Non-Redundant Spectral Dimensionality Reduction

TL;DR

This paper introduces a novel spectral dimensionality reduction method that eliminates redundant embedding coordinates by enforcing unpredictability constraints, resulting in more informative and compact data representations.

Contribution

It proposes replacing orthogonality constraints with unpredictability constraints in spectral algorithms to prevent redundancy and improve data embedding quality.

Findings

Redundancy in spectral embeddings is effectively reduced.

The method enhances visualization and classification performance.

It produces more compact and informative data representations.

Abstract

Spectral dimensionality reduction algorithms are widely used in numerous domains, including for recognition, segmentation, tracking and visualization. However, despite their popularity, these algorithms suffer from a major limitation known as the "repeated Eigen-directions" phenomenon. That is, many of the embedding coordinates they produce typically capture the same direction along the data manifold. This leads to redundant and inefficient representations that do not reveal the true intrinsic dimensionality of the data. In this paper, we propose a general method for avoiding redundancy in spectral algorithms. Our approach relies on replacing the orthogonality constraints underlying those methods by unpredictability constraints. Specifically, we require that each embedding coordinate be unpredictable (in the statistical sense) from all previous ones. We prove that these constraints necessarily prevent redundancy, and provide a simple technique to incorporate them into existing methods. As we illustrate on challenging high-dimensional scenarios, our approach produces significantly more informative and compact representations, which improve visualization and classification tasks.