Neural network modeling of data with gaps: method of principal curves, Carleman's formula, and other

TL;DR

This paper introduces a neural network-based method for modeling data with gaps using principal curves, Carleman's formulas, and iterative techniques, enabling data gap filling, repair, and correction.

Contribution

It generalizes existing methods by developing three versions of principal curve-based models with physical interpretations and links to neural network architectures.

Findings

Effective data gap filling demonstrated

Method improves data repair and correction

Connections to self-organizing maps established

Abstract

A method of modeling data with gaps by a sequence of curves has been developed. The new method is a generalization of iterative construction of singular expansion of matrices with gaps. Under discussion are three versions of the method featuring clear physical interpretation: linear - modeling the data by a sequence of linear manifolds of small dimension; quasilinear - constructing "principal curves: (or "principal surfaces"), univalently projected on the linear principal components; essentially non-linear - based on constructing "principal curves": (principal strings and beams) employing the variation principle; the iteration implementation of this method is close to Kohonen self-organizing maps. The derived dependencies are extrapolated by Carleman's formulas. The method is interpreted as a construction of neural network conveyor designed to solve the following problems: to fill gaps in data; to repair data - to correct initial data values in such a way as to make the constructed models work best; to construct a calculator to fill gaps in the data line fed to the input.