calculate distance of 2 list of points in numpy

Question

I have 2 lists of points as numpy.ndarray, each row is the coordinate of a point, like:

a = np.array([[1,0,0],[0,1,0],[0,0,1]])
b = np.array([[1,1,0],[0,1,1]         


        

Answer 1

To compute the squared euclidean distance for each pair of elements off them - x and y, we need to find :

(Xik-Yjk)**2 = Xik**2 + Yjk**2 - 2*Xik*Yjk

and then sum along k to get the distance at coressponding point as dist(Xi,Yj).

Using associativity, it reduces to :

dist(Xi,Yj) = sum_k(Xik**2) + sum_k(Yjk**2) - 2*sum_k(Xik*Yjk)

Bringing in matrix-multiplication for the last part, we would have all the distances, like so -

dist = sum_rows(X^2), sum_rows(Y^2), -2*matrix_multiplication(X, Y.T)

Hence, putting into NumPy terms, we would end up with the euclidean distances for our case with a and b as the inputs, like so -

np.sqrt((a**2).sum(1)[:,None] + (b**2).sum(1) - 2*a.dot(b.T))

Leveraging np.einsum, we could replace the first two summation-reductions with -

np.einsum('ij,ij->i',a,a)[:,None] + np.einsum('ij,ij->i',b,b) 

More info could be found on eucl_dist package's wiki page (disclaimer: I am its author).