Local High-order Regularization on Data Manifolds

TL;DR

This paper introduces a new sparse, high-order regularizer for data manifolds that overcomes the limitations of traditional graph Laplacian regularizers, improving efficiency and effectiveness in semi-supervised learning tasks.

Contribution

A novel high-order regularizer based on local derivatives that is both globally effective and computationally efficient for large-scale problems.

Findings

Effective in human body shape and pose analysis

Outperforms traditional graph Laplacian regularizers

Maintains computational efficiency with local approximations

Abstract

The common graph Laplacian regularizer is well-established in semi-supervised learning and spectral dimensionality reduction. However, as a first-order regularizer, it can lead to degenerate functions in high-dimensional manifolds. The iterated graph Laplacian enables high-order regularization, but it has a high computational complexity and so cannot be applied to large problems. We introduce a new regularizer which is globally high order and so does not suffer from the degeneracy of the graph Laplacian regularizer, but is also sparse for efficient computation in semi-supervised learning applications. We reduce computational complexity by building a local first-order approximation of the manifold as a surrogate geometry, and construct our high-order regularizer based on local derivative evaluations therein. Experiments on human body shape and pose analysis demonstrate the effectiveness and efficiency of our method.