Low-Rank Isomap Algorithm

TL;DR

The paper introduces Low-Rank Isomap, an improved nonlinear dimensionality reduction method that significantly reduces computational complexity by using a low-rank projection, maintaining accuracy and structural integrity.

Contribution

It proposes a novel low-rank projection technique for Isomap, reducing eigenvalue decomposition complexity from quadratic to linear order while preserving data structure.

Findings

Outperforms state-of-the-art algorithms in speed and accuracy

Maintains structural information during dimensionality reduction

Effective for facial image clustering

Abstract

The Isomap is a well-known nonlinear dimensionality reduction method that highly suffers from computational complexity. Its computational complexity mainly arises from two stages; a) embedding a full graph on the data in the ambient space, and b) a complete eigenvalue decomposition. Although the reduction of the computational complexity of the graphing stage has been investigated, yet the eigenvalue decomposition stage remains a bottleneck in the problem. In this paper, we propose the Low-Rank Isomap algorithm by introducing a projection operator on the embedded graph from the ambient space to a low-rank latent space to facilitate applying the partial eigenvalue decomposition. This approach leads to reducing the complexity of Isomap to a linear order while preserving the structural information during the dimensionality reduction process. The superiority of the Low-Rank Isomap algorithm compared to some state-of-art algorithms is experimentally verified on facial image clustering in terms of speed and accuracy.