# The Morse Transform for Discrete Shape Analysis

arXiv: https://arxiv.org/abs/2503.04507 · HTML: https://mlsanity.com/paper/2503.04507/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 81 references that are in the map.
- First posted on arXiv 6 Mar 2025.

## Builds on (most cited first, 11 of 11)

- [Tune: A Research Platform for Distributed Model Selection and Training](https://mlsanity.com/paper/1807.05118/) (2018)
- [GEOM: Energy-annotated molecular conformations for property prediction and molecular generation](https://mlsanity.com/paper/2006.05531/) (2020)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [More Asymmetry Yields Faster Matrix Multiplication](https://mlsanity.com/paper/2404.16349/) (2024)
- [ToDD: Topological Compound Fingerprinting in Computer-Aided Drug Discovery](https://mlsanity.com/paper/2211.03808/) (2022)
- [Euler Characteristic Curves and Profiles: a stable shape invariant for big data problems](https://mlsanity.com/paper/2212.01666/) (2022)
- [The Weighted Euler Characteristic Transform for Image Shape Classification](https://mlsanity.com/paper/2307.13940/) (2023)
- [Hybrid transforms of constructible functions](https://mlsanity.com/paper/2111.07829/) (2021)
- [Topological fingerprints for audio identification](https://mlsanity.com/paper/2309.03516/) (2023)
- [Efficient computation of topological integral transforms](https://mlsanity.com/paper/2405.02256/) (2024)
- [Shoving tubes through shapes gives a sufficient and efficient shape statistic](https://mlsanity.com/paper/2412.18452/) (2024)

## Abstract (arXiv)

The geometry of an object plays a vital role in modulating its interactions with the physical world. It nevertheless remains difficult to describe geometric information numerically for the purposes of statistical inference or classification tasks. Here, we introduce a new topological transform which leverages directional piecewise-linear Morse theory to quantify the geometry of an embedded object by cataloguing critical points across multiple height-functions. The output of this Morse transform records both the heights and the local topological type (peak, trough or saddle) of the critical points that characterise the underlying shape, retaining finer information than the Euler characteristic transform whilst naturally prioritising a shape's outermost regions. Crucially, this output can be further compressed into a rich but compact feature vector. We benchmark the Morse feature vector as a descriptor for ligand-based virtual screening (LBVS), which intrinsically depends on the shape of molecules. Under a common gradient-boosted tree classification pipeline, Morse descriptors achieve the highest mean AUROC when compared to other topological transform descriptors and to standard shape-based LBVS descriptors.