# Coarse Graining of Data via Inhomogeneous Diffusion Condensation

arXiv: https://arxiv.org/abs/1907.04463 · HTML: https://mlsanity.com/paper/1907.04463/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jul 2019.

## Built on by (most cited first, 9 of 9)

- [Taxonomy of Benchmarks in Graph Representation Learning](https://mlsanity.com/paper/2206.07729/) (2022)
- [Learning shared neural manifolds from multi-subject FMRI data](https://mlsanity.com/paper/2201.00622/) (2021)
- [CUTS: A Deep Learning and Topological Framework for Multigranular Unsupervised Medical Image Segmentation](https://mlsanity.com/paper/2209.11359/) (2022)
- [HUMAP: Hierarchical Uniform Manifold Approximation and Projection](https://mlsanity.com/paper/2106.07718/) (2021)
- [Time-inhomogeneous diffusion geometry and topology](https://mlsanity.com/paper/2203.14860/) (2022)
- [Persistent Homology for High-dimensional Data Based on Spectral Methods](https://mlsanity.com/paper/2311.03087/) (2023)
- [Cover Learning for Large-Scale Topology Representation](https://mlsanity.com/paper/2503.09767/) (2025)
- [Unsupervised Segmentation of Colonoscopy Images](https://mlsanity.com/paper/2312.12599/) (2023)
- [scShapeBench: Discovering geometry from high dimensional scRNAseq data](https://mlsanity.com/paper/2605.12662/) (2026)

## Abstract (arXiv)

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.