# Probabilistic Watershed: Sampling all spanning forests for seeded segmentation and semi-supervised learning

arXiv: https://arxiv.org/abs/1911.02921 · HTML: https://mlsanity.com/paper/1911.02921/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Nov 2019.

## Built on by (most cited first, 2 of 2)

- [A Novel Approach to Regularising 1NN classifier for Improved Generalization](https://mlsanity.com/paper/2402.08405/) (2024)
- [Extensions of Karger's Algorithm: Why They Fail in Theory and How They Are Useful in Practice](https://mlsanity.com/paper/2110.02750/) (2021)

## Abstract (arXiv)

The seeded Watershed algorithm / minimax semi-supervised learning on a graph computes a minimum spanning forest which connects every pixel / unlabeled node to a seed / labeled node. We propose instead to consider all possible spanning forests and calculate, for every node, the probability of sampling a forest connecting a certain seed with that node. We dub this approach "Probabilistic Watershed". Leo Grady (2006) already noted its equivalence to the Random Walker / Harmonic energy minimization. We here give a simpler proof of this equivalence and establish the computational feasibility of the Probabilistic Watershed with Kirchhoff's matrix tree theorem. Furthermore, we show a new connection between the Random Walker probabilities and the triangle inequality of the effective resistance. Finally, we derive a new and intuitive interpretation of the Power Watershed.