Low‐rank isomap algorithm

Abstract

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 by graph processing methods, the eigenvalue decomposition stage remains a bottleneck in the problem. In this paper, we propose the Low-Rank Isomap (LRI) 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 as long as the number of observations remains extensively larger than the dimensionality of the ambient space. The superiority of the LRI algorithm compared to some state-of-art algorithms is experimentally verified on facial image clustering in terms of speed and accuracy.