# Coarse Graining of Data via Inhomogeneous Diffusion Condensation

## TL;DR

This paper introduces a multiresolution data analysis method using inhomogeneous diffusion processes to uncover hierarchical structures and groupings in complex datasets, demonstrated on neuronal data.

## Contribution

It presents a novel inhomogeneous diffusion-based approach for multilevel data coarse-graining and hierarchical clustering, revealing nested structures at multiple granularities.

## Key findings

- Effectively uncovers hierarchical data structures.
- Visualizes multilevel clustering and variation elimination.
- Demonstrates utility on neuronal data for organization and connectivity insights.

## Abstract

Big data often has emergent structure that exists at multiple levels of abstraction, which are useful for characterizing complex interactions and dynamics of the observations. Here, we consider multiple levels of abstraction via a multiresolution geometry of data points at different granularities. To construct this geometry we define a time-inhomogeneous diffusion process that effectively condenses data points together to uncover nested groupings at larger and larger granularities. This inhomogeneous process creates a deep cascade of intrinsic low pass filters on the data affinity graph that are applied in sequence to gradually eliminate local variability while adjusting the learned data geometry to increasingly coarser resolutions. We provide visualizations to exhibit our method as a continuously-hierarchical clustering with directions of eliminated variation highlighted at each step. The utility of our algorithm is demonstrated via neuronal data condensation, where the constructed multiresolution data geometry uncovers the organization, grouping, and connectivity between neurons.