arXiv subjects

Gianluca Finocchio

Publications and source records attributed to Gianluca Finocchio.

5 recordsLinked to original sources

arXiv2026

On identification in ill-posed linear regression

A novel framework is introduced to formalize identifiability in well-specified but ill-posed linear regression models. The framework is distribution-free and accommodates highly correlated features that may or may not relate to the response, reflecting typical real-data structures. First, the identifiable parameter is defined as the least-squares solution obtained by regressing the response on the largest subset of relevant features whose condition number does not exceed a specified threshold, and the relative risk incurred by using this predictor instead of the optimal one is quantified. Second, simple, verifiable conditions are provided under which a broad class of linear dimensionality reduction algorithms can estimate identifiable parameters; algorithms satisfying these conditions are termed statistically interpretable. Third, sharp high-probability error bounds are derived for these algorithms, with rates explicitly reflecting the degree of ill-posedness. With heavy-tailed features and sufficiently low effective rank, these algorithms achieve convergence rates that improve upon both the minimax least-squares rate and lower bounds for sparse estimation under sub-Gaussian features. Results are illustrated via simulations and a real-data application, in which effective rank grows logarithmically with dimension. The framework may extend to algorithms modeling nonlinear response-feature dependence.

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arXiv2025

An extended latent factor framework for ill-posed linear regression

In many applications, particularly in the natural sciences, the available high-dimensional set of features may contain variables that are not correlated with the response under consideration. Such irrelevant features can, in certain cases, hinder both the accurate estimation and meaningful interpretation of the effects of the relevant features on the response. At the same time, the relevant features may also be well-approximated within a low-dimensional linear subspace, rendering the problem ill-posed. These observations motivate an extension of the classical latent factor model for linear regression. In this extended framework, it is assumed that, up to an unknown orthogonal transformation, the feature set comprises two subsets: one relevant and one irrelevant to the response. A joint low-dimensionality is imposed solely on the relevant features and the response variable. This setting enables the analysis of arbitrary linear dimensionality reduction techniques under a random design setting. In particular, it is demonstrated why principal component regression (PCR) is generally unsuitable for most applications. The framework also allows for a comprehensive analysis of the partial least squares (PLS) algorithm under random design. High-probability convergence rates are established for the sample PLS estimator with respect to an oracle latent coefficient vector, along with the corresponding linear prediction risk. Additionally, it is shown that early stopping can be guided by the empirical condition numbers of the projected design matrix. The theoretical results are validated through numerical studies on both real and simulated datasets.

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arXiv2022

Posterior contraction for deep Gaussian process priors

We study posterior contraction rates for a class of deep Gaussian process priors applied to the nonparametric regression problem under a general composition assumption on the regression function. It is shown that the contraction rates can achieve the minimax convergence rate (up to $\log n$ factors), while being adaptive to the underlying structure and smoothness of the target function. The proposed framework extends the Bayesian nonparametric theory for Gaussian process priors.

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arXiv2019

Bayesian variance estimation in the Gaussian sequence model with partial information on the means

Consider the Gaussian sequence model under the additional assumption that a fixed fraction of the means is known. We study the problem of variance estimation from a frequentist Bayesian perspective. The maximum likelihood estimator (MLE) for $σ^2$ is biased and inconsistent. This raises the question whether the posterior is able to correct the MLE in this case. By developing a new proving strategy that uses refined properties of the posterior distribution, we find that the marginal posterior is inconsistent for any i.i.d. prior on the mean parameters. In particular, no assumption on the decay of the prior needs to be imposed. Surprisingly, we also find that consistency can be retained for a hierarchical prior based on Gaussian mixtures. In this case we also establish a limiting shape result and determine the limit distribution. In contrast to the classical Bernstein-von Mises theorem, the limit is non-Gaussian. We show that the Bayesian analysis leads to new statistical estimators outperforming the correctly calibrated MLE in a numerical simulation study.

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arXiv2018

SciSports: Learning football kinematics through two-dimensional tracking data

SciSports is a Dutch startup company specializing in football analytics. This paper describes a joint research effort with SciSports, during the Study Group Mathematics with Industry 2018 at Eindhoven, the Netherlands. The main challenge that we addressed was to automatically process empirical football players' trajectories, in order to extract useful information from them. The data provided to us was two-dimensional positional data during entire matches. We developed methods based on Newtonian mechanics and the Kalman filter, Generative Adversarial Nets and Variational Autoencoders. In addition, we trained a discriminator network to recognize and discern different movement patterns of players. The Kalman-filter approach yields an interpretable model, in which a small number of player-dependent parameters can be fit; in theory this could be used to distinguish among players. The Generative-Adversarial-Nets approach appears promising in theory, and some initial tests showed an improvement with respect to the baseline, but the limits in time and computational power meant that we could not fully explore it. We also trained a Discriminator network to distinguish between two players based on their trajectories; after training, the network managed to distinguish between some pairs of players, but not between others. After training, the Variational Autoencoders generated trajectories that are difficult to distinguish, visually, from the data. These experiments provide an indication that deep generative models can learn the underlying structure and statistics of football players' trajectories. This can serve as a starting point for determining player qualities based on such trajectory data.

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