ECLayr: Fast and Robust Topological Layer based on Differentiable Euler Characteristic Curve

Hajin Lee, Jisu Kim, Kwangho Kim

Keywords: Topological Data Analysis, Deep Learning, Euler Characteristic Curve

TL;DR: We introduce a novel topological layer for deep learning that is computationally efficient and enables stable backpropagation.

Abstract: In the realm of Topological Data Analysis, persistent homology has traditionally served as a primary tool for extracting topological features. However, approaches relying on persistent homology often encounter practical challenges due to their high computational costs. To address this issue, we propose a computationally efficient novel topological layer tailored for general deep learning architectures, leveraging the Euler Characteristic Curve (ECC). Unlike methods based on persistent homology, ECC offers computational advantages by circumventing the need for persistent homology calculation, while still allowing access to crucial information about the underlying topological structure. The proposed layer can readily adapt to diverse data modalities by allowing appropriate filtration according to the user's preference, enabling its application across various learning problems without data preprocessing. We present a novel technique for stable backpropagation that effectively mitigates the vanishing gradient problems commonly encountered in existing methods, allowing for seamless integration of our layer into deep learning models. We go on to present stability analysis, showing that the proposed layer is robust against noise and outliers. We apply our method to topological autoencoders, showing that the standard loss function can effectively regularize topological structures of the latent space. Through classification experiments across various datasets, we illustrate the benefits of our approach in mitigating information loss under conditions of data scarcity or data contamination.

Primary Area: learning on graphs and other geometries & topologies

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Submission Number: 4291