Directional Sign Loss: A Topology-Preserving Loss Function that Approximates the Sign of Finite Differences

Abstract

Preserving topological features in learned latent spaces is a fundamental challenge in representation learning, particularly for topology-sensitive data. This paper introduces directional sign loss (DSL), an efficient, differentiable loss function that approximates the number of mismatches in the signs of finite differences between corresponding elements of two arrays. By penalizing discrepancies in critical points between input and reconstructed data, DSL encourages autoencoders and other learnable compressors to retain the topological features of the original data. We present the formulation and complexity analysis of DSL, comparing it to other non-differentiable topological measures. Experiments on multidimensional array data show that combining DSL with traditional loss functions preserves topological features more effectively than traditional losses alone. DSL serves as a differentiable, efficient proxy for common topology-based metrics, enabling topological feature preservation on previously impractical problem sizes and in a wider range of gradient-based optimization frameworks.