# Distance Matrix Vectorization Trick

A common problem that comes up in machine learning is to find the l2-distance between two sets of vectors. For example, in implementing the K nearest neighbors algorithm, we have to find the l2-distance between the a set of test vectors, held in a matrix X (MxD), and a set of training vectors, held in a matrix X_train (NxD). Our goal is to create a distance matrix D (MxN) that contains the l2-distance from every test vector to every training vector. How can we do this efficiently?

### Single Loop

There is the really stupid way of constructing the distance matrix using using two loops — but let’s not even go there. That is known inefficient. The first thing I tried was to use a single loop over the test vectors. The code is below. The logic is simple — for each test vector I subtract it from the entire training matrix. I use a trick called numpy addition broadcasting. As long as the dimensions match up, numpy knows to do a row-wise subtraction if the element on the right is one-dimensional. After this subtraction, I simply element-wise square and sum along the column dimension to get a single row of the distance matrix for test vector i.

With M=500 (test vectors) and N=5000 (training vectors) running this function takes about 50.8 seconds. Too long!

`def compute_distances_one_loop(self, X):    dists = np.zeros((num_test, num_train))    for i in xrange(num_test):        dists[i, :] = np.sum((self.X_train - X[i, :])**2, axis=1)    return dists    1 loops, best of 3: 50.8 s per loop`

### No Loops

I need to vectorize more. How? After some Googling and old-fashioned pen and paper I figured out better way. The “trick” is to expand the l2 distance formula. For each vector x and y, the l2 distance between them can be expressed as:

To vectorize efficiently, we need to express this operation for ALL the vectors at once in numpy. The first two terms are easy — just take the l2 norm of every row in the matrices X and X_train. The last term can be expressed as a matrix multiply between X and transpose(X_train). Numpy can do all of these things super efficiently.

The final code is below. It took a second to get it right. The trickiest part was figuring out how to transpose the last row vector that comes out of np.sum() back into a column vector so the sum broadcast operation works easily, i.e. using [:, np.newaxis] (thank you StackOverflow). Without that trick, I was transposing the larger matrix and transposing back at the end.

`def compute_distances_no_loops(self, X):    dists = -2 * np.dot(X, self.X_train.T) + np.sum(self.X_train**2,    axis=1) + np.sum(X**2, axis=1)[:, np.newaxis]    return dists`
`1 loops, best of 3: 190 ms per loop`

So, what does it buy me? It takes only 190 ms for this code to run on the same example above. That’s a speedup of OVER 267 times! A dirty vectorization trick — but well worth the effort. Now I don’t have to sit around while my distance matrix is computed.

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