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The Morse Transform for Discrete Shape Analysis

This paper introduces the Morse Transform, a new topological descriptor that quantifies object geometry by cataloging critical points across multiple height-functions, demonstrating superior performance in ligand-based virtual screening compared to existing topological and standard shape-based methods.

Original authors: Alexander M. Tanaka, Aras T. Asaad, Richard Cooper, Vidit Nanda

Published 2026-06-17

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CC BY 4.0

Original authors: Alexander M. Tanaka, Aras T. Asaad, Richard Cooper, Vidit Nanda

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine you are trying to describe a complex, 3D object—like a molecule that might become a medicine—to a computer. You want the computer to understand its shape so it can predict if it will fit into a specific "lock" (a protein in the body) to do its job.

Usually, scientists try to describe shapes by counting holes, bumps, and tunnels, or by taking a single "shadow" of the object from one angle. But the authors of this paper, Alexander Tanaka and his team, argue that these methods miss the most important details. They introduce a new tool called the Morse Transform.

Here is how it works, using simple analogies:

1. The "Hiking Map" Analogy

Imagine the molecule is a mountain range.

Old Methods: Some methods just look at the total area of the mountains or count how many lakes (holes) are in the valleys. Others look at the mountain from just one direction, like a single photograph.
The Morse Transform: This method is like sending a hiker to walk over the mountain from every possible direction (North, South, East, West, and every angle in between). As the hiker walks, they don't just record the height; they specifically note the special spots:
- Peaks: The very top of a mountain.
- Valleys: The bottom of a dip.
- Saddles: The pass between two peaks (where you go up one side and down the other).

The Morse Transform creates a detailed "hiking log" for every single direction. It records the height of these special spots and what kind of spot they are.

2. Why Focus on the "Outer Skin"?

The paper makes a crucial point: The outside of the shape matters most.
When a drug molecule tries to bind to a protein, it's the outer surface that touches the protein, not the atoms deep inside.

The Morse Transform is smart because it prioritizes the outermost peaks and valleys. It ignores the deep, internal details that don't affect how the molecule interacts with the world.
It does this by only looking at the "top 20" special spots it finds in each direction, rather than getting bogged down in every single tiny bump on the surface.

3. Turning the Map into a "Fingerprint"

A list of hiking logs for 100 different directions is too big for a computer to use directly. So, the authors created a way to compress this massive amount of data into a single, compact "fingerprint" (a vector of numbers).

Think of it like summarizing a whole novel into a single sentence that captures the main plot points.
This fingerprint is robust. Even if you shake the molecule slightly (adding a little "noise" or jitter), the fingerprint stays mostly the same, which is great for real-world data that isn't perfect.

4. The Big Test: Virtual Drug Screening

The team tested this new tool on a classic problem: Virtual Screening.

The Goal: Find which molecules in a giant library are likely to stick to a specific protein target (like finding the right key for a lock).
The Competition: They compared their Morse Transform against:
- Other topological methods (like the Euler Characteristic Transform, which is like a simpler, less detailed map).
- Standard shape-based methods used by chemists for years.
The Result: The Morse Transform won. It was better at predicting which molecules would bind to the protein than the other shape-based methods.
The Secret Sauce: When they added simple chemical data (like the electrical charge of atoms) to their shape "fingerprint," the results became even stronger, beating almost every other method they tested.

Summary

The authors didn't just invent a new math trick; they built a better way to describe the shape of molecules. By focusing on the "peaks and valleys" of a molecule from every angle, and by prioritizing the outer surface, they created a digital fingerprint that helps computers predict drug behavior more accurately than previous shape-based tools.

What the paper does NOT claim:

They do not claim this method has been tested on actual patients or in clinical trials.
They do not claim this will immediately cure diseases.
They strictly tested this on computer simulations (virtual screening) using existing datasets of known molecules.

In short: They found a better way to describe the shape of a molecule so computers can guess if it will work as a drug, and it worked better than the old ways of describing shapes.

Technical Summary: The Morse Transform for Discrete Shape Analysis

Problem Statement
The geometry of an object is fundamental to its physical interactions, yet numerically describing this geometry for statistical inference or classification remains challenging. While Topological Data Analysis (TDA) offers tools like persistent homology and directional transforms (e.g., Persistent Homology Transform [PHT] and Euler Characteristic Transform [ECT]), these methods often lack explicit local information or prioritize global structures over surface geometry. Specifically, in ligand-based virtual screening (LBVS), where protein-ligand binding is governed by molecular shape complementarity, existing descriptors may not sufficiently capture the critical surface features that drive binding affinity.

Methodology
The authors introduce the Morse Transform (MT), a new topological transform based on piecewise-linear (PL) Morse theory, designed to summarize the shape of a simplicial complex $K$ embedded in Euclidean space $\mathbb{R}^n$ .

Mathematical Foundation:
- The transform utilizes a PL Morse function $f_\xi(v) = \langle v, \xi \rangle$ defined by a direction vector $\xi$ on the unit sphere.
- It identifies critical points (vertices) where the homology of the "upper link" (the subcomplex of simplices connected to the vertex with higher or equal function values) changes.
- Unlike the ECT, which tracks the Euler characteristic of superlevel sets, the MT records the reduced Betti numbers of the upper links at these critical points. This provides a finer topological invariant, categorizing critical points as peaks, troughs, or saddles with greater specificity.
The Transform Output:
- For a chosen depth $d$ , the MT maps a direction $\xi$ to a matrix where rows correspond to the top $d$ critical points.
- Each row contains the critical value (height) and the vector of reduced Betti numbers of the corresponding upper link.
- The authors prove that at maximal depth, the MT determines the ECT, inheriting its injectivity (the ability to uniquely reconstruct the shape).
Vectorisation (Morse Statistics):
- To make the transform amenable to machine learning, the authors propose a vectorisation scheme called Morse statistics.
- This involves computing summary statistics (percentiles at 0, 10, 25, 40, 50, 60, 75, 90, 100) of the critical values and Betti numbers across a finite set of directions.
- The resulting vector is compact, invariant to the labelling of directions, and independent of the specific depth or number of directions used during generation.
Application to Molecular Data:
- Molecules are represented as pruned weighted Delaunay triangulations derived from atom centers and van der Waals radii.
- The pipeline generates a feature vector by combining the Morse statistics of the molecular shape with chemical descriptors (partial charge, molar refractivity, lipophilicity) for each atom.
- A Light Gradient Boosting Machine (LGBM) is trained on these vectors for binary classification (active ligands vs. decoys).

Key Contributions

Novel Transform: Definition of the Morse Transform, which leverages PL Morse theory to provide explicit local topological information (Betti numbers of upper links) rather than just global Euler characteristics or persistence diagrams.
Surface Prioritization: The transform naturally prioritizes the outermost regions of a shape (the first few critical points), which is crucial for surface-dependent tasks like molecular binding.
Efficiency: For dimensions $n \le 4$ , computing the MT along one direction has a time complexity comparable to the ECT ( $O(|K_0| \log |K_0|)$ ) and is asymptotically faster than the PHT.
Robust Vectorisation: A method to compress the transform into a fixed-length feature vector that retains rich structural information while being robust to noise and variations in direction sampling.

Results
The authors benchmarked the Morse feature vector on two standard virtual screening datasets: DUD-E (specifically the D8 diverse subset) and MUV.

Comparison with Topological Transforms: On both datasets, the plain Morse features (MT-P) achieved the highest mean Area Under the Receiver Operating Characteristic Curve (AUROC) compared to ECT-based descriptors (ECT-F and ECT-P) and the Morse Euler-critical transform (MET-P).
- D8: MT-P mean AUROC = 0.84 (vs. ECT-F 0.79).
- MUV: MT-P mean AUROC = 0.65 (vs. ECT-F 0.58).
Comparison with Specialized LBVS Descriptors:
- Plain Morse features (M) outperformed other shape-based descriptors (USR, WHIMu, RGMolSA, KQMolSA) on average.
- Chemistry-supplemented Morse features (M+C), which combine shape statistics with chemical properties, achieved the highest mean AUROC across all evaluated descriptors (0.97 on D8 and 0.74 on MUV), outperforming hybrid descriptors like USRCAT and WHIM.
Ablation Studies:
- Performance plateaued around a depth of 10 and 32 directions, indicating that the method is robust and does not require exhaustive tuning of these parameters.
- The method demonstrated robustness to positional noise (Gaussian perturbations) and sensitivity to the choice of critical vs. non-critical vertices (focusing on critical vertices yielded better results).
Deep Learning Context: While the authors note that some deep learning models report higher AUROCs, they highlight that these often use different validation protocols (e.g., single splits vs. cross-validation). When re-evaluated with rigorous cross-validation, the performance of some deep learning baselines decreased, suggesting the Morse-based approach is competitive with complex learned representations under fair comparison.

Significance and Claims
The paper claims that the Morse Transform offers a superior balance between locality and global structure for discrete shape analysis. By focusing on critical points and their local topological types, it captures the "outermost" geometry essential for tasks like protein-ligand binding more effectively than the ECT.

The authors emphasize that their work demonstrates the utility of concentrating on local critical point information without needing to reconstruct the entire superlevel set structure. The success of the chemistry-supplemented Morse vector in virtual screening suggests that topological descriptors can effectively complement chemical data, providing a "rich but compact" feature set that outperforms both traditional shape descriptors and other topological transforms in classification tasks. The authors provide a codebase to facilitate the adoption of this transform for general discrete shape analysis beyond drug discovery.