Magnitude Matters: a Superior Class of Similarity Metrics for Holistic Semantic Understanding

Vector comparison in high dimensions is a fundamental task in NLP, yet it is dominated by two baselines: the raw dot product, which is unbounded and sensitive to vector norms, and the cosine similarity, which discards magnitude information entirely. This paper challenges both standards by proposing and rigorously evaluating a new class of parameter-free, magnitude-aware similarity metrics. I introduce two such functions, Overlap Similarity (OS) and Hyperbolic Tangent Similarity (HTS), designed to integrate vector magnitude and alignment in a more principled manner. To ensure that my findings are robust and generalizable, I conducted a comprehensive evaluation using four state-of-the-art sentence embedding models (all-MiniLM-L6-v2, all-mpnet-base-v2, paraphrase-mpnet-base-v2, and BAAI/bge-large-en-v1.5) across a diverse suite of eight standard NLP benchmarks, including STS-B, SICK, Quora, and PAWS. Using the Wilcoxon signed-rank test for statistical significance, my results are definitive: on the tasks requiring holistic semantic understanding (paraphrase and inference), both OS and HTS provide a statistically significant improvement in Mean Squared Error over both the raw dot product and cosine similarity, regardless of the underlying embedding model.Crucially, my findings delineate the specific domain of advantage for these metrics: for tasks requiring holistic semantic understanding like paraphrase and inference, my magnitude-aware metrics offer a statistically superior alternative. This significant improvement was not observed on benchmarks designed to test highly nuanced compositional semantics (SICK, STS-B), identifying the challenge of representing compositional text as a distinct and important direction for future work.