Source-linked AI summary
A Survey of Vectorization Methods in Topological Data Analysis
Dashti Ali, Aras Asaad, Maria-Jose Jimenez, Vidit Nanda, Eduardo Paluzo-Hidalgo, Manuel Soriano-Trigueros
TL;DR
The paper addresses the difficulty of choosing among many persistent-homology barcode vectorizations for supervised learning. It organizes and benchmarks thirteen methods across three classification datasets, finding that the simple Persistence Statistics vectorization performs best in all three cases, and supplies a web application for experimentation. The study cautions that its comparative conclusions are not exhaustive.
Problem
It is difficult to know which of many barcode vectorization techniques suit particular data, and stable representations may lack discriminative power.
Method
The paper develops a five-category taxonomy, benchmarks thirteen vectorizations on three image-classification datasets, and provides a web application for computing and visualizing them.
Results
Persistence Statistics is the best-performing vectorization across Outex, SHREC14, and Fashion-MNIST classification experiments.
Takeaways & Limitations
The paper provides a taxonomy and reproducible empirical evidence for exploring barcode vectorizations rather than a theorem determining the best method for every classification problem.
Takeaways & Limitations
The comparison is not exhaustive, and different datasets or method choices could change which vectorizations perform best.
Abstract
from arXiv · showhide
Attempts to incorporate topological information in supervised learning tasks have resulted in the creation of several techniques for vectorizing persistent homology barcodes. In this paper, we study thirteen such methods. Besides describing an organizational framework for these methods, we comprehensively benchmark them against three well-known classification tasks. Surprisingly, we discover that the best-performing method is a simple vectorization, which consists only of a few elementary summary statistics. Finally, we provide a convenient web application which has been designed to facilitate exploration and experimentation with various vectorization methods.
Introduction
Persistent homology converts multiscale topological structure into barcodes, but choosing barcode vectorizations for supervised learning remains difficult because the methods face both selection and representational challenges.
- Introduction: Persistent homology builds scale-indexed cell complexes, computes homology across dimensions, and decomposes the resulting vector-space families into barcodes.The resulting intervals summarize geometric features across scales, while the pipeline itself is unsupervised once the cell-complex construction is fixed.
- Introduction: Barcode vectorization is needed to turn topological summaries into Euclidean vectors usable as inputs to neural-network classifiers.Several dozen vectorization methods have been proposed, with thirteen examined in this paper.
- Introduction: Practitioners lack reliable guidance about which vectorization suits which data, because the literature contains many techniques with limited comparative evidence.Even established users may not know all available methods or their appropriate data regimes.
- Introduction: The barcode space is infinite-dimensional and highly nonlinear under bottleneck distance, so faithful finite-dimensional embeddings do not exist.This geometric constraint complicates attempts to represent barcodes without information loss.
- Introduction: Stable vectorizations can preserve distances yet remain weakly discriminative when datasets share coarse structure and differ at finer scales.Thus, stability alone does not guarantee useful classification features.
In This Paper.
The paper organizes barcode vectorizations into a five-part taxonomy, benchmarks thirteen methods on three image-classification datasets, and provides a companion exploration tool.
- In This Paper.: The taxonomy comprises statistical, algebraic, curve, functional, and ensemble vectorizations.Overlaps are assigned to the earliest applicable category, while curve methods remain separate because of their number.
- In This Paper.: The benchmark evaluates thirteen vectorization techniques across five categories and three well-known image-classification datasets.The datasets are Outex, SHREC14, and Fashion-MNIST, selected to increase difficulty for topological methods while remaining recognizable to machine-learning audiences.
- In This Paper.: Across all three classification tasks, the best-performing method is Persistence Statistics, a simple vectorization based on elementary interval summary statistics.The paper presents this result as surprising relative to the more sophisticated alternatives.
- In This Paper.: A companion web application computes and visualizes all thirteen investigated vectorizations and can also be downloaded for local use.The local version is intended to support experiments on more challenging datasets.
Not In This Paper.
The study excludes several related approaches to keep its comparison focused on implemented methods that produce genuine barcode vectors and meet a minimum empirical-performance threshold.
- Not In This Paper.: The comparison restricts attention to methods that produce genuine vectors from barcodes and have accessible, trusted implementations.This excludes kernel methods and path-signature vectorizations.
- Not In This Paper.: The study does not compare architectures designed to infer persistent homology or neural networks built using topological-data-analysis tools.These approaches are outside the paper’s vectorization-focused scope.
- Not In This Paper.: Even methods satisfying the first criteria were discarded if they regularly achieved classification accuracy below fifty percent.This creates an explicit empirical-performance cutoff for inclusion.
- Not In This Paper.: Earlier work has summarized or partially compared barcode vectorizations, but this paper distinguishes itself by organizing and benchmarking thirteen methods comprehensively.The related literature includes summaries, metric-focused comparisons, and smaller method comparisons.
Outline.
The paper introduces barcode preliminaries, presents its taxonomy and datasets, reports experiments, describes the web application, and closes with concluding remarks.
- Outline.: Sections 1–3 establish notation, introduce thirteen organized vectorizations, and describe the three datasets.The paper then reports experimental results in Section 4, with finer details in Appendices A and B.
- Outline.: Section 5 describes the web application, and Section 6 provides brief concluding remarks.This structure follows the experimental results and their supporting appendices.
1. Persistence Barcodes from Data
Persistent homology converts filtered cell complexes into persistence modules whose interval decompositions form barcodes, providing robust but statistically nonlinear summaries of data.
- Every persistence module decomposes uniquely into interval modules, so its barcode and multiplicities recover the module up to isomorphism.
- Barcodes inherit stability through an isometry between interleaving distance on persistence modules and bottleneck distance on barcodes.
- Although barcodes tolerate certain perturbations, their bottleneck metric space is nonlinear, making even averages undefined for arbitrary barcode collections.
- Filtered simplicial or cubical complexes arise from datasets such as point clouds and grayscale images before homology produces persistence modules.
- The paper motivates vectorization because barcode spaces are difficult statistically, despite admitting many mappings into Hilbert spaces.
2. Vectorization Methods for Barcodes
The paper organizes barcode vectorizations into statistical, algebraic, curve-based, and ensemble approaches, each converting interval data into finite-dimensional features or functions.
- Statistical Vectorizations: Persistence Statistics summarizes births, deaths, midpoints, lifespans, bar counts, and entropy using elementary statistics.The listed statistics include means, dispersion measures, percentiles, and full ranges.
- Algebraic Vectorizations: Algebraic methods evaluate polynomial or complex-polynomial features of barcode endpoints and lifespans, including Adcock–Carlsson coordinates and tropical coordinates.Tropical methods replace ordinary polynomial operations with max and plus to address instability in algebraic functions.
- Stability: Small bottleneck changes can cause large fluctuations in algebraic, Betti, and lifespan representations, motivating stable alternatives and design adjustments.
- Algebraic Vectorizations: Complex-polynomial vectorizations often retain only selected coefficients or pad by powers of z so all feature vectors share one dimension.
- Curve Vectorizations: Curve vectorizations map barcodes to functions such as Betti, lifespan, persistence landscape, entropy summary, and weighted silhouette curves.The Betti curve counts intervals containing t, while weighted silhouettes emphasize shorter or longer intervals according to α.
- Ensemble Vectorizations: Ensemble approaches learn a finite-dimensional representation from collections of training barcodes, as in the ATOL vectorization mapping samples into R^b.
3. Datasets
The benchmark uses three classification datasets of increasing difficulty for topological methods: texture images, 3D human shapes, and Fashion-MNIST images.
- The experiments compare vectorizations on Outex, SHREC14, and Fashion-MNIST, three standard datasets arranged by increasing difficulty for topological methods.
- Outex: Outex contains texture images from 68 classes, with Outex10 using 10 randomly selected classes and both variants applying a 70/30 train/test split.Each class has 20 images of size 128 × 128 pixels, totaling 1,360 images.
- Outex: Outex images are treated as cubical complexes, with pixel-intensity filtrations and persistent homology computed in dimensions 0 and 1 using GUDHI.
- SHREC14: SHREC14 uses 300 synthetic human shapes across 15 classes, with Heat Kernel Signature filtrations and a 70/30 training/testing split.
- Fashion-MNIST: Fashion-MNIST provides 60,000 training and 10,000 testing grayscale images across 10 classes, using preprocessing, Canny edges, distance filtrations, and lower-star inheritance.
4. Results
Across three classification tasks, Persistence Statistics consistently performs best or near-best, while dataset difficulty and experimental scope materially affect accuracy. The results also show that algebraic methods can perform strongly on the harder Outex68 task, and that Adaptive Template Systems can underperform despite extensive training data.
- 99.2% accuracy is shared by Persistence Statistics and Algebraic Functions on the easier Outex10 experiment, with Persistent Silhouettes at 98.3%.
- Persistence Statistics is the strongest overall method, achieving the top reported accuracy on Outex68, SHREC14, and FMNIST.It earns 93.4% on Outex68, 94.7% on SHREC14, and 74.9% on FMNIST.
- 93.4% accuracy makes Persistence Statistics the clear Outex68 winner, ahead of Tropical Coordinates at 88.7%, while every method degrades from Outex10.
- 74.9% accuracy puts Persistence Statistics ahead of Template Functions at 74.7% on FMNIST, where all methods score lower than on the preceding datasets.These experiments used only 0-dimensional barcodes and a Random Forest classifier with 100 trees.
- Adaptive Template Systems perform far worse than ordinary Template Functions despite using 60,000 training barcodes, and the authors report no clear explanation.
5. Web Application
The web application interactively computes, visualizes, and exports the paper’s barcode vectorizations without requiring users to write code.
- The interactive web application computes and visualizes all thirteen vectorization techniques in a modern browser.It supports different visualization formats depending on the vectorization method.
- Users can upload their own data or use pre-loaded samples from the three benchmark datasets.Specifications, formatting guidelines, and downloading instructions are provided through the accompanying repository.
- Barcode intervals in dimensions 0 and 1 are displayed by default, while vectorizations use tables, bar graphs, scatter plots, piecewise-linear graphs, and heat maps.Persistence statistics appear in tables; algebraic, template, adaptive template, and ATOL methods use bar graphs; landscapes use piecewise-linear graphs; persistence images use heat maps.
- The application visualizes template functions through bar heights that indicate the values of selected functions.The same bar-graph convention is used for adaptive template systems and ATOL.
- Users can download vectors generated by each method to support downstream analysis.The application is intended to provide these visualizations without requiring users to write code.
6. Concluding Remarks
The paper offers taxonomy, benchmarking, and a web application as empirical guidance for choosing barcode vectorizations, while cautioning that its conclusions are not exhaustive.
- The paper organizes thirteen vectorization methods into five categories and provides reproducible empirical evidence because method selection remains difficult to guide theoretically.The five categories and web application are intended to support further experimentation.
- Persistence Statistics achieved the best performance in all experiments and is faster to compute than the alternatives.The authors do not necessarily recommend dispensing with more sophisticated methods.
- The comparative conclusion may change with different classification datasets or additional polynomials in an algebraic vectorization.The authors emphasize that no comparative study such as theirs can be truly exhaustive.
Appendix A. Implementation and Parameter Details
The appendix documents implementations, tuned parameter ranges, data dimensions, classifiers, and heat-kernel parameter dependence for the benchmark.
- The benchmark uses existing packages alongside custom implementations, with method-specific parameters optimized over explicitly listed ranges.These include resolutions, polynomial counts, Gaussian bandwidths, grid parameters, cluster counts, and function counts.
- Curve, image, and template-based methods are tuned with dataset-specific resolution and structural parameters.Persistence landscapes vary resolution and the number of landscapes; silhouettes vary resolution and weight; persistence images vary resolution and Gaussian bandwidth; template systems vary grid, padding, or cluster parameters.
- Outex combines dimensions 0 and 1, SHREC14 reports dimension 1 only, and FMNIST uses dimension 0 only.SVM and Random Forest classifiers are considered, but only Random Forest is used for FMNIST because of convergence issues.
- Parameters for models and vectorizations are selected by random or grid search with 5-fold cross-validation on the training data.Each method receives the parameter setting with the best average cross-validation score.
- For SHREC14, ensemble methods perform best at small heat-kernel times, while ensemble and functional methods perform well at intermediate and larger values.Algebraic and curve methods perform poorly for low t-values but become more competitive between t5 and t8.