Manifold Data Imputation

TL;DR

This paper introduces a unified framework for reconstructing missing data on smooth manifolds from incomplete samples, combining spectral and variational methods for accurate, stable, and flexible manifold data imputation.

Contribution

It develops a novel approach that integrates Fourier-based and local variational techniques with a moving least-squares framework for effective manifold data imputation.

Findings

The spectral method enforces high-order smoothness via Fourier coefficient decay.

The variational method ensures stability by analyzing conditioning related to missing region geometry.

Numerical experiments demonstrate accurate recovery on surfaces with large missing regions.

Abstract

We consider the problem of reconstructing missing data on a smooth manifold from incomplete and nonuniform samples. While classical methods for manifold approximation typically assume quasi-uniform data, their performance deteriorates significantly in the presence of large gaps or holes. We propose a unified framework for manifold data imputation that reduces the problem to function reconstruction on locally defined tangent spaces. The approach combines two complementary strategies. The first is a Fourier-based method that determines missing values by prescribing a decay rate of the discrete Fourier coefficients, thereby enforcing high-order smoothness through a global spectral criterion. The second is a local variational method based on minimizing high-order central differences, leading to sparse least-squares systems with favorable stability and conditioning properties. We establish a discrete inverse estimate linking decay of Fourier coefficients to uniform bounds on high-order divided differences, providing a theoretical foundation for the spectral approach. For the variational method, we analyze existence, uniqueness, and scaling behavior, showing that conditioning depends primarily on the geometry of the missing region. These functional reconstruction techniques are integrated with a moving least-squares projection framework to yield a practical algorithm for manifold completion. Numerical experiments, including reconstruction on surfaces with significant missing regions, demonstrate accurate and stable recovery without requiring a global parameterization. The proposed framework provides a flexible and effective approach to manifold data imputation in challenging settings with incomplete data.