Ax+ε$, where $A\colon X\to Y$ is a known linear operator between the separable Hilbert spaces $X$ and $Y$, $x$ is a random variable in $X$ and $ε$ is a zero-mean random process in $Y$. This setting covers several inverse problems in imaging including denoising, deblurring, and X-ray tomography. Within the classical framework of regularization, we focus on the case where the regularization functional is not given a priori but learned from data. Our first result is a characterization of the optimal generalized Tikhonov regularizer, with respect to the mean squared error. We find that it is completely independent of the forward operator $A$ and depends only on the mean and covariance of $x$. Then, we consider the problem of learning the regularizer from a finite training set in two different frameworks: one supervised, based on samples of both $x$ and $y$, and one unsupervised, based only on samples of $x$. In both cases, we prove generalization bounds, under some weak assumptions on the distribution of $x$ and $ε$, including the case of sub-Gaussian variables. Our bounds hold in infinite-dimensional spaces, thereby showing that finer and finer discretizations do not make this learning problem harder. The results are validated through numerical simulations.

Andrea Della Vecchia, Ernesto De Vito, Jaouad Mourtada et al.

We investigate an extension of classical empirical risk minimization, where the hypothesis space consists of a random subspace within a given Hilbert space. Specifically, we examine the Nyström method where the subspaces are defined by a random subset of the data. This approach recovers Nyström approximations used in kernel methods as a specific case. Using random subspaces naturally leads to computational advantages, but a key question is whether it compromises the learning accuracy. Recently, the tradeoffs between statistics and computation have been explored for the square loss and self-concordant losses, such as the logistic loss. In this paper, we extend these analyses to general convex Lipschitz losses, which may lack smoothness, such as the hinge loss used in support vector machines. Our main results show the existence of various scenarios where computational gains can be achieved without sacrificing learning performance. When specialized to smooth loss functions, our analysis recovers most previous results. Moreover, it allows to consider classification problems and translate the surrogate risk bounds into classification error bounds. Indeed, this gives the opportunity to compare the effect of Nyström approximations when combined with different loss functions such as the hinge or the square loss.

Nicolò Pagliana, Alessandro Rudi, Ernesto De Vito et al.

We study the learning properties of nonparametric ridge-less least squares. In particular, we consider the common case of estimators defined by scale dependent kernels, and focus on the role of the scale. These estimators interpolate the data and the scale can be shown to control their stability through the condition number. Our analysis shows that are different regimes depending on the interplay between the sample size, its dimensions, and the smoothness of the problem. Indeed, when the sample size is less than exponential in the data dimension, then the scale can be chosen so that the learning error decreases. As the sample size becomes larger, the overall error stop decreasing but interestingly the scale can be chosen in such a way that the variance due to noise remains bounded. Our analysis combines, probabilistic results with a number of analytic techniques from interpolation theory.

Ernesto De Vito, Zeljko Kereta, Valeriya Naumova et al.

We introduce a construction of multiscale tight frames on general domains. The frame elements are obtained by spectral filtering of the integral operator associated with a reproducing kernel. Our construction extends classical wavelets as well as generalized wavelets on both continuous and discrete non-Euclidean structures such as Riemannian manifolds and weighted graphs. Moreover, it allows to study the relation between continuous and discrete frames in a random sampling regime, where discrete frames can be seen as Monte Carlo estimates of the continuous ones. Pairing spectral regularization with learning theory, we show that a sample frame tends to its population counterpart, and derive explicit finite-sample rates on spaces of Sobolev and Besov regularity. Our results prove the stability of frames constructed on empirical data, in the sense that all stochastic discretizations have the same underlying limit regardless of the set of initial training samples.

Enrico Cecini, Ernesto De Vito, Lorenzo Rosasco

We propose and study a multi-scale approach to vector quantization. We develop an algorithm, dubbed reconstruction trees, inspired by decision trees. Here the objective is parsimonious reconstruction of unsupervised data, rather than classification. Contrasted to more standard vector quantization methods, such as K-means, the proposed approach leverages a family of given partitions, to quickly explore the data in a coarse to fine-- multi-scale-- fashion. Our main technical contribution is an analysis of the expected distortion achieved by the proposed algorithm, when the data are assumed to be sampled from a fixed unknown distribution. In this context, we derive both asymptotic and finite sample results under suitable regularity assumptions on the distribution. As a special case, we consider the setting where the data generating distribution is supported on a compact Riemannian sub-manifold. Tools from differential geometry and concentration of measure are useful in our analysis.

Ernesto De Vito, Nicole Mücke, Lorenzo Rosasco

We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with a number of detailed examples.

Zeljko Kereta, Stefano Vigogna, Valeriya Naumova et al.

In this paper we propose and study a family of continuous wavelets on general domains, and a corresponding stochastic discretization that we call Monte Carlo wavelets. First, using tools from the theory of reproducing kernel Hilbert spaces and associated integral operators, we define a family of continuous wavelets by spectral calculus. Then, we propose a stochastic discretization based on Monte Carlo estimates of integral operators. Using concentration of measure results, we establish the convergence of such a discretization and derive convergence rates under natural regularity assumptions.

Ernesto de Vito, Zeljko Kereta, Valeria Naumova

Despite recent advances in regularisation theory, the issue of parameter selection still remains a challenge for most applications. In a recent work the framework of statistical learning was used to approximate the optimal Tikhonov regularisation parameter from noisy data. In this work, we improve their results and extend the analysis to the elastic net regularisation, providing explicit error bounds on the accuracy of the approximated parameter and the corresponding regularisation solution in a simplified case. Furthermore, in the general case we design a data-driven, automated algorithm for the computation of an approximate regularisation parameter. Our analysis combines statistical learning theory with insights from regularisation theory. We compare our approach with state-of-the-art parameter selection criteria and illustrate its superiority in terms of accuracy and computational time on simulated and real data sets.

Miguel A. Duval-Poo, Nicoletta Noceti, Francesca Odone et al.

Shearlets are a relatively new directional multi-scale framework for signal analysis, which have been shown effective to enhance signal discontinuities such as edges and corners at multiple scales. In this work we address the problem of detecting and describing blob-like features in the shearlets framework. We derive a measure which is very effective for blob detection and closely related to the Laplacian of Gaussian. We demonstrate the measure satisfies the perfect scale invariance property in the continuous case. In the discrete setting, we derive algorithms for blob detection and keypoint description. Finally, we provide qualitative justifications of our findings as well as a quantitative evaluation on benchmark data. We also report an experimental evidence that our method is very suitable to deal with compressed and noisy images, thanks to the sparsity property of shearlets.

Ernesto De Vito, Lorenzo Rosasco, Alessandro Toigo

We consider the problem of learning a set from random samples. We show how relevant geometric and topological properties of a set can be studied analytically using concepts from the theory of reproducing kernel Hilbert spaces. A new kind of reproducing kernel, that we call separating kernel, plays a crucial role in our study and is analyzed in detail. We prove a new analytic characterization of the support of a distribution, that naturally leads to a family of provably consistent regularized learning algorithms and we discuss the stability of these methods with respect to random sampling. Numerical experiments show that the approach is competitive, and often better, than other state of the art techniques.