How to compute cosine similarity using two matrices

Question

I have two matrices, A (dimensions M x N) and B (N x P). In fact, they are collections of vectors - row vectors in A, column vectors in B. I want to get cosine similarity scores for every pair a and b, where a is a vector (row) from matrix A and b is a vector (column) from matrix B.

I have started by multiplying the matrices, which results in matrix C (dimensions M x P).

C = A*B

However, to obtain cosine similarity scores, I need to divide each value C(i,j) by the norm of the two corresponding vectors. Could you suggest the easiest way to do this in Matlab?

Answer 1

The simplest solution would be computing the norms first using element-wise multiplication and summation along the desired dimensions:

normA = sqrt(sum(A .^ 2, 2));
normB = sqrt(sum(B .^ 2, 1));

normA and normB are now a column vector and row vector, respectively. To divide corresponding elements in A * B by normA and normB, use bsxfun like so:

C = bsxfun(@rdivide, bsxfun(@rdivide, A * B, normA), normB);