User defined SVM kernel with scikit-learn - python-2.7

Question

I encounter a problem when defining a kernel by myself in scikit-learn.
I define by myself the gaussian kernel and was able to fit the SVM but not to use it to make a prediction.
More precisely I have the following code
from sklearn.datasets import load_digits
from sklearn.svm import SVC
from sklearn.utils import shuffle
import scipy.sparse as sparse
import numpy as np
digits = load_digits(2)
X, y = shuffle(digits.data, digits.target)
gamma = 1.0
X_train, X_test = X[:100, :], X[100:, :]
y_train, y_test = y[:100], y[100:]
m1 = SVC(kernel='rbf',gamma=1)
m1.fit(X_train, y_train)
m1.predict(X_test)
def my_kernel(x,y):
d = x - y
c = np.dot(d,d.T)
return np.exp(-gamma*c)
m2 = SVC(kernel=my_kernel)
m2.fit(X_train, y_train)
m2.predict(X_test)
m1 and m2 should be the same, but m2.predict(X_test) return the error :
operands could not be broadcast together with shapes (260,64) (100,64)
I don't understand the problem.
Furthermore if x is one data point, the m1.predict(x) gives a +1/-1 result, as expexcted, but m2.predict(x) gives an array of +1/-1...
No idea why.

Answer 1

The error is at the x - y line. You cannot subtract the two like that, because the first dimensions of both may not be equal. Here is how the rbf kernel is implemented in scikit-learn, taken from here (only keeping the essentials):
def row_norms(X, squared=False):
if issparse(X):
norms = csr_row_norms(X)
else:
norms = np.einsum('ij,ij->i', X, X)
if not squared:
np.sqrt(norms, norms)
return norms
def euclidean_distances(X, Y=None, Y_norm_squared=None, squared=False):
"""
Considering the rows of X (and Y=X) as vectors, compute the
distance matrix between each pair of vectors.
[...]
Returns
-------
distances : {array, sparse matrix}, shape (n_samples_1, n_samples_2)
"""
X, Y = check_pairwise_arrays(X, Y)
if Y_norm_squared is not None:
YY = check_array(Y_norm_squared)
if YY.shape != (1, Y.shape[0]):
raise ValueError(
"Incompatible dimensions for Y and Y_norm_squared")
else:
YY = row_norms(Y, squared=True)[np.newaxis, :]
if X is Y: # shortcut in the common case euclidean_distances(X, X)
XX = YY.T
else:
XX = row_norms(X, squared=True)[:, np.newaxis]
distances = safe_sparse_dot(X, Y.T, dense_output=True)
distances *= -2
distances += XX
distances += YY
np.maximum(distances, 0, out=distances)
if X is Y:
# Ensure that distances between vectors and themselves are set to 0.0.
# This may not be the case due to floating point rounding errors.
distances.flat[::distances.shape[0] + 1] = 0.0
return distances if squared else np.sqrt(distances, out=distances)
def rbf_kernel(X, Y=None, gamma=None):
X, Y = check_pairwise_arrays(X, Y)
if gamma is None:
gamma = 1.0 / X.shape[1]
K = euclidean_distances(X, Y, squared=True)
K *= -gamma
np.exp(K, K) # exponentiate K in-place
return K
You might want to dig deeper into the code, but look at the comments for the euclidean_distances function. A naive implementation of what you're trying to achieve would be this:
def my_kernel(x,y):
d = np.zeros((x.shape[0], y.shape[0]))
for i, row_x in enumerate(x):
for j, row_y in enumerate(y):
d[i, j] = np.exp(-gamma * np.linalg.norm(row_x - row_y))
return d