# From graph cuts to isoperimetric inequalities: Convergence rates of Cheeger cuts on data clouds

arXiv: https://arxiv.org/abs/2004.09304 · HTML: https://mlsanity.com/paper/2004.09304/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Apr 2020.

## Built on by (most cited first, 6 of 6)

- [Gamma-convergence of a nonlocal perimeter arising in adversarial machine learning](https://mlsanity.com/paper/2211.15223/) (2022)
- [On the existence of solutions to adversarial training in multiclass classification](https://mlsanity.com/paper/2305.00075/) (2023)
- [Rates of Convergence for Regression with the Graph Poly-Laplacian](https://mlsanity.com/paper/2209.02305/) (2022)
- [Robust Certification for Laplace Learning on Geometric Graphs](https://mlsanity.com/paper/2104.10837/) (2021)
- [Higher-Order Regularization Learning on Hypergraphs](https://mlsanity.com/paper/2510.26533/) (2025)
- [Affinity Graph Connectivity in Convex Clustering](https://mlsanity.com/paper/2605.24673/) (2026)

## Abstract (arXiv)

In this work we study statistical properties of graph-based clustering algorithms that rely on the optimization of balanced graph cuts, the main example being the optimization of Cheeger cuts. We consider proximity graphs built from data sampled from an underlying distribution supported on a generic smooth compact manifold M. In this setting, we obtain high probability convergence rates for both the Cheeger constant and the associated Cheeger cuts towards their continuum counterparts. The key technical tools are careful estimates of interpolation operators which lift empirical Cheeger cuts to the continuum, as well as continuum stability estimates for isoperimetric problems. To our knowledge the quantitative estimates obtained here are the first of their kind.