def test_randomized_svd_sign_flip_with_transpose():
# Check if the randomized_svd sign flipping is always done based on u
# irrespective of transpose.
# See https://github.com/scikit-learn/scikit-learn/issues/5608
# for more details.
def max_loading_is_positive(u, v):
"""
returns bool tuple indicating if the values maximising np.abs
are positive across all rows for u and across all columns for v.
"""
u_based = (np.abs(u).max(axis=0) == u.max(axis=0)).all()
v_based = (np.abs(v).max(axis=1) == v.max(axis=1)).all()
return u_based, v_based
mat = np.arange(10 * 8).reshape(10, -1)
# Without transpose
u_flipped, _, v_flipped = randomized_svd(mat, 3, flip_sign=True)
u_based, v_based = max_loading_is_positive(u_flipped, v_flipped)
assert u_based
assert not v_based
# With transpose
u_flipped_with_transpose, _, v_flipped_with_transpose = randomized_svd(
mat, 3, flip_sign=True, transpose=True)
u_based, v_based = max_loading_is_positive(u_flipped_with_transpose,
v_flipped_with_transpose)
assert u_based
assert not v_based
def test_randomized_svd_sign_flip():
a = np.array([[2.0, 0.0], [0.0, 1.0]])
u1, s1, v1 = randomized_svd(a, 2, flip_sign=True, random_state=41)
for seed in range(10):
u2, s2, v2 = randomized_svd(a, 2, flip_sign=True, random_state=seed)
assert_almost_equal(u1, u2)
assert_almost_equal(v1, v2)
assert_almost_equal(np.dot(u2 * s2, v2), a)
assert_almost_equal(np.dot(u2.T, u2), np.eye(2))
assert_almost_equal(np.dot(v2.T, v2), np.eye(2))
def svd_timing(X,
n_comps,
n_iter,
n_oversamples,
power_iteration_normalizer='auto',
method=None):
"""
Measure time for decomposition
"""
print("... running SVD ...")
if method is not 'fbpca':
gc.collect()
t0 = time()
U, mu, V = randomized_svd(X,
n_comps,
n_oversamples,
n_iter,
power_iteration_normalizer,
random_state=random_state,
transpose=False)
call_time = time() - t0
else:
gc.collect()
t0 = time()
# There is a different convention for l here
U, mu, V = fbpca.pca(X,
n_comps,
raw=True,
n_iter=n_iter,
l=n_oversamples + n_comps)
call_time = time() - t0
return U, mu, V, call_time
def test_randomized_svd_power_iteration_normalizer():
# randomized_svd with power_iteration_normalized='none' diverges for
# large number of power iterations on this dataset
rng = np.random.RandomState(42)
X = make_low_rank_matrix(100, 500, effective_rank=50, random_state=rng)
X += 3 * rng.randint(0, 2, size=X.shape)
n_components = 50
# Check that it diverges with many (non-normalized) power iterations
U, s, V = randomized_svd(X,
n_components,
n_iter=2,
power_iteration_normalizer='none')
A = X - U.dot(np.diag(s).dot(V))
error_2 = linalg.norm(A, ord='fro')
U, s, V = randomized_svd(X,
n_components,
n_iter=20,
power_iteration_normalizer='none')
A = X - U.dot(np.diag(s).dot(V))
error_20 = linalg.norm(A, ord='fro')
assert np.abs(error_2 - error_20) > 100
for normalizer in ['LU', 'QR', 'auto']:
U, s, V = randomized_svd(X,
n_components,
n_iter=2,
power_iteration_normalizer=normalizer,
random_state=0)
A = X - U.dot(np.diag(s).dot(V))
error_2 = linalg.norm(A, ord='fro')
for i in [5, 10, 50]:
U, s, V = randomized_svd(X,
n_components,
n_iter=i,
power_iteration_normalizer=normalizer,
random_state=0)
A = X - U.dot(np.diag(s).dot(V))
error = linalg.norm(A, ord='fro')
assert 15 > np.abs(error_2 - error)
def test_randomized_svd_transpose_consistency():
# Check that transposing the design matrix has limited impact
n_samples = 100
n_features = 500
rank = 4
k = 10
X = make_low_rank_matrix(n_samples=n_samples,
n_features=n_features,
effective_rank=rank,
tail_strength=0.5,
random_state=0)
assert X.shape == (n_samples, n_features)
U1, s1, V1 = randomized_svd(X,
k,
n_iter=3,
transpose=False,
random_state=0)
U2, s2, V2 = randomized_svd(X, k, n_iter=3, transpose=True, random_state=0)
U3, s3, V3 = randomized_svd(X,
k,
n_iter=3,
transpose='auto',
random_state=0)
U4, s4, V4 = linalg.svd(X, full_matrices=False)
assert_almost_equal(s1, s4[:k], decimal=3)
assert_almost_equal(s2, s4[:k], decimal=3)
assert_almost_equal(s3, s4[:k], decimal=3)
assert_almost_equal(np.dot(U1, V1),
np.dot(U4[:, :k], V4[:k, :]),
decimal=2)
assert_almost_equal(np.dot(U2, V2),
np.dot(U4[:, :k], V4[:k, :]),
decimal=2)
# in this case 'auto' is equivalent to transpose
assert_almost_equal(s2, s3)
def test_randomized_svd_low_rank_with_noise():
# Check that extmath.randomized_svd can handle noisy matrices
n_samples = 100
n_features = 500
rank = 5
k = 10
# generate a matrix X wity structure approximate rank `rank` and an
# important noisy component
X = make_low_rank_matrix(n_samples=n_samples,
n_features=n_features,
effective_rank=rank,
tail_strength=0.1,
random_state=0)
assert X.shape == (n_samples, n_features)
# compute the singular values of X using the slow exact method
_, s, _ = linalg.svd(X, full_matrices=False)
for normalizer in ['auto', 'none', 'LU', 'QR']:
# compute the singular values of X using the fast approximate
# method without the iterated power method
_, sa, _ = randomized_svd(X,
k,
n_iter=0,
power_iteration_normalizer=normalizer,
random_state=0)
# the approximation does not tolerate the noise:
assert np.abs(s[:k] - sa).max() > 0.01
# compute the singular values of X using the fast approximate
# method with iterated power method
_, sap, _ = randomized_svd(X,
k,
power_iteration_normalizer=normalizer,
random_state=0)
# the iterated power method is helping getting rid of the noise:
assert_almost_equal(s[:k], sap, decimal=3)
def compute_bench(samples_range, features_range, n_iter=3, rank=50):
it = 0
results = defaultdict(lambda: [])
max_it = len(samples_range) * len(features_range)
for n_samples in samples_range:
for n_features in features_range:
it += 1
print('====================')
print('Iteration %03d of %03d' % (it, max_it))
print('====================')
X = make_low_rank_matrix(n_samples,
n_features,
effective_rank=rank,
tail_strength=0.2)
gc.collect()
print("benchmarking scipy svd: ")
tstart = time()
svd(X, full_matrices=False)
results['scipy svd'].append(time() - tstart)
gc.collect()
print("benchmarking scikit-learn randomized_svd: n_iter=0")
tstart = time()
randomized_svd(X, rank, n_iter=0)
results['scikit-learn randomized_svd (n_iter=0)'].append(time() -
tstart)
gc.collect()
print("benchmarking scikit-learn randomized_svd: n_iter=%d " %
n_iter)
tstart = time()
randomized_svd(X, rank, n_iter=n_iter)
results['scikit-learn randomized_svd (n_iter=%d)' %
n_iter].append(time() - tstart)
return results
def test_randomized_svd_infinite_rank():
# Check that extmath.randomized_svd can handle noisy matrices
n_samples = 100
n_features = 500
rank = 5
k = 10
# let us try again without 'low_rank component': just regularly but slowly
# decreasing singular values: the rank of the data matrix is infinite
X = make_low_rank_matrix(n_samples=n_samples,
n_features=n_features,
effective_rank=rank,
tail_strength=1.0,
random_state=0)
assert X.shape == (n_samples, n_features)
# compute the singular values of X using the slow exact method
_, s, _ = linalg.svd(X, full_matrices=False)
for normalizer in ['auto', 'none', 'LU', 'QR']:
# compute the singular values of X using the fast approximate method
# without the iterated power method
_, sa, _ = randomized_svd(X,
k,
n_iter=0,
power_iteration_normalizer=normalizer)
# the approximation does not tolerate the noise:
assert np.abs(s[:k] - sa).max() > 0.1
# compute the singular values of X using the fast approximate method
# with iterated power method
_, sap, _ = randomized_svd(X,
k,
n_iter=5,
power_iteration_normalizer=normalizer)
# the iterated power method is still managing to get most of the
# structure at the requested rank
assert_almost_equal(s[:k], sap, decimal=3)
def check_randomized_svd_low_rank(dtype):
# Check that extmath.randomized_svd is consistent with linalg.svd
n_samples = 100
n_features = 500
rank = 5
k = 10
decimal = 5 if dtype == np.float32 else 7
dtype = np.dtype(dtype)
# generate a matrix X of approximate effective rank `rank` and no noise
# component (very structured signal):
X = make_low_rank_matrix(n_samples=n_samples,
n_features=n_features,
effective_rank=rank,
tail_strength=0.0,
random_state=0).astype(dtype, copy=False)
assert X.shape == (n_samples, n_features)
# compute the singular values of X using the slow exact method
U, s, V = linalg.svd(X, full_matrices=False)
# Convert the singular values to the specific dtype
U = U.astype(dtype, copy=False)
s = s.astype(dtype, copy=False)
V = V.astype(dtype, copy=False)
for normalizer in ['auto', 'LU', 'QR']: # 'none' would not be stable
# compute the singular values of X using the fast approximate method
Ua, sa, Va = randomized_svd(X,
k,
power_iteration_normalizer=normalizer,
random_state=0)
# If the input dtype is float, then the output dtype is float of the
# same bit size (f32 is not upcast to f64)
# But if the input dtype is int, the output dtype is float64
if dtype.kind == 'f':
assert Ua.dtype == dtype
assert sa.dtype == dtype
assert Va.dtype == dtype
else:
assert Ua.dtype == np.float64
assert sa.dtype == np.float64
assert Va.dtype == np.float64
assert Ua.shape == (n_samples, k)
assert sa.shape == (k, )
assert Va.shape == (k, n_features)
# ensure that the singular values of both methods are equal up to the
# real rank of the matrix
assert_almost_equal(s[:k], sa, decimal=decimal)
# check the singular vectors too (while not checking the sign)
assert_almost_equal(np.dot(U[:, :k], V[:k, :]),
np.dot(Ua, Va),
decimal=decimal)
# check the sparse matrix representation
X = sparse.csr_matrix(X)
# compute the singular values of X using the fast approximate method
Ua, sa, Va = \
randomized_svd(X, k, power_iteration_normalizer=normalizer,
random_state=0)
if dtype.kind == 'f':
assert Ua.dtype == dtype
assert sa.dtype == dtype
assert Va.dtype == dtype
else:
assert Ua.dtype.kind == 'f'
assert sa.dtype.kind == 'f'
assert Va.dtype.kind == 'f'
assert_almost_equal(s[:rank], sa[:rank], decimal=decimal)