# On The Radon-Nikodym Spectral Approach With Optimal Clustering

## TL;DR

This paper introduces a Radon-Nikodym spectral approach for interpolation, classification, and clustering that constructs probabilities and the probability space from data, using eigenproblems and Lebesgue quadrature techniques.

## Contribution

It presents a novel Radon-Nikodym method that dynamically updates both outcome probabilities and the probability space with new data, unlike traditional Bayesian approaches.

## Key findings

- The approach solves interpolation via a Radon-Nikodym derivative.
- Classification uses Lebesgue quadrature for probability estimation.
- Clustering is achieved through Gaussian quadrature on Lebesgue measure.

## Abstract

Problems of interpolation, classification, and clustering are considered. In the tenets of Radon--Nikodym approach $\langle f(\mathbf{x})\psi^2 \rangle / \langle\psi^2\rangle$, where the $\psi(\mathbf{x})$ is a linear function on input attributes, all the answers are obtained from a generalized eigenproblem $|f|\psi^{[i]}\rangle = \lambda^{[i]} |\psi^{[i]}\rangle$. The solution to the interpolation problem is a regular Radon-Nikodym derivative. The solution to the classification problem requires prior and posterior probabilities that are obtained using the Lebesgue quadrature[1] technique. Whereas in a Bayesian approach new observations change only outcome probabilities, in the Radon-Nikodym approach not only outcome probabilities but also the probability space $|\psi^{[i]}\rangle$ change with new observations. This is a remarkable feature of the approach: both the probabilities and the probability space are constructed from the data. The Lebesgue quadrature technique can be also applied to the optimal clustering problem. The problem is solved by constructing a Gaussian quadrature on the Lebesgue measure. A distinguishing feature of the Radon-Nikodym approach is the knowledge of the invariant group: all the answers are invariant relatively any non-degenerated linear transform of input vector $\mathbf{x}$ components. A software product implementing the algorithms of interpolation, classification, and optimal clustering is available from the authors.