def test_eigensystem():
spmat = scipy.sparse.coo_matrix(
([0.3, 0.6, 0.1, 0.3, 0.1, 0.6, 0.6, 0.1, 0.3, 0.1, 0.6, 0.3],
([0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3
], [1, 2, 3, 0, 2, 3, 0, 1, 3, 0, 1, 2])),
shape=(4, 4))
# Eigenvalues are as expected; note that we get the largest magnitudes,
# but in algebraic order
u1, s1 = eigensystem(spmat, 3, strip_a0=False)
assert np.allclose(s1, [1, -0.4, -0.8])
# Eigenvectors are eigenvectors
for i, si in enumerate(s1):
ui = u1[:, i]
assert np.allclose(spmat.dot(ui), si * ui)
# Eigenvectors are orthonormal
assert np.allclose(u1.T.dot(u1), np.identity(s1.shape[0]))
# Stripping a0 removes the right eigenvalue
u2, s2 = eigensystem(spmat, 2, strip_a0=True)
assert np.allclose(s2, s1[1:])
compare_cols_within_sign(u2, u1[:, 1:])
# Asking for way too many eigenvalues is okay
u3, s3 = eigensystem(spmat, 5)
assert np.allclose(s3, s1)
compare_cols_within_sign(u3, u1)
def test_eigensystem():
spmat = scipy.sparse.coo_matrix(
([0.3, 0.6, 0.1, 0.3, 0.1, 0.6, 0.6, 0.1, 0.3, 0.1, 0.6, 0.3],
([0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3],
[1, 2, 3, 0, 2, 3, 0, 1, 3, 0, 1, 2])),
shape=(4, 4))
# Eigenvalues are as expected; note that we get the largest magnitudes,
# but in algebraic order
u1, s1 = eigensystem(spmat, 3, strip_a0=False)
assert np.allclose(s1, [1, -0.4, -0.8])
# Eigenvectors are eigenvectors
for i, si in enumerate(s1):
ui = u1[:, i]
assert np.allclose(spmat.dot(ui), si * ui)
# Eigenvectors are orthonormal
assert np.allclose(u1.T.dot(u1), np.identity(s1.shape[0]))
# Stripping a0 removes the right eigenvalue
u2, s2 = eigensystem(spmat, 2, strip_a0=True)
assert np.allclose(s2, s1[1:])
compare_cols_within_sign(u2, u1[:, 1:])
# Asking for way too many eigenvalues is okay
u3, s3 = eigensystem(spmat, 5)
assert np.allclose(s3, s1)
compare_cols_within_sign(u3, u1)
def from_matrix(cls, matrix, labels, k, offset_weight=8e-6, strip_a0=True, normalize_gm=True):
"""
Build an AssocSpace from a SciPy sparse matrix and a LabelSet.
Pass k to specify the number of dimensions; otherwise a value will be
chosen for you based on the size of the matrix.
strip_a0=True (on by default) removes the first eigenvector, which is
often uninformative.
normalize_gm=True (on by default) divides each entry by the geometric
mean of the sum of the column and the sum of the row. This is one
iteration of a process that might eventually yield a Markov matrix.
However, in order to suppress sufficiently rare terms, we add an offset
to the row and column sums computed from the number of dimensions and
the overall sum of the matrix.
"""
# Immediately reject empty inputs
if not labels:
return None
sums = matrix.sum(0)
matrix_sum = np.sum(sums)
logger.info("Building space with k=%d (sum=%.6f)." % (k, matrix_sum))
if normalize_gm:
offset = matrix_sum * offset_weight
normalizer = spdiags(1.0 / np.sqrt(sums + offset), 0, matrix.shape[0], matrix.shape[0])
matrix = normalizer * matrix * normalizer
u, s = eigenmath.eigensystem(matrix, k=k, strip_a0=strip_a0)
# This ensures that the normalization step is sane
if s.shape[0] == 0 or s[0] <= 0:
return None
return cls(np.asarray(u, ">f4"), np.asarray(s, ">f4"), labels)
