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

Preserving critical 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), a novel 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 mathematical formulation, complexity analysis, and practical implementation of DSL, comparing its behavior to its non-differentiable counterpart and to other topological measures. Experiments on one-, two-, and three-dimensional data show that combining DSL with traditional loss functions preserves topological features more effectively than traditional losses alone. Moreover, DSL serves as a differentiable, efficient proxy for common topology-based metrics, enabling its use in gradient-based optimization frameworks.