Norm-Free Radon-Nikodym Approach to Machine Learning
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For Machine Learning (ML) classification problem, where a vector of 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 ψ^2(x) are considered and "Cluster Centers", corresponding to the extremums of <yψ^2(x)>/<ψ^2(x)>, are found from generalized eigenvalues problem. The eigenvalues give possible y^[i] outcomes and corresponding to them eigenvectors ψ^[i](x) define "Cluster Centers". The projection of a ψ state, localized at given 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.
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