Is "norm" equivalent to "Euclidean distance"?

A norm is a function that takes a vector as an input and returns a scalar value that can be interpreted as the "size", "length" or "magnitude" of that vector. More formally, norms are defined as having the following mathematical properties:

• They scale multiplicatively, i.e. Norm(a·v) = |a|·Norm(v) for any scalar a
• They satisfy the triangle inequality, i.e. Norm(u + v) ≤ Norm(u) + Norm(v)
• The norm of a vector is zero if and only if it is the zero vector, i.e. Norm(v) = 0 ⇔ v = 0

The Euclidean norm (also known as the L² norm) is just one of many different norms - there is also the max norm, the Manhattan norm etc. The L² norm of a single vector is equivalent to the Euclidean distance from that point to the origin, and the L² norm of the difference between two vectors is equivalent to the Euclidean distance between the two points.

As @nobar's answer says, np.linalg.norm(x - y, ord=2) (or just np.linalg.norm(x - y)) will give you Euclidean distance between the vectors x and y.

Since you want to compute the Euclidean distance between a[1, :] and every other row in a, you could do this a lot faster by eliminating the for loop and broadcasting over the rows of a:

dist = np.linalg.norm(a[1:2] - a, axis=1)

It's also easy to compute the Euclidean distance yourself using broadcasting:

dist = np.sqrt(((a[1:2] - a) ** 2).sum(1))

The fastest method is probably scipy.spatial.distance.cdist:

from scipy.spatial.distance import cdistdist = cdist(a[1:2], a)[0]

Some timings for a (1000, 1000) array:

a = np.random.randn(1000, 1000)%timeit np.linalg.norm(a[1:2] - a, axis=1)# 100 loops, best of 3: 5.43 ms per loop%timeit np.sqrt(((a[1:2] - a) ** 2).sum(1))# 100 loops, best of 3: 5.5 ms per loop%timeit cdist(a[1:2], a)[0]# 1000 loops, best of 3: 1.38 ms per loop# check that all 3 methods return the same resultd1 = np.linalg.norm(a[1:2] - a, axis=1)d2 = np.sqrt(((a[1:2] - a) ** 2).sum(1))d3 = cdist(a[1:2], a)[0]assert np.allclose(d1, d2) and np.allclose(d1, d3)

The concept of a "norm" is a generalized idea in mathematics which, when applied to vectors (or vector differences), broadly represents some measure of length. There are various different approaches to computing a norm, but the one called Euclidean distance is called the "2-norm" and is based on applying an exponent of 2 (the "square"), and after summing applying an exponent of 1/2 (the "square root").

It's a bit cryptic in the docs, but you get Euclidean distance between two vectors by setting the parameter ord=2.

sum(abs(x)**ord)**(1./ord)

becomes sqrt(sum(x**2)).

Note: as pointed out by @Holt, the default value is ord=None, which is documented to compute the "2-norm" for vectors. This is, therefore, equivalent to ord=2 (Euclidean distance).