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

TL;DR

This paper introduces a novel functional independent component analysis framework using Sobolev spaces and penalized kurtosis, enabling near-perfect classification of high-dimensional functional data.

Contribution

It develops a new theory for functional ICA with penalized kurtosis in Sobolev spaces, linking it to Fisher discriminant analysis for improved classification.

Findings

Proposes a new functional ICA method with competitive classification accuracy.

Demonstrates near-perfect classification in high-dimensional functional data.

Validates approach through simulations and real EEG datasets.

Abstract

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.