提出新相似度度量recos，更好捕捉语义向量间的非线性关系。
Beyond Cosine Similarity
- 基于更紧的点积上界，设计新相似度recos，用排序后分量归一化。
- 在11种嵌入模型上测试，recos与人工判断相关性更高。
- 适合需要精准语义匹配的场景，如文本相似度分析。
余弦相似度作为向量空间中语义相似性的标准度量，其数学基础源于柯西-施瓦茨不等式，本质上仅能捕捉线性关系，难以建模真实语义空间中的复杂非线性结构。本文推导出比经典柯西-施瓦茨界更紧的点积上界，由此直接导出recos：一种将点积按向量分量排序后归一化的相似度度量。recos将完全相似的条件从严格的线性依赖放宽为序数一致性，从而能捕捉更广泛的关联模式。在11种嵌入模型（涵盖静态、上下文和通用类型）上的大量实验表明，recos在标准语义文本相似性（STS）基准上始终优于传统余弦相似度，与人类判断的相关性更高。本工作确立了recos作为一种数学严谨且实证优越的替代方案，显著提升复杂嵌入空间中的语义分析精度。
原文摘要 · Abstract (English)
Cosine similarity, the standard metric for measuring semantic similarity in vector spaces, is mathematically grounded in the Cauchy-Schwarz inequality, which inherently limits it to capturing linear relationships--a constraint that fails to model the complex, nonlinear structures of real-world semantic spaces. We advance this theoretical underpinning by deriving a tighter upper bound for the dot product than the classical Cauchy-Schwarz bound. This new bound leads directly to recos, a similarity metric that normalizes the dot product by the sorted vector components. recos relaxes the condition for perfect similarity from strict linear dependence to ordinal concordance, thereby capturing a broader class of relationships. Extensive experiments across 11 embedding models--spanning static, contextualized, and universal types--demonstrate that recos consistently outperforms traditional cosine similarity, achieving higher correlation with human judgments on standard Semantic Textual Similarity (STS) benchmarks. Our work establishes recos as a mathematically principled and empirically superior alternative, offering enhanced accuracy for semantic analysis in complex embedding spaces.
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