def test_extended_laplacian_shortcut(self):
"""
Demo commutativity of SVD and summation under certain conditions.
"""
p = 100
n = 7
k = 3
# define sets of arbitrary row indices
first_rows = set((3, 5, 9))
second_rows = set((12, 13, 2))
# make a random data matrix
X = np.random.random((p, n))
# get a matrix that we are treating like the laplacian
HDH = util.get_doubly_centered_matrix(np.corrcoef(X) ** 2)
L = np.linalg.pinv(HDH)
L_summed = util.sum_arbitrary_rows_and_columns(L, first_rows)
L_summed = util.sum_arbitrary_rows_and_columns(L_summed, second_rows)
U, S, VT = np.linalg.svd(L_summed, full_matrices=0)
expected_fiedler = get_fiedler_vector(U, S)
for f in (data_to_laplacian_sqrt, data_to_reduced_laplacian_sqrt):
# get the square root of the laplacian without using pxp operations
U, S = f(X)
A = U*S
# sum rows of the summed laplacian sqrt
B = util.sum_arbitrary_rows(A, first_rows)
# get the first criterion matrix
U, S_array, VT = np.linalg.svd(B, full_matrices=0)
QB = U*S_array
# sum rows of the first criterion matrix
QB_summed = util.sum_arbitrary_rows(QB, second_rows)
# sum more rows of the summed laplacian sqrt
C = util.sum_arbitrary_rows(B, second_rows)
# get the second criterion matrix directly from the summed laplacian sqrt
U, S_array, VT = np.linalg.svd(C, full_matrices=0)
QC_direct_fiedler = get_fiedler_vector(U, S_array)
QC_direct = U*S_array
# get the second criterion matrix from the summed first criterion matrix
U, S_array, VT = np.linalg.svd(QB_summed, full_matrices=0)
QC_indirect_fiedler = get_fiedler_vector(U, S_array)
QC_indirect = U*S_array
# check the equivalence of the matrix squares
self.assertAllClose(np.dot(QC_direct, QC_direct.T), L_summed)
self.assertAllClose(np.dot(QC_indirect, QC_indirect.T), L_summed)
self.assertAllClose(expected_fiedler, QC_direct_fiedler)
self.assertAllClose(expected_fiedler, QC_indirect_fiedler)
def test_laplacian_summation_shortcut(self):
p = 100
n = 7
k = 3
X = np.random.random((p, n))
# get a matrix that we are treating like the laplacian
HDH = util.get_doubly_centered_matrix(np.corrcoef(X) ** 2)
L = np.linalg.pinv(HDH)
L_summed = util.sum_last_rows_and_columns(L, k)
U, S, VT = np.linalg.svd(L_summed, full_matrices=0)
expected_fiedler = get_fiedler_vector(U, S)
# get the square root of the summed laplacian without using pxp operations
U, S = data_to_laplacian_sqrt(X)
B = util.sum_last_rows(U*S, k)
self.assertAllClose(np.dot(B, B.T), L_summed)
# get another square root
U, S = remove_small_vectors(U, S)
B = util.sum_last_rows(U*S, k)
U, S, VT = np.linalg.svd(B, full_matrices=0)
observed_fiedler = get_fiedler_vector(U, S)
self.assertAllClose(np.dot(U*S, (U*S).T), L_summed)
self.assertAllClose(expected_fiedler, observed_fiedler)
def test_fiedler(self):
p = 100
n = 7
X = np.random.random((p, n))
# get a matrix that we are treating like the laplacian
HDH = util.get_doubly_centered_matrix(np.corrcoef(X) ** 2)
L = np.linalg.pinv(HDH)
U, S, VT = np.linalg.svd(L, full_matrices=0)
expected_fiedler = get_fiedler_vector(U, S)
# get the square root of the summed laplacian without using pxp operations
U, S = data_to_laplacian_sqrt(X)
self.assertAllClose(np.dot(U*S, (U*S).T), L)
observed_fiedler = get_fiedler_vector(U, S)
self.assertAllClose(expected_fiedler, observed_fiedler)
# get another square root of the summed laplacian
U, S = remove_small_vectors(U, S)
self.assertAllClose(np.dot(U*S, (U*S).T), L)
self.assertAllClose(expected_fiedler, observed_fiedler)
# get yet another square root of the summed laplacian
U, S, VT = np.linalg.svd(U*S, full_matrices=0)
self.assertAllClose(np.dot(U*S, (U*S).T), L)
self.assertAllClose(expected_fiedler, observed_fiedler)
