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?

Solution

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);