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

2025-04-05

AI Summary

Preserving critical topological structures—such as extrema and monotonicity transitions—in the latent space remains challenging in representation learning. To address this, we propose Directional Sign Loss (DSL), a differentiable and computationally efficient topology-preserving loss function that, for the first time, models sign-level finite-difference consistency as a gradient-compatible supervisory signal. DSL leverages tanh-smoothed sign approximation, per-dimension directional weighting, and finite-difference computation, supporting arbitrary-dimensional arrays and seamless integration into architectures such as autoencoders. Experiments on 1D–3D topologically sensitive data demonstrate that DSL, when combined with standard reconstruction losses, significantly improves topological fidelity of reconstructions: Betti number errors decrease by 37–62% compared to baselines, while computational overhead remains substantially lower than persistent homology-based methods.

Technology Category

• Machine Learning: Representation Learning
• Computer Vision: Representation Learning for Vision
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

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.

Problem

• Preserves topological features in learned latent spaces
• Approximates sign mismatches in finite differences between arrays
• Enhances autoencoders by retaining data topology

Innovation

• Directional sign loss preserves topology
• Penalizes critical point discrepancies
• Differentiable proxy for topology metrics