Principal Ellipsoid Analysis (PEA): Efficient non-linear dimension reduction & clustering

TL;DR

Principal Elliptical Analysis (PEA) offers a computationally efficient, non-linear alternative to PCA that captures complex data structures and shapes, improving clustering performance in diverse real-world applications.

Contribution

The paper introduces PEA, a novel framework that extends PCA to fit elliptical data structures, with theoretical guarantees and practical advantages over traditional methods.

Findings

PEA performs comparably to k-means on simple datasets.

PEA significantly outperforms in complex, non-linear data structures.

Theoretical guarantees ensure PEA's consistency and reliability.

Abstract

Even with the rise in popularity of over-parameterized models, simple dimensionality reduction and clustering methods, such as PCA and k-means, are still routinely used in an amazing variety of settings. A primary reason is the combination of simplicity, interpretability and computational efficiency. The focus of this article is on improving upon PCA and k-means, by allowing non-linear relations in the data and more flexible cluster shapes, without sacrificing the key advantages. The key contribution is a new framework for Principal Elliptical Analysis (PEA), defining a simple and computationally efficient alternative to PCA that fits the best elliptical approximation through the data. We provide theoretical guarantees on the proposed PEA algorithm using Vapnik-Chervonenkis (VC) theory to show strong consistency and uniform concentration bounds. Toy experiments illustrate the performance of PEA, and the ability to adapt to non-linear structure and complex cluster shapes. In a rich variety of real data clustering applications, PEA is shown to do as well as k-means for simple datasets, while dramatically improving performance in more complex settings.