def eval(self, X, Y):
"""Evalute the kernel on data X and Y """
b = self.b
c = self.c
D2 = util.dist2_matrix(X, Y)
K = (c**2 + D2)**b
return K
def gradXY_sum(self, X, Y):
"""
Compute
\sum_{i=1}^d \frac{\partial^2 k(X, Y)}{\partial x_i \partial y_i}
evaluated at each x_i in X, and y_i in Y.
X: nx x d numpy array.
Y: ny x d numpy array.
Return a nx x ny numpy array of the derivatives.
"""
b = self.b
c = self.c
D2 = util.dist2_matrix(X, Y)
# d = input dimension
d = X.shape[1]
c2D2 = c**2 + D2
T1 = -4.0 * b * (b - 1) * D2 * (c2D2**(b - 2))
T2 = -2.0 * b * d * c2D2**(b - 1)
return T1 + T2
def gradX_Y(self, X, Y, dim):
"""
Compute the gradient with respect to the dimension dim of X in k(X, Y).
X: nx x d
Y: ny x d
Return a numpy array of size nx x ny.
"""
D2 = util.dist2_matrix(X, Y)
# 1d array of length nx
Xi = X[:, dim]
# 1d array of length ny
Yi = Y[:, dim]
# nx x ny
dim_diff = Xi[:, np.newaxis] - Yi[np.newaxis, :]
b = self.b
c = self.c
Gdim = (2.0 * b * (c**2 + D2)**(b - 1)) * dim_diff
assert Gdim.shape[0] == X.shape[0]
assert Gdim.shape[1] == Y.shape[0]
return Gdim