# Compressed Diffusion

## TL;DR

This paper introduces a novel compressed diffusion method that reduces computational complexity in diffusion maps by using data regions instead of individual data points, enabling efficient analysis of large datasets.

## Contribution

It presents a new approach to compute diffusion geometry via data region compression, improving scalability and efficiency over traditional diffusion maps.

## Key findings

- Outperforms existing methods on large datasets
- Reduces computational complexity from cubic to lower orders
- Provides accurate diffusion embeddings through data region approximation

## Abstract

Diffusion maps are a commonly used kernel-based method for manifold learning, which can reveal intrinsic structures in data and embed them in low dimensions. However, as with most kernel methods, its implementation requires a heavy computational load, reaching up to cubic complexity in the number of data points. This limits its usability in modern data analysis. Here, we present a new approach to computing the diffusion geometry, and related embeddings, from a compressed diffusion process between data regions rather than data points. Our construction is based on an adaptation of the previously proposed measure-based Gaussian correlation (MGC) kernel that robustly captures the local geometry around data points. We use this MGC kernel to efficiently compress diffusion relations from pointwise to data region resolution. Finally, a spectral embedding of the data regions provides coordinates that are used to interpolate and approximate the pointwise diffusion map embedding of data. We analyze theoretical connections between our construction and the original diffusion geometry of diffusion maps, and demonstrate the utility of our method in analyzing big datasets, where it outperforms competing approaches.