Cosine-similarity is the cosine of the angle between two vectors, or
equivalently the dot product between their normalizations. A popular
application is to quantify semantic similarity between high-dimensional objects
by applying cosine-similarity to a learned low-dimensional feature embedding.
This can work better but sometimes also worse than the unnormalized dot-product
between embedded vectors in practice. To gain insight into this empirical
observation, we study embeddings derived from regularized linear models, where
closed-form solutions facilitate analytical insights. We derive analytically
how cosine-similarity can yield arbitrary and therefore meaningless
`similarities.' For some linear models the similarities are not even unique,
while for others they are implicitly controlled by the regularization. We
discuss implications beyond linear models: a combination of different
regularizations are employed when learning deep models; these have implicit and
unintended effects when taking cosine-similarities of the resulting embeddings,
rendering results opaque and possibly arbitrary. Based on these insights, we
caution against blindly using cosine-similarity and outline alternatives.

用于量化高维对象之间语义相似度的余弦相似度在实践中比未归一化的嵌入向量点积表现有时更好、有时更差。通过研究基于正则化线性模型的嵌入，我们得出了余弦相似度可以产生任意且无意义的相似度的结论。因此，我们提醒不要盲目使用余弦相似度，并提出替代方法。