def remove_small_vectors(U, S):
"""
Remove negligible values in S and corresponding columns in U.
The inputs are not directly modified, but the results are returned.
@param U: columns are eigenvectors
@param S: conformant singular values
@return: U_reduced, S_reduced
"""
columns = [U.T[i] for i, w in enumerate(S) if not util.is_small(w)]
U_reduced = np.vstack(columns).T
S_reduced = [w for w in S if not util.is_small(w)]
return U_reduced, S_reduced
def get_fiedler_vector(U, S):
"""
The first element of the output vector is forced to be nonnegative.
@param U: columns are eigenvectors
@param S: conformant singular values
@return: the eigenvector corresponding to the smallest nonsmall eigenvalue
"""
best_w, best_i = min((w, i) for i, w in enumerate(S) if not util.is_small(w))
v = U.T[best_i]
if v[0] < 0:
return -v
else:
return v
def data_to_laplacian_sqrt(X):
"""
If the output is U, S then (U*S)(U*S)' is like a Laplacian.
@param X: a data matrix
@return: U, S
"""
logging.debug('data_to_laplacian_sqrt: creating the standardized matrix')
Z = get_standardized_matrix(X)
logging.debug('data_to_laplacian_sqrt: creating the augmented matrix')
Q = standardized_to_augmented_C(Z)
logging.debug('data_to_laplacian_sqrt: creating the column centered matrix')
W = util.get_column_centered_matrix(Q)
logging.debug('data_to_laplacian_sqrt: manually cleaning up old matrices')
del Z
del Q
logging.debug('data_to_laplacian_sqrt: doing a singular value decomposition')
U, S_array, VT = np.linalg.svd(W, full_matrices=0)
S_pinv_array = np.array([0 if util.is_small(x) else 1/x for x in S_array])
return U, S_pinv_array
