def test_analytic_gradient_matches_numerical_gradient_with_zero_delta(
self):
"""Test analytical gradient matches finite difference approximation."""
random_seed = 0
random_state = check_random_state(random_seed)
n_features = 3
n_components = 5
n_samples = 10
X = random_state.uniform(size=(n_samples, n_features))
K = X.dot(X.T)
C = right_stochastic_matrix((n_components, n_samples),
random_state=random_state)
S = right_stochastic_matrix((n_samples, n_components),
random_state=random_state)
self.assertTrue(np.allclose(C.sum(axis=1), 1, 1e-12))
self.assertTrue(np.allclose(S.sum(axis=1), 1, 1e-12))
delta = 0
aa = KernelAA(n_components=n_components, delta=delta)
aa.K = K
aa.C = C
aa.S = S
aa._initialize_workspace()
aa._update_weights_gradient()
analytic_grad_S = aa.grad_S
def central_difference_deriv(i, j, h=1e-4):
old_x = S[i, j]
xmh = old_x - h
aa.S[i, j] = xmh
aa._initialize_workspace()
fmh = aa._evaluate_cost()
xph = old_x + h
aa.S[i, j] = xph
aa._initialize_workspace()
fph = aa._evaluate_cost()
aa.S[i, j] = old_x
return (fph - fmh) / (2 * h)
numeric_grad_S = np.zeros((n_samples, n_components))
for i in range(n_samples):
for j in range(n_components):
numeric_grad_S[i, j] = central_difference_deriv(i, j)
self.assertTrue(np.allclose(analytic_grad_S, numeric_grad_S, 1e-4))
def test_repeated_updates_converge_with_nonzero_delta(self):
"""Test repeated updates converge to a fixed point with non-zero delta."""
random_seed = 0
random_state = check_random_state(random_seed)
n_features = 30
n_components = 11
n_samples = 320
max_iter = 1000
tolerance = 1e-4
X = random_state.uniform(size=(n_samples, n_features))
K = X.dot(X.T)
C = right_stochastic_matrix((n_components, n_samples),
random_state=random_state)
S = right_stochastic_matrix((n_samples, n_components),
random_state=random_state)
self.assertTrue(np.allclose(C.sum(axis=1), 1, tolerance))
self.assertTrue(np.allclose(S.sum(axis=1), 1, tolerance))
delta = 3.2
aa = KernelAA(n_components=n_components, delta=delta)
aa.K = K
aa.C = C
aa.S = S
aa._initialize_workspace()
cost_delta = 1 + tolerance
old_cost = aa._evaluate_cost()
new_cost = old_cost
n_iter = 0
while abs(cost_delta) > tolerance and n_iter < max_iter:
old_cost = new_cost
error = aa._update_dictionary()
self.assertEqual(error, 0)
new_cost = aa._evaluate_cost()
cost_delta = new_cost - old_cost
self.assertTrue(cost_delta <= 0)
n_iter += 1
self.assertTrue(n_iter < max_iter)
updated_C = aa.C
self.assertTrue(np.allclose(updated_C.sum(axis=1), 1, 1e-12))
updated_alpha = aa.alpha
for i in range(n_components):
self.assertTrue(1 - delta <= updated_alpha[i] <= 1 + delta)
def test_repeated_updates_converge_with_zero_delta(self):
"""Test repeated updates converge to a fixed point with delta = 0."""
random_seed = 0
random_state = check_random_state(random_seed)
n_features = 10
n_components = 3
n_samples = 600
max_iter = 100
tolerance = 1e-6
X = random_state.uniform(size=(n_samples, n_features))
K = X.dot(X.T)
C = right_stochastic_matrix((n_components, n_samples),
random_state=random_state)
S = right_stochastic_matrix((n_samples, n_components),
random_state=random_state)
self.assertTrue(np.allclose(C.sum(axis=1), 1, tolerance))
self.assertTrue(np.allclose(S.sum(axis=1), 1, tolerance))
delta = 0
aa = KernelAA(n_components=n_components, delta=delta)
aa.K = K
aa.C = C
aa.S = S
aa._initialize_workspace()
cost_delta = 1 + tolerance
old_cost = aa._evaluate_cost()
new_cost = old_cost
n_iter = 0
while abs(cost_delta) > tolerance and n_iter < max_iter:
old_cost = new_cost
error = aa._update_weights()
self.assertEqual(error, 0)
new_cost = aa._evaluate_cost()
cost_delta = new_cost - old_cost
self.assertTrue(cost_delta <= 0)
n_iter += 1
self.assertTrue(n_iter < max_iter)
updated_S = aa.S
self.assertTrue(np.allclose(updated_S.sum(axis=1), 1, 1e-12))
def test_single_dictionary_update_reduces_cost_with_nonzero_delta(self):
"""Test single update step reduces cost function."""
random_seed = 0
random_state = check_random_state(random_seed)
n_features = 10
n_components = 5
n_samples = 400
X = random_state.uniform(size=(n_samples, n_features))
K = X.dot(X.T)
C = right_stochastic_matrix((n_components, n_samples),
random_state=random_state)
S = right_stochastic_matrix((n_samples, n_components),
random_state=random_state)
self.assertTrue(np.allclose(C.sum(axis=1), 1, 1e-12))
self.assertTrue(np.allclose(S.sum(axis=1), 1, 1e-12))
delta = 1.2
aa = KernelAA(n_components=n_components, delta=delta)
aa.K = K
aa.C = C
aa.S = S
aa._initialize_workspace()
initial_cost = aa._evaluate_cost()
error = aa._update_dictionary()
self.assertEqual(error, 0)
final_cost = aa._evaluate_cost()
self.assertTrue(final_cost <= initial_cost)
updated_C = aa.C
self.assertTrue(np.allclose(updated_C.sum(axis=1), 1, 1e-12))
def test_exact_solution_with_zero_delta_is_fixed_point(self):
"""Test exact solution for weights is fixed point of update step."""
random_seed = 0
random_state = check_random_state(random_seed)
n_features = 30
n_components = 10
n_samples = 130
tolerance = 1e-12
basis = random_state.uniform(size=(n_components, n_features))
S = right_stochastic_matrix((n_samples, n_components),
random_state=random_state)
archetype_indices = np.zeros(n_components, dtype='i8')
for i in range(n_components):
new_index = False
current_index = 0
while not new_index:
new_index = True
current_index = random_state.randint(low=0, high=n_samples)
for index in archetype_indices:
if current_index == index:
new_index = False
archetype_indices[i] = current_index
C = np.zeros((n_components, n_samples))
component = 0
for index in archetype_indices:
C[component, index] = 1.0
for i in range(n_components):
if i == component:
S[index, i] = 1.0
else:
S[index, i] = 0.0
component += 1
X = S.dot(basis)
basis_projection = C.dot(X)
self.assertTrue(np.allclose(basis_projection, basis, tolerance))
self.assertTrue(np.linalg.norm(X - S.dot(C.dot(X))) < tolerance)
K = X.dot(X.T)
delta = 0
aa = KernelAA(n_components=n_components, delta=delta)
aa.K = K
aa.C = C
aa.S = S
aa._initialize_workspace()
initial_cost = aa._evaluate_cost()
error = aa._update_weights()
self.assertEqual(error, 0)
final_cost = aa._evaluate_cost()
self.assertTrue(abs(final_cost - initial_cost) < tolerance)
updated_S = aa.S
self.assertTrue(np.allclose(updated_S.sum(axis=1), 1, 1e-12))
self.assertTrue(np.allclose(updated_S, S, tolerance))