arXiv subjects

Marc Vidal

Publications and source records attributed to Marc Vidal.

5 recordsLinked to original sources

arXiv2026

An Operator-Theoretic Characterization of Gaussian Measure Singularity under Mean Shifts

For two Gaussian random elements on a separable Hilbert space with common covariance operator, the fourth-order moment tensor structure of their mixture admits an operator-valued representation obtained from the purely quadratic component of the class-conditional second moment. We identify the tensor mechanism generating the completely diagonal coefficients of this fourth-order representation after covariance standardization, and show that these coefficients are governed entirely by the coordinatewise Cameron-Martin energy of the mean shift. This yields a necessary and sufficient condition, together with an explicit eigenvalue, for the Fisher discriminant to be an eigenfunction of the induced coordinatewise operator. We further introduce aggregated fourth-order spectral functionals for two complementary constructions, one built from fourth-order moments and the other from a product-of-expectations counterpart, and prove that they are asymptotically equivalent precisely when the underlying Gaussian measures are mutually singular. These results provide a spectral realization of the classical Cameron-Martin criterion through higher-order moment operators, explaining the probabilistic origin of fourth-order spectral quantities previously proposed for Gaussian discrimination and functional data classification and relating them to the "near-perfect classification" regime.

math.PR↗

arXiv2026

Noise-resilient penalty operators based on statistical differentiation schemes

Penalized smoothing is a standard tool in regression analysis. Classical approaches often rely on basis or kernel expansions, which constrain the estimator to a fixed span and impose smoothness assumptions that may be restrictive for discretely observed data. We introduce a class of penalized estimators that operate directly on the data grid, denoising sampled trajectories under minimal smoothness assumptions by penalizing local roughness through statistically calibrated difference operators. Some distributional and asymptotic properties of sample-based contrast statistics associated with the resulting linear smoothers are established under Hellinger differentiability of the model, without requiring Fr\'echet differentiability in function space. Simulation results confirm that the proposed estimators perform competitively across both smooth and locally irregular settings.

math.ST↗

arXiv2024

Functional independent component analysis by choice of norm: a framework for near-perfect classification

We develop a theory for functional independent component analysis in an infinite-dimensional framework using Sobolev spaces that accommodate smoother functions. The notion of penalized kurtosis is introduced motivated by Silverman's method for smoothing principal components. This approach allows for a classical definition of independent components obtained via projection onto the eigenfunctions of a smoothed kurtosis operator mapping a whitened functional random variable. We discuss the theoretical properties of this operator in relation to a generalized Fisher discriminant function and the relationship it entails with the Feldman-H\'ajek dichotomy for Gaussian measures, both of which are critical to the principles of functional classification. The proposed estimators are a particularly competitive alternative in binary classification of functional data and can eventually achieve the so-called near-perfect classification, which is a genuine phenomenon of high-dimensional data. Our methods are illustrated through simulations, various real datasets, and used to model electroencephalographic biomarkers for the diagnosis of depressive disorder.

math.ST↗

arXiv2021

New views of old proteins: clarifying the enigmatic proteome

All human diseases involve proteins, yet our current tools to characterize and quantify them are limited. To better elucidate proteins across space, time, and molecular composition, we provide provocative projections for technologies to meet the challenges that protein biology presents. With a broad perspective, we discuss grand opportunities to transition the science of proteomics into a more propulsive enterprise. Extrapolating recent trends, we offer potential futures for a next generation of disruptive approaches to define, quantify and visualize the multiple dimensions of the proteome, thereby transforming our understanding and interactions with human disease in the coming decade.

q-bio.BM↗

arXiv2021

Bi-Smoothed Functional Independent Component Analysis for EEG Artifact Removal

Motivated by mapping adverse artifactual events caused by body movements in electroencephalographic (EEG) signals, we present a functional independent component analysis based on the spectral decomposition of the kurtosis operator of a smoothed principal component expansion. A discrete roughness penalty is introduced in the orthonormality constraint of the covariance eigenfunctions in order to obtain the smoothed basis for the proposed independent component model. To select the tuning parameters, a cross-validation method that incorporates shrinkage is used to enhance the performance on functional representations with large basis dimension. This method provides an estimation strategy to determine the penalty parameter and the optimal number of components. Our independent component approach is applied to real EEG data to estimate genuine brain potentials from a contaminated signal. As a result, it is possible to control high-frequency remnants of neural origin overlapping artifactual sources to optimize their removal from the signal. An R package implementing our methods is available at CRAN.

stat.ME↗