arXiv subjects

Alberto Bordino

Publications and source records attributed to Alberto Bordino.

5 recordsLinked to original sources

arXiv2026

Estimation of Linear Functionals in Multilayer Panels under Staggered Adoption

We study the estimation of bilinear forms from noisy, partially observed multilayer data. The signal follows a Tucker2 model, with shared unit and time factors across tensor layers and slice-specific cores. The missingness pattern is structured and motivated by staggered adoption designs, which are common in causal inference and related applications. We first analyze the four-block missingness pattern, the basic building block for general staggered adoption, and propose a spectral algorithm that pools information across layers and targets the functional directly. We prove a non-asymptotic mean squared error bound that exhibits a phase transition in the number of layers, showing when pooling improves estimation, and match it with a local minimax lower bound up to constants when ranks and logarithmic factors are treated as constant-order quantities. We then extend the construction to general staggered adoption designs via an anchored four-block reduction, and derive analogous theoretical guarantees. Finally, we validate our theoretical findings using synthetic and real-world data on Castle Doctrine laws and COVID-19 policies

math.ST↗

arXiv2026

Nonparametric inference for ratios of densities via uniformly valid and powerful permutation tests

We propose the density ratio permutation test, a hypothesis test that assesses whether the ratio between two densities is proportional to a known function based on independent samples from each distribution. The test uses an efficient Markov Chain Monte Carlo scheme to draw weighted permutations of the pooled data, yielding exchangeable samples and finite sample validity. For power, if the statistic is an integral probability metric, our procedure is consistent under mild assumptions on the defining function class; specializing to a reproducing kernel Hilbert space, we introduce the shifted maximum mean discrepancy and prove minimax optimality of our test when a normalized difference between the densities lies in a Sobolev ball. We extend to the case of an unknown density ratio by estimating it on an independent training sample and derive type~I error bounds in terms of the estimation error as well as power results. This allows adapting our method to conditional two sample testing, making it a versatile tool for assessing covariate-shift and related assumptions, which frequently arise in transfer learning and causal inference. Finally, we validate our theoretical findings through experiments on both simulated and real-world datasets.

stat.ME↗

arXiv2025

Non-asymptotic approximations of Gaussian neural networks via second-order Poincaré inequalities

There is a recent and growing literature on large-width asymptotic and non-asymptotic properties of deep Gaussian neural networks (NNs), namely NNs with weights initialized as Gaussian distributions. For a Gaussian NN of depth $L\geq1$ and width $n\geq1$, it is well-known that, as $n\rightarrow+\infty$, the NN's output converges (in distribution) to a Gaussian process. Recently, some quantitative versions of this result, also known as quantitative central limit theorems (QCLTs), have been obtained, showing that the rate of convergence is $n^{-1}$, in the $2$-Wasserstein distance, and that such a rate is optimal. In this paper, we investigate the use of second-order Poincaré inequalities as an alternative approach to establish QCLTs for the NN's output. Previous approaches consist of a careful analysis of the NN, by combining non-trivial probabilistic tools with ad-hoc techniques that rely on the recursive definition of the network, typically by means of an induction argument over the layers, and it is unclear if and how they still apply to other NN's architectures. Instead, the use of second-order Poincaré inequalities rely only on the fact that the NN is a functional of a Gaussian process, reducing the problem of establishing QCLTs to the algebraic problem of computing the gradient and Hessian of the NN's output, which still applies to other NN's architectures. We show how our approach is effective in establishing QCLTs for the NN's output, though it leads to suboptimal rates of convergence. We argue that such a worsening in the rates is peculiar to second-order Poincaré inequalities, and it should be interpreted as the "cost" for having a straightforward, and general, procedure for obtaining QCLTs.

cs.LG↗

arXiv2024

Tests of Missing Completely At Random based on sample covariance matrices

We study the problem of testing whether the missing values of a potentially high-dimensional dataset are Missing Completely at Random (MCAR). We relax the problem of testing MCAR to the problem of testing the compatibility of a collection of covariance matrices, motivated by the fact that this procedure is feasible when the dimension grows with the sample size. Our first contributions are to define a natural measure of the incompatibility of a collection of correlation matrices, which can be characterised as the optimal value of a Semi-definite Programming (SDP) problem, and to establish a key duality result allowing its practical computation and interpretation. By analysing the concentration properties of the natural plug-in estimator for this measure, we propose a novel hypothesis test, which is calibrated via a bootstrap procedure and demonstrates power against any distribution with incompatible covariance matrices. By considering key examples of missingness structures, we demonstrate that our procedures are minimax rate optimal in certain cases. We further validate our methodology with numerical simulations that provide evidence of validity and power, even when data are heavy tailed. Furthermore, tests of compatibility can be used to test the feasibility of positive semi-definite matrix completion problems with noisy observations, and thus our results may be of independent interest.

math.ST↗

arXiv2023

Infinitely wide limits for deep Stable neural networks: sub-linear, linear and super-linear activation functions

There is a growing literature on the study of large-width properties of deep Gaussian neural networks (NNs), i.e. deep NNs with Gaussian-distributed parameters or weights, and Gaussian stochastic processes. Motivated by some empirical and theoretical studies showing the potential of replacing Gaussian distributions with Stable distributions, namely distributions with heavy tails, in this paper we investigate large-width properties of deep Stable NNs, i.e. deep NNs with Stable-distributed parameters. For sub-linear activation functions, a recent work has characterized the infinitely wide limit of a suitable rescaled deep Stable NN in terms of a Stable stochastic process, both under the assumption of a ``joint growth" and under the assumption of a ``sequential growth" of the width over the NN's layers. Here, assuming a ``sequential growth" of the width, we extend such a characterization to a general class of activation functions, which includes sub-linear, asymptotically linear and super-linear functions. As a novelty with respect to previous works, our results rely on the use of a generalized central limit theorem for heavy tails distributions, which allows for an interesting unified treatment of infinitely wide limits for deep Stable NNs. Our study shows that the scaling of Stable NNs and the stability of their infinitely wide limits may depend on the choice of the activation function, bringing out a critical difference with respect to the Gaussian setting.

cs.LG↗