Multi-Frequency Vector Diffusion Maps

TL;DR

Multi-Frequency Vector Diffusion Maps (MFVDM) is a novel framework that enhances high-dimensional data analysis by integrating multiple nonlinear embeddings, improving neighbor search and alignment in noisy datasets.

Contribution

MFVDM generalizes vector diffusion maps by combining multiple irreducible representations, offering a more robust method for organizing complex datasets.

Findings

Outperforms VDM and diffusion maps on noisy data

Improves nearest neighbor search accuracy

Effective on synthetic and cryo-EM datasets

Abstract

We introduce multi-frequency vector diffusion maps (MFVDM), a new framework for organizing and analyzing high dimensional datasets. The new method is a mathematical and algorithmic generalization of vector diffusion maps (VDM) and other non-linear dimensionality reduction methods. MFVDM combines different nonlinear embeddings of the data points defined with multiple unitary irreducible representations of the alignment group that connect two nodes in the graph. We illustrate the efficacy of MFVDM on synthetic data generated according to a random graph model and cryo-electron microscopy image dataset. The new method achieves better nearest neighbor search and alignment estimation than the state-of-the-arts VDM and diffusion maps (DM) on extremely noisy data.