def test_multiqr(self):
shape = (self.k, self.m, self.m)
A_real = np.random.normal(size=shape)
q, r = multiqr(A_real)
np_testing.assert_allclose(q @ r, A_real)
A_complex = np.random.normal(
size=shape) + 1j * np.random.normal(size=shape)
q, r = multiqr(A_complex)
np_testing.assert_allclose(q @ r, A_complex)
def random_point(self):
q, _ = multiqr(
np.random.normal(size=(self._k, self._n, self._p)) +
1j * np.random.normal(size=(self._k, self._n, self._p)))
if self._k == 1:
return q[0]
return q
def exp(self, point, tangent_vector):
U, S, VH = np.linalg.svd(tangent_vector, full_matrices=False)
cos_S = np.expand_dims(np.cos(S), -2)
sin_S = np.expand_dims(np.sin(S), -2)
Y = point @ (multihconj(VH) * cos_S) @ VH + (U * sin_S) @ VH
# From numerical experiments, it seems necessary to
# re-orthonormalize. This is overall quite expensive.
q, _ = multiqr(Y)
return q
def random_point(self):
# Generate eigenvalues between 1 and 2.
d = 1.0 + np.random.uniform(size=(self._k, self._n, 1))
# Generate an orthogonal matrix.
q, _ = multiqr(np.random.normal(size=(self._n, self._n)))
point = q @ (d * multitransp(q))
if self._k == 1:
return point[0]
return point
def exp(self, point, tangent_vector):
u, s, vt = np.linalg.svd(tangent_vector, full_matrices=False)
cos_s = np.expand_dims(np.cos(s), -2)
sin_s = np.expand_dims(np.sin(s), -2)
Y = point @ (multitransp(vt) * cos_s) @ vt + (u * sin_s) @ vt
# From numerical experiments, it seems necessary to re-orthonormalize.
# This is quite expensive.
q, _ = multiqr(Y)
return q
def random_point(self):
n, k = self._n, self._k
if n == 1:
point = np.ones((k, 1, 1))
else:
point, _ = multiqr(np.random.normal(size=(k, n, n)))
# Swap the first two columns of matrices where det(point) < 0 to
# flip the sign of their determinants.
negative_det, *_ = np.where(np.linalg.det(point) < 0)
slice_ = np.arange(point.shape[1])
point[np.ix_(negative_det, slice_,
[0, 1])] = point[np.ix_(negative_det, slice_, [1, 0])]
if k == 1:
return point[0]
return point
def _retraction_qr(self, point, tangent_vector):
a = point + tangent_vector
point, _ = multiqr(a)
return point
def random_point(self):
point, _ = multiqr(np.random.normal(size=(self._k, self._n, self._p)))
if self._k == 1:
return point[0]
return point
def _retraction_qr(self, point, tangent_vector):
Y = point + point @ tangent_vector
q, _ = multiqr(Y)
return q