Trajan Murphy | Searcharxiv

arXiv2026

Karhunen Lo\`eve Expansions of Hilbert Space-Valued Random Elements

The Karhunen-Lo\`eve Expansion (KLE) of a stochastic process is a well understood eigenfunction expansion used widely in time series analysis, stochastic PDEs, and signal processing. Karhunen-Lo\`eve expansions have also been proven to exist for other types of stochastic elements whose values lie in certain $L^2$ spaces. This article provides a concise proof about the necessary and sufficient conditions for a function $v$ defined on some sample space $\Omega$ and whose values lie in some Hilbert space $\mathcal H$ to admit an eigenfunction expansion like the well-known KLE. We draw on the existing theory of Bochner spaces and Hilbert-Schmidt spaces and construct an isomorphism between them. Furthermore, this isomorphism is natural, which has important computational consequences. Finally, we demonstrate with an example the computational advantages conferred by considering the KLE in this generalized setting.

math.FA↗

arXiv2025

FINDER: Feature Inference on Noisy Datasets using Eigenspace Residuals

''Noisy'' datasets (regimes with low signal to noise ratios, small sample sizes, faulty data collection, etc) remain a key research frontier for classification methods with both theoretical and practical implications. We introduce FINDER, a rigorous framework for analyzing generic classification problems, with tailored algorithms for noisy datasets. FINDER incorporates fundamental stochastic analysis ideas into the feature learning and inference stages to optimally account for the randomness inherent to all empirical datasets. We construct ''stochastic features'' by first viewing empirical datasets as realizations from an underlying random field (without assumptions on its exact distribution) and then mapping them to appropriate Hilbert spaces. The Kosambi-Karhunen-Lo\'eve expansion (KLE) breaks these stochastic features into computable irreducible components, which allow classification over noisy datasets via an eigen-decomposition: data from different classes resides in distinct regions, identified by analyzing the spectrum of the associated operators. We validate FINDER on several challenging, data-deficient scientific domains, producing state of the art breakthroughs in: (i) Alzheimer's Disease stage classification, (ii) Remote sensing detection of deforestation. We end with a discussion on when FINDER is expected to outperform existing methods, its failure modes, and other limitations.

cs.LG↗