def test_feature_map_derivative_d_2n():
X = np.array([[1.], [3.]])
u = np.array([2.])
omega = np.array([[2.]])
d = 0
phi_derivative = feature_map_derivative_d(X, omega, u, d)
phi_derivative_manual = -np.sin(X * omega + u) * omega[:, d] * np.sqrt(2.)
assert_close(phi_derivative, phi_derivative_manual)
def test_compute_C_1d2n():
X = np.array([[1.], [2.]])
u = np.array([2.])
omega = np.array([[2.]])
d = 0
C_manual = np.mean(feature_map_derivative_d(X, omega, u, d)**2)
C = compute_C_memory(X, omega, u)
assert_allclose(C_manual, C)
def test_compute_C_1d1n():
X = np.array([[1.]])
u = np.array([2.])
omega = np.array([[2.]])
d = 0
phi = feature_map_derivative_d(X, omega, u, d).flatten()
C_manual = np.outer(phi, phi)
C = compute_C_memory(X, omega, u)
assert_allclose(C_manual, C)
def test_feature_map_derivatives_loop_equals_map_derivative_d():
N = 10
D = 20
m = 3
X = np.random.randn(N, D)
omega = np.random.randn(D, m)
u = np.random.uniform(0, 2 * np.pi, m)
derivatives = feature_map_derivatives_loop(X, omega, u)
for d in range(D):
derivative = feature_map_derivative_d(X, omega, u, d)
assert_allclose(derivatives[d], derivative)
def test_objective_sym_equals_completely_manual_manually():
N = 100
D = 3
m = 3
omega = np.random.randn(D, m)
u = np.random.uniform(0, 2 * np.pi, m)
X = np.random.randn(N, D)
lmbda = 1.
theta = np.random.randn(m)
J_manual = 0.
for n in range(N):
b_manual = np.zeros(m)
C_manual = np.zeros((m, m))
J_n_manual = 0.
for d in range(D):
b_term_manual = -np.sqrt(2. / m) * np.cos(np.dot(X[n], omega) +
u) * (omega[d, :]**2)
b_term = feature_map_derivative2_d(X[n], omega, u, d)
assert_allclose(b_term_manual, b_term)
b_manual -= b_term_manual
J_manual += np.dot(b_term_manual, theta)
J_n_manual += np.dot(b_term_manual, theta)
c_vec_manual = -np.sqrt(2. / m) * np.sin(np.dot(X[n], omega) +
u) * omega[d, :]
c_vec = feature_map_derivative_d(X[n], omega, u, d)
assert_allclose(c_vec_manual, c_vec)
C_term = np.outer(c_vec_manual, c_vec_manual)
C_manual += C_term
# not regularised here, done afterwards
J_manual += 0.5 * np.dot(theta, np.dot(C_term, theta))
J_n_manual += 0.5 * np.dot(theta, np.dot(C_term, theta))
b = compute_b_memory(X[n].reshape(1, m), omega, u)
C = compute_C_memory(X[n].reshape(1, m), omega, u)
assert_allclose(b_manual, b)
assert_allclose(C_manual, C)
# discard regularisation for these internal checks
J_n = objective(X[n].reshape(1, m), theta, 0, omega, u)
J_n_2 = 0.5 * np.dot(theta, np.dot(C, theta)) - np.dot(theta, b)
assert_allclose(J_n_2, J_n, rtol=1e-4)
assert_allclose(J_n_manual, J_n, rtol=1e-4)
J_manual /= N
J_manual += 0.5 * lmbda * np.dot(theta, theta)
J = objective(X, theta, lmbda, omega, u)
assert_close(J, J_manual, decimal=5)
def test_feature_map_grad_single_equals_feature_map_derivative_d():
D = 2
m = 3
omega = np.random.randn(D, m)
u = np.random.uniform(0, 2 * np.pi, m)
x = np.random.randn(D)
grad = feature_map_grad_single(x, omega, u)
grad_manual = np.zeros((D, m))
for d in range(D):
grad_manual[d, :] = feature_map_derivative_d(x, omega, u, d)
assert_allclose(grad_manual, grad)
# sample random feature
np.random.seed(seed_offset + i)
omega, u = sample_basis(D=D, m=1, gamma=gamma)
phi_x = feature_map(x, omega, u)
# sample data point and predict
f = predict(x, alphas[:i]) * phi_x
# gradient of f at x
f_grad = feature_map_grad_single(x, omega, u) * f
# gradient
grad = 0
for d in range(D):
phi_derivative_d = feature_map_derivative_d(x, omega, u, d)
phi_derivative2_d = feature_map_derivative2_d(x, omega, u, d)
grad += phi_derivative_d * f_grad[d] + phi_derivative2_d
# take gradient step
r = learning_rate(i + 1)
alphas[i] = -r * grad * phi_x
# down-weight past
alphas[:i] *= (1 - r * lmbda)
# visualise log pdf
log_pdf = lambda x: predict(x, alphas)
res = 20
Xs = np.linspace(-3, 3, res)