Norm-Free Radon-Nikodym Approach to Machine Learning

TL;DR

This paper introduces a norm-free, Radon-Nikodym-based method for machine learning classification that leverages quantum-mechanics-like probability states and eigenvalue problems to identify class outcomes and estimate distributions without traditional norms.

Contribution

It proposes a novel Radon-Nikodym approach for ML classification that avoids norms and uses eigenvalue problems to determine class outcomes and probabilities.

Findings

Eigenvalue-based class outcome identification.

Norm-free probability estimation method.

Effective in non-Gaussian distribution cases.

Abstract

For Machine Learning (ML) classification problem, where a vector of $\mathbf{x}$--observations (values of attributes) is mapped to a single $y$ value (class label), a generalized Radon--Nikodym type of solution is proposed. Quantum--mechanics --like probability states $\psi^2(\mathbf{x})$ are considered and "Cluster Centers", corresponding to the extremums of $<y\psi^2(\mathbf{x})>/<\psi^2(\mathbf{x})>$, are found from generalized eigenvalues problem. The eigenvalues give possible $y^{[i]}$ outcomes and corresponding to them eigenvectors $\psi^{[i]}(\mathbf{x})$ define "Cluster Centers". The projection of a $\psi$ state, localized at given $\mathbf{x}$ to classify, on these eigenvectors define the probability of $y^{[i]}$ outcome, thus avoiding using a norm ($L^2$ or other types), required for "quality criteria" in a typical Machine Learning technique. A coverage of each `Cluster Center" is calculated, what potentially allows to separate system properties (described by $y^{[i]}$ outcomes) and system testing conditions (described by $C^{[i]}$ coverage). As an example of such application $y$ distribution estimator is proposed in a form of pairs $(y^{[i]},C^{[i]})$, that can be considered as Gauss quadratures generalization. This estimator allows to perform $y$ probability distribution estimation in a strongly non--Gaussian case.