euclidean_distances

sklearn.metrics.pairwise.euclidean_distances(X, Y=None, *, Y_norm_squared=None, squared=False, X_norm_squared=None)

Compute the distance matrix between each pair from a vector array X and Y.

For efficiency reasons, the euclidean distance between a pair of row vector x and y is computed as:

dist(x, y) = sqrt(dot(x, x) - 2 * dot(x, y) + dot(y, y))

This formulation has two advantages over other ways of computing distances. First, it is computationally efficient when dealing with sparse data. Second, if one argument varies but the other remains unchanged, then dot(x, x) and/or dot(y, y) can be pre-computed.

However, this is not the most precise way of doing this computation, because this equation potentially suffers from “catastrophic cancellation”. Also, the distance matrix returned by this function may not be exactly symmetric as required by, e.g., scipy.spatial.distance functions.

Read more in the User guide.

Parameters:

X{array-like, sparse matrix} of shape (n_samples_X, n_features)

An array where each row is a sample and each column is a feature.

Y{array-like, sparse matrix} of shape (n_samples_Y, n_features), default=None

An array where each row is a sample and each column is a feature. If None, method uses Y=X.

Y_norm_squaredarray-like of shape (n_samples_Y,) or (n_samples_Y, 1) or (1, n_samples_Y), default=None

Pre-computed dot-products of vectors in Y (e.g., (Y**2).sum(axis=1)) May be ignored in some cases, see the note below.

squaredbool, default=False

Return squared Euclidean distances.

X_norm_squaredarray-like of shape (n_samples_X,) or (n_samples_X, 1) or (1, n_samples_X), default=None

Pre-computed dot-products of vectors in X (e.g., (X**2).sum(axis=1)) May be ignored in some cases, see the note below.

Returns:

distancesndarray of shape (n_samples_X, n_samples_Y)

Returns the distances between the row vectors of X and the row vectors of Y.

See also

paired_distances

Distances between pairs of elements of X and Y.

Notes

To achieve a better accuracy, X_norm_squared and Y_norm_squared may be unused if they are passed as np.float32.

Examples

>>> from sklearn.metrics.pairwise import euclidean_distances
>>> X = [[0, 1], [1, 1]]
>>> # distance between rows of X
>>> euclidean_distances(X, X)
array([[0., 1.],
       [1., 0.]])
>>> # get distance to origin
>>> euclidean_distances(X, [[0, 0]])
array([[1.        ],
       [1.41421356]])