def second_order_grad(self, x):
g2 = np.sum(gaussian_kernel_dx_dx(x, self.X, sigma=self.bandwidth),
axis=0) / np.sum(gaussian_kernel(
x[None, :], self.X, sigma=self.bandwidth),
axis=-1)
g2 -= self.grad(x)**2
return g2
def test_gaussian_kernel_dx_dx_component_equals_gaussian_kernel_dx_dx():
D = 4
x = np.random.randn(D)
y = np.random.randn(D)
sigma = 0.5
dx_dx = gaussian_kernel_dx_dx(x, np.atleast_2d(y), sigma)[0]
for i in range(D):
dxi = gaussian_kernel_dx_dx_component(x, y, i, sigma)
assert_allclose(dx_dx[i], dxi)
def test_gaussian_kernel_dx_dx_equals_SE_dx_dx():
D = 4
x = np.random.randn(D)
Y = np.random.randn(1, D)
sigma = 0.5
implementation = gaussian_kernel_dx_dx(x, Y, sigma)
reference = SE_dx_dx(x.reshape(-1, 1), Y.T, l=np.sqrt(sigma / 2.0))
assert_close(implementation, reference.T)
def log_pdf_naive(x, X, sigma, alpha, beta):
N, D = X.shape
xi = 0
betasum = 0
for a in range(N):
x_a = np.atleast_2d(X[a, :])
gradient_x_xa = np.squeeze(gaussian_kernel_grad(x, x_a, sigma))
xi_grad = np.squeeze(gaussian_kernel_dx_dx(x, x_a, sigma))
for i in range(D):
xi += xi_grad[i] / N
betasum += gradient_x_xa[i] * beta[a, i]
return alpha * xi + betasum
def test_gaussian_kernel_dx_dx_multiple_ys():
D = 4
N = 3
x = np.random.randn(D)
Y = np.random.randn(N, D)
sigma = 0.5
implementation = gaussian_kernel_dx_dx(x, Y, sigma)
for i in range(N):
reference = SE_dx_dx(x.reshape(-1, 1),
Y[i].reshape(-1, 1),
l=np.sqrt(sigma / 2.0))
assert_close(implementation[i], np.squeeze(reference.T))
def log_pdf(x, basis, sigma, alpha, beta):
m, D = basis.shape
assert_array_shape(x, ndim=1, dims={0: D})
SE_dx_dx_l = lambda x, y: gaussian_kernel_dx_dx(x, y.reshape(1, -1), sigma)
SE_dx_l = lambda x, y: gaussian_kernel_grad(x, y.reshape(1, -1), sigma)
xi = 0
betasum = 0
for a in range(m):
x_a = basis[a]
xi += np.sum(SE_dx_dx_l(x, x_a)) / m
gradient_x_xa = np.squeeze(SE_dx_l(x, x_a))
betasum += np.dot(gradient_x_xa, beta[a, :])
return np.float(alpha * xi + betasum)