Optimization Dynamics Imprint Semantic Specificity in Contrastive Embedding Norms

[Submitted on 29 Jun 2026]

View PDF HTML (experimental)

Abstract:Contrastive embedding models trained with scale-invariant losses are typically paired with distance metrics like cosine similarity, effectively ignoring embedding magnitudes. However, surprisingly, empirical studies reveal that despite this, these “discarded” norms seem to correlate with semantic properties such as concept specificity, token frequency, and human uncertainty. In this work, we provide a formal theoretical framework explaining this phenomenon. By analyzing the optimization dynamics, we derive an analytic formula demonstrating that embedding length naturally encodes this information as a byproduct of the training process. We also show how this gives rise to signals that can serve as “free” calibration tools in specific models and retrieval tasks, providing a grounded explanation for a previously heuristic observation.

Submission history

From: Ziwei Su [view email]
[v1] Mon, 29 Jun 2026 17:55:40 UTC (220 KB)