def test_gaussian_kernel_dx_i_dx_j_component_equals_gaussian_kernel_dx_i_dx_j(
):
D = 4
x = np.random.randn(D)
y = np.random.randn(D)
sigma = 0.5
dx_i_dx_j = gaussian_kernel_dx_i_dx_j(x, y, sigma)
for i in range(D):
a = gaussian_kernel_dx_i_dx_j_component(x, y, i, sigma)
assert_allclose(dx_i_dx_j[i], a)
def grad(x, basis, sigma, alpha, beta):
m, D = basis.shape
assert_array_shape(x, ndim=1, dims={0: D})
xi_grad = 0
betasum_grad = 0
for a, x_a in enumerate(basis):
xi_grad += np.sum(gaussian_kernel_dx_i_dx_i_dx_j(x, x_a, sigma), axis=0) / m
left_arg_hessian = gaussian_kernel_dx_i_dx_j(x, x_a, sigma)
betasum_grad += beta[a, :].dot(left_arg_hessian)
return alpha * xi_grad + betasum_grad
def test_gaussian_kernel_dx_i_dx_j_equals_SE_dx_i_dx_j():
D = 4
x = np.random.randn(D)
y = np.random.randn(D)
sigma = 0.5
implementation = gaussian_kernel_dx_i_dx_j(x, y, sigma)
reference = SE_dx_i_dx_j(x.reshape(-1, 1),
y.reshape(-1, 1),
l=np.sqrt(sigma / 2.0))
assert_close(implementation, reference)
def grad_naive(x, X, sigma, alpha, beta):
N, D = X.shape
xi_grad = 0
betasum_grad = 0
for a, x_a in enumerate(X):
xi_gradient_vec = gaussian_kernel_dx_i_dx_i_dx_j(x, x_a, sigma)
left_arg_hessian = gaussian_kernel_dx_i_dx_j(x, x_a, sigma)
for i in range(D):
xi_grad += xi_gradient_vec[i] / N
betasum_grad += beta[a, i] * left_arg_hessian[i]
return alpha * xi_grad + betasum_grad