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Stefano Vigogna

Publications and source records attributed to Stefano Vigogna.

At least 19 recordsLinked to original sources
arXiv2025

Fractal and Regular Geometry of Deep Neural Networks

We study the geometric properties of random neural networks by investigating the boundary volumes of their excursion sets for different activation functions, as the depth increases. More specifically, we show that, for activations which are not very regular (e.g., the Heaviside step function), the boundary volumes exhibit fractal behavior, with their Hausdorff dimension monotonically increasing with the depth. On the other hand, for activations which are more regular (e.g., ReLU, logistic and $\tanh$), as the depth increases, the expected boundary volumes can either converge to zero, remain constant or diverge exponentially, depending on a single spectral parameter which can be easily computed. Our theoretical results are confirmed in some numerical experiments based on Monte Carlo simulations.

math.PR↗
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

Spectral complexity of deep neural networks

It is well-known that randomly initialized, push-forward, fully-connected neural networks weakly converge to isotropic Gaussian processes, in the limit where the width of all layers goes to infinity. In this paper, we propose to use the angular power spectrum of the limiting field to characterize the complexity of the network architecture. In particular, we define sequences of random variables associated with the angular power spectrum, and provide a full characterization of the network complexity in terms of the asymptotic distribution of these sequences as the depth diverges. On this basis, we classify neural networks as low-disorder, sparse, or high-disorder; we show how this classification highlights a number of distinct features for standard activation functions, and in particular, sparsity properties of ReLU networks. Our theoretical results are also validated by numerical simulations.

stat.ML↗
arXiv2024

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↗
arXiv2023

How many samples are needed to leverage smoothness?

A core principle in statistical learning is that smoothness of target functions allows to break the curse of dimensionality. However, learning a smooth function seems to require enough samples close to one another to get meaningful estimate of high-order derivatives, which would be hard in machine learning problems where the ratio between number of data and input dimension is relatively small. By deriving new lower bounds on the generalization error, this paper formalizes such an intuition, before investigating the role of constants and transitory regimes which are usually not depicted beyond classical learning theory statements while they play a dominant role in practice.

stat.ML↗
arXiv2023

A Quantitative Functional Central Limit Theorem for Shallow Neural Networks

We prove a Quantitative Functional Central Limit Theorem for one-hidden-layer neural networks with generic activation function. The rates of convergence that we establish depend heavily on the smoothness of the activation function, and they range from logarithmic in non-differentiable cases such as the Relu to $\sqrt{n}$ for very regular activations. Our main tools are functional versions of the Stein-Malliavin approach; in particular, we exploit heavily a quantitative functional central limit theorem which has been recently established by Bourguin and Campese (2020).

math.PR↗
arXiv2023

A Case of Exponential Convergence Rates for SVM

Classification is often the first problem described in introductory machine learning classes. Generalization guarantees of classification have historically been offered by Vapnik-Chervonenkis theory. Yet those guarantees are based on intractable algorithms, which has led to the theory of surrogate methods in classification. Guarantees offered by surrogate methods are based on calibration inequalities, which have been shown to be highly sub-optimal under some margin conditions, failing short to capture exponential convergence phenomena. Those "super" fast rates are becoming to be well understood for smooth surrogates, but the picture remains blurry for non-smooth losses such as the hinge loss, associated with the renowned support vector machines. In this paper, we present a simple mechanism to obtain fast convergence rates and we investigate its usage for SVM. In particular, we show that SVM can exhibit exponential convergence rates even without assuming the hard Tsybakov margin condition.

stat.ML↗
arXiv2022

ParK: Sound and Efficient Kernel Ridge Regression by Feature Space Partitions

We introduce ParK, a new large-scale solver for kernel ridge regression. Our approach combines partitioning with random projections and iterative optimization to reduce space and time complexity while provably maintaining the same statistical accuracy. In particular, constructing suitable partitions directly in the feature space rather than in the input space, we promote orthogonality between the local estimators, thus ensuring that key quantities such as local effective dimension and bias remain under control. We characterize the statistical-computational tradeoff of our model, and demonstrate the effectiveness of our method by numerical experiments on large-scale datasets.

stat.ML↗
arXiv2022

Conditional regression for single-index models

The single-index model is a statistical model for intrinsic regression where responses are assumed to depend on a single yet unknown linear combination of the predictors, allowing to express the regression function as $ \mathbb{E} [ Y | X ] = f ( \langle v , X \rangle ) $ for some unknown \emph{index} vector $v$ and \emph{link} function $f$. Conditional methods provide a simple and effective approach to estimate $v$ by averaging moments of $X$ conditioned on $Y$, but depend on parameters whose optimal choice is unknown and do not provide generalization bounds on $f$. In this paper we propose a new conditional method converging at $\sqrt{n}$ rate under an explicit parameter characterization. Moreover, we prove that polynomial partitioning estimates achieve the $1$-dimensional min-max rate for regression of Hölder functions when combined to any $\sqrt{n}$-convergent index estimator. Overall this yields an estimator for dimension reduction and regression of single-index models that attains statistical optimality in quasilinear time.

math.ST↗
arXiv2022

Multiclass learning with margin: exponential rates with no bias-variance trade-off

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.

stat.ML↗
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

Construction and Monte Carlo estimation of wavelet frames generated by a reproducing kernel

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.

math.FA↗
arXiv2021

Multiscale regression on unknown manifolds

We consider the regression problem of estimating functions on $\mathbb{R}^D$ but supported on a $d$-dimensional manifold $ \mathcal{M} \subset \mathbb{R}^D $ with $ d \ll D $. Drawing ideas from multi-resolution analysis and nonlinear approximation, we construct low-dimensional coordinates on $\mathcal{M}$ at multiple scales, and perform multiscale regression by local polynomial fitting. We propose a data-driven wavelet thresholding scheme that automatically adapts to the unknown regularity of the function, allowing for efficient estimation of functions exhibiting nonuniform regularity at different locations and scales. We analyze the generalization error of our method by proving finite sample bounds in high probability on rich classes of priors. Our estimator attains optimal learning rates (up to logarithmic factors) as if the function was defined on a known Euclidean domain of dimension $d$, instead of an unknown manifold embedded in $\mathbb{R}^D$. The implemented algorithm has quasilinear complexity in the sample size, with constants linear in $D$ and exponential in $d$. Our work therefore establishes a new framework for regression on low-dimensional sets embedded in high dimensions, with fast implementation and strong theoretical guarantees.

stat.ML↗
arXiv2020

Estimating multi-index models with response-conditional least squares

The multi-index model is a simple yet powerful high-dimensional regression model which circumvents the curse of dimensionality assuming $ \mathbb{E} [ Y | X ] = g(A^\top X) $ for some unknown index space $A$ and link function $g$. In this paper we introduce a method for the estimation of the index space, and study the propagation error of an index space estimate in the regression of the link function. The proposed method approximates the index space by the span of linear regression slope coefficients computed over level sets of the data. Being based on ordinary least squares, our approach is easy to implement and computationally efficient. We prove a tight concentration bound that shows $N^{-1/2}$-convergence, but also faithfully describes the dependence on the chosen partition of level sets, hence giving indications on the hyperparameter tuning. The estimator's competitiveness is confirmed by extensive comparisons with state-of-the-art methods, both on synthetic and real data sets. As a second contribution, we establish minimax optimal generalization bounds for k-nearest neighbors and piecewise polynomial regression when trained on samples projected onto any $N^{-1/2}$-consistent estimate of the index space, thus providing complete and provable estimation of the multi-index model.

math.ST↗
arXiv2019

Monte Carlo wavelets: a randomized approach to frame discretization

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.

math.FA↗
arXiv2014

Intrinsic Localization of Anisotropic Frames II: $α$-Molecules

This article is a continuation of the recent paper [Grohs, Intrinsic localization of anisotropic frames, ACHA, 2013], where off-diagonal-decay properties (often referred to as 'localization' in the literature) of Moore-Penrose pseudoinverses of (bi-infinite) matrices are established, whenever the latter possess similar off-diagonal-decay properties. This problem is especially interesting if the matrix arises as a discretization of an operator with respect to a frame or basis. Previous work on this problem has been restricted to wavelet- or Gabor frames. In the previous work we extended these results to frames of parabolic molecules, including curvelets or shearlets as special cases. The present paper extends and unifies these results by establishing analogous properties for frames of $α$-molecules as introduced in recent work [Grohs, Keiper, Kutyniok, Schäfer, Alpha molecules: curvelets, shearlets, ridgelets, and beyond, Proc. SPIE. 8858, 2013]. Since wavelets, curvelets, shearlets, ridgelets and hybrid shearlets all constitute instances of $α$-molecules, our results establish localization properties for all these systems simultaneously.

math.FA↗
arXiv2014

Geometric classification of semidirect products in the maximal parabolic subgroup of $\operatorname{Sp}(2,\mathbb{R})$

We classify up to conjugation by $\operatorname{GL}(2,\mathbb{R})$ (more precisely, block diagonal symplectic matrices) all the semidirect products inside the maximal parabolic of $\operatorname{Sp}(2,\mathbb{R})$ by means of an essentially geometric argument. This classification has already been established without geometry, under a stricter notion of equivalence, namely conjugation by arbitrary symplectic matrices. The present approach might be useful in higher dimensions and provides some insight.

math.GR↗