Jaccard Similarity is a lower-bound for Cosine Similarity

up:: Jaccard Similarity, Cosine Similarity

Given an undirected Simple Graph with Adjacency Matrix and nodes , then their Jaccard similarity is

Their cosine similarity is

Multiplying and dividing by ¹, we have

since

with .

We need to consider two cases:

1. , in which case we simply have

2. without loss of generality, for which we have

Since

we have that as defined above.

Thus, we conclude that

That is, the Jaccard similarity of two nodes is more stringent than their cosine similarity.

Footnotes

1. Assuming both have degree greater than . ↩