Role-Aware Neural Convex Divergence Heads for Asymmetric Representation Learning

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.

Several SLOG test categories explicitly involve directional distinctions (modifier position shifts, argument extraction positions), yet AM-Parser, the previous SOTA, uses an AM algebra whose operations do not encode direction. We redesign the symbolic backend around CCG directed types (deterministic CKY + single linear decoder, 30K learnable parameters). Under the same BERT-base encoder, the system achieves 75.9$\pm$6.4% LF exact match, surpassing AM-Parser (70.8$\pm$4.3%). Per SLOG's own category groupings, gains are highly directional: the CCG system outperforms AM-Parser on all 5 position-shift categories (+29.9pp), while AM-Parser outperforms on all 6 recursive-depth categories. Replacing the encoder with DeBERTa-v3-large yields 90.7$\pm$4.9%, with the largest encoder gains in recursive-depth categories, complementary to directionality's gains. Directional representations shift the bottleneck from the symbolic layer (AM-Parser's 0% category ceiling) to the neural layer, which improves with encoder upgrades.

Understanding why independently trained neural networks from different modalities converge toward shared representations, and where this convergence leads, remains an open question in representation learning. All existing evidence relies on symmetric similarity measures, which can detect convergence but are structurally blind to its direction. We introduce directional convergence analysis using cycle-kNN, an asymmetric alignment measure, applied across dozens of independently trained unimodal models spanning point clouds, vision, and language. We uncover a consistent directional asymmetry: non-language modalities move toward the neighborhood structure of language significantly more than the reverse, and this pattern holds across all model families and scales--yet is entirely invisible to symmetric measures. Mechanistic analysis traces the directionality to feature density asymmetry, whereby language representations occupy the most compact regions of representational space. The Information Bottleneck framework provides a principled interpretation: optimization under compression drives representations toward discrete, compositional structures characteristic of language. We formalize this as the Wittgensteinian Representation Hypothesis: the semantic structure of language is the asymptotic attractor of multimodal representation convergence.