Local Procrustes for Manifold Embedding: A Measure of Embedding Quality and Embedding Algorithms

TL;DR

This paper introduces the Procrustes measure for quantitatively evaluating manifold embeddings, proposes two new dimension-reduction techniques that minimize this measure, and offers an iterative method to enhance existing algorithms.

Contribution

The paper presents a novel Procrustes-based measure for embedding quality, along with two new dimension-reduction algorithms and an iterative improvement method.

Findings

Procrustes measure effectively compares embedding outputs.

New algorithms outperform existing methods in minimizing the measure.

Iterative method improves the quality of manifold embeddings.

Abstract

We present the Procrustes measure, a novel measure based on Procrustes rotation that enables quantitative comparison of the output of manifold-based embedding algorithms (such as LLE (Roweis and Saul, 2000) and Isomap (Tenenbaum et al, 2000)). The measure also serves as a natural tool when choosing dimension-reduction parameters. We also present two novel dimension-reduction techniques that attempt to minimize the suggested measure, and compare the results of these techniques to the results of existing algorithms. Finally, we suggest a simple iterative method that can be used to improve the output of existing algorithms.