# Component Tree Loss Function: Definition and Optimization

arXiv: https://arxiv.org/abs/2101.08063 · HTML: https://mlsanity.com/paper/2101.08063/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 24 references that are in the map.
- First posted on arXiv 20 Jan 2021.

## Builds on (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Rich feature hierarchies for accurate object detection and semantic segmentation](https://mlsanity.com/paper/1311.2524/) (2013)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology](https://mlsanity.com/paper/1910.01877/) (2019)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [Explicit topological priors for deep-learning based image segmentation using persistent homology](https://mlsanity.com/paper/1901.10244/) (2019)
- [Ultrametric Fitting by Gradient Descent](https://mlsanity.com/paper/1905.10566/) (2019)

## Built on by (most cited first, 1 of 1)

- [On Hamming-Lipschitz Type Stability of the Subdominant (Minmax) Ultrametric: Theory and Simple Proofs](https://mlsanity.com/paper/2608.04014/) (2026)

## Abstract (arXiv)

In this article, we propose a method to design loss functions based on component trees which can be optimized by gradient descent algorithms and which are therefore usable in conjunction with recent machine learning approaches such as neural networks. We show how the altitudes associated to the nodes of such hierarchical image representations can be differentiated with respect to the image pixel values. This feature is used to design a generic loss function that can select or discard image maxima based on various attributes such as extinction values. The possibilities of the proposed method are demonstrated on simulated and real image filtering.