Challenges in 3D Data Synthesis for Training Neural Networks on Topological Features
This addresses a data gap for researchers in computational topology, though it is incremental as it focuses on dataset generation rather than a new paradigm.
The paper tackles the lack of labeled 3D data for training neural networks in Topological Data Analysis by introducing a novel method to generate synthetic datasets with controlled topological features, resulting in a dataset used to train a genus estimator network that shows decreased accuracy with increased deformations.
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.