Challenges in 3D Data Synthesis for Training Neural Networks on Topological Features

TL;DR

This paper introduces a new method for generating labeled 3D datasets with controllable topological features to improve training and benchmarking of neural network estimators in Topological Data Analysis, addressing a key data scarcity challenge.

Contribution

It presents a systematic approach for creating diverse labeled 3D datasets with controllable topological invariants, facilitating supervised learning and evaluation in TDA tasks.

Findings

Decreased estimator accuracy with increased deformations

Geometric complexity impacts topological estimator performance

Synthetic dataset enables benchmarking of neural TDA methods

Abstract

Topological Data Analysis (TDA) involves techniques of analyzing the underlying structure and connectivity of data. However, traditional methods like persistent homology can be computationally demanding, motivating the development of neural network-based estimators capable of reducing computational overhead and inference time. A key barrier to advancing these methods is the lack of labeled 3D data with class distributions and diversity tailored specifically for supervised learning in TDA tasks. To address this, we introduce a novel approach for systematically generating labeled 3D datasets using the Repulsive Surface algorithm, allowing control over topological invariants, such as hole count. The resulting dataset offers varied geometry with topological labeling, making it suitable for training and benchmarking neural network estimators. This paper uses a synthetic 3D dataset to train a genus estimator network, created using a 3D convolutional transformer architecture. An observed decrease in accuracy as deformations increase highlights the role of not just topological complexity, but also geometric complexity, when training generalized estimators. This dataset fills a gap in labeled 3D datasets and generation for training and evaluating models and techniques for TDA.