Elastic principal manifolds and their practical applications

TL;DR

This paper introduces a fast, flexible algorithm for constructing principal manifolds using an elastic membrane analogy, enabling efficient data approximation and visualization across various dimensions and topologies.

Contribution

The paper presents a novel quadratic functional-based algorithm for rapid construction of principal manifolds, adaptable to different dimensions and topologies, with practical implementation in C++ and visualization tools.

Findings

Algorithm is highly effective and suitable for parallel implementation.

Flexible approach allows adaptive strategies like principal graph construction.

Demonstrated applications show good speed and versatility.

Abstract

Principal manifolds serve as useful tool for many practical applications. These manifolds are defined as lines or surfaces passing through "the middle" of data distribution. We propose an algorithm for fast construction of grid approximations of principal manifolds with given topology. It is based on analogy of principal manifold and elastic membrane. The first advantage of this method is a form of the functional to be minimized which becomes quadratic at the step of the vertices position refinement. This makes the algorithm very effective, especially for parallel implementations. Another advantage is that the same algorithmic kernel is applied to construct principal manifolds of different dimensions and topologies. We demonstrate how flexibility of the approach allows numerous adaptive strategies like principal graph constructing, etc. The algorithm is implemented as a C++ package elmap and as a part of stand-alone data visualization tool VidaExpert, available on the web. We describe the approach and provide several examples of its application with speed performance characteristics.