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

Alexander N. Gorban, A. A. Rossiev+1 more|arXiv (Cornell University)|May 21, 2003

Neural Networks and Applications6 references3 citations

TL;DR

This paper proposes a neural network-based method for modeling incomplete data with gaps using principal curves, Carleman's formula, and iterative optimization. It introduces three variants—linear, quasilinear, and nonlinear—using dimensionality reduction and variational principles to reconstruct missing data, with extrapolation via Carleman's formula, offering a robust framework for data repair and gap-filling in noisy or sparse datasets.

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.

Motivation & Objective

• To address the challenge of modeling multivariate data with missing values or gaps using neural network-inspired techniques.
• To develop a systematic approach for reconstructing incomplete data by modeling underlying low-dimensional manifolds.
• To provide a physically interpretable method that combines dimensionality reduction with extrapolation for improved data repair.
• To extend traditional matrix decomposition techniques to handle data with structural gaps through iterative refinement.
• To enable the construction of a neural network conveyor that dynamically fills gaps and corrects data values for optimal model fitting.

Proposed method

• Employs iterative construction of singular expansions for matrices with missing entries, generalizing low-rank approximation to incomplete data.
• Introduces three methodological variants: linear (low-dimensional linear manifolds), quasilinear (principal curves univocally projected onto principal components), and nonlinear (principal curves via variational principles).
• Uses a variational principle to define principal curves as minimizing a functional related to data fidelity and smoothness, analogous to energy minimization.
• Implements the nonlinear method through an iterative algorithm closely resembling Kohonen self-organizing maps, enabling adaptive curve fitting.
• Applies Carleman's formula to extrapolate derived dependencies beyond observed data ranges, ensuring continuity and smoothness.
• Treats the entire process as a neural network conveyor that takes input data with gaps and outputs repaired, gap-filled data sequences.

Experimental results

Research questions

• RQ1 How can neural network principles be adapted to model multivariate data with missing values or gaps?
• RQ2 What is the optimal way to represent incomplete data using low-dimensional manifolds while preserving data structure?
• RQ3 Can variational principles be used to define and construct principal curves that serve as effective models for incomplete data?
• RQ4 How can extrapolation be reliably performed beyond observed data ranges using analytical formulas like Carleman’s?
• RQ5 To what extent can iterative, self-organizing algorithms emulate the behavior of principal curve construction in data repair tasks?

Key findings

• The method successfully reconstructs missing data points by modeling the underlying data manifold through iterative refinement of principal curves.
• The quasilinear variant provides a stable and interpretable model by projecting curves univocally onto principal components, ensuring uniqueness and smoothness.
• The nonlinear variant, based on a variational principle, demonstrates strong convergence properties similar to Kohonen self-organizing maps, enabling robust fitting to complex data structures.
• Carleman's formula enables reliable extrapolation of derived dependencies beyond the observed data range, enhancing model generalization.
• The framework functions as a neural network conveyor capable of real-time gap-filling and data correction, improving model performance on incomplete datasets.
• The approach generalizes classical matrix decomposition techniques to handle data with structural gaps, offering a unified solution for data repair and imputation.