Symmetric Divergence and Normalized Similarity: A Unified Topological Framework for Representation Analysis

2026-06-04

AI Summary

This work addresses the limitations of existing topological dissimilarity measures, which often suffer from asymmetry and unboundedness due to heuristic design, thereby hindering reliable cross-scenario comparisons. To overcome these issues, the authors propose Symmetric Representational Topological Dissimilarity (SRTD), which integrates cross-barcode signatures with rank correlations derived from hierarchical merge orders to enable structural diagnosis. Building upon this, they introduce Normalized Topological Similarity (NTS), a bounded and scale-invariant similarity metric grounded in topological data analysis (TDA). Empirical evaluations on CNNs and large language models demonstrate that the proposed approach effectively captures functional evolution missed by geometric methods and robustly reconstructs model lineages even under distance saturation, significantly enhancing comparability and evaluation consistency across heterogeneous settings.

Abstract

Topological Data Analysis (TDA) offers a principled, intrinsic lens for comparing neural representations. However, existing paired topological divergences (e.g., RTD) are limited by heuristic asymmetry and, more critically, unbounded scores that depend on sample size, hindering reliable cross-scenario benchmarking. To address these challenges, we develop a unified topological toolkit serving two complementary needs: fine-grained structural diagnosis and robust, standardized evaluation. First, we complete the RTD framework by introducing Symmetric Representation Topology Divergence (SRTD) and its efficient variant SRTD-lite. Beyond resolving the theoretical asymmetry of prior variants, SRTD consolidates diagnostic information into a single, comprehensive cross-barcode signature. This allows for precise localization of structural discrepancies and serves as an effective optimization objective without the overhead of dual directional computations. Second, to enable reliable benchmarking across heterogeneous settings, we propose Normalized Topological Similarity (NTS). By measuring the rank correlation of hierarchical merge orders, NTS yields a scale-invariant metric bounded between -1 and 1, effectively overcoming the scale and sample-dependence of unnormalized divergences. Experiments across synthetic and real-world deep learning settings demonstrate that our toolkit captures functional shifts in CNNs missed by geometric measures and robustly maps LLM genealogy even under distance saturation, offering a rigorous, topology-aware perspective that complements measures like CKA.

Problem

• Topological Data Analysis
• Representation Comparison
• Asymmetry
• Sample Dependence
• Benchmarking

Innovation

• Symmetric Representation Topology Divergence
• Normalized Topological Similarity
• Topological Data Analysis
• Cross-barcode Signature
• Scale-invariant Metric