def vector_transport(self, u, vec1, vec2):
"""Returns a vector tranported along an another vector
via vector transport.
Args:
u: complex valued tensor of shape (..., n, n),
a set of points from the manifold, starting points.
vec1: complex valued tensor of shape (..., n, n),
a set of vectors to be transported.
vec2: complex valued tensor of shape (..., n, n),
a set of direction vectors.
Returns:
complex valued tensor of shape (..., n, n),
a set of transported vectors.
Note:
The complexity O(n^3)."""
if self._metric == 'log_euclidean':
lmbd, U = tf.linalg.eigh(u)
# geoidesic in S
Su = U @ tf.linalg.diag(tf.math.log(lmbd)) @ adj(U)
Svec2 = _pull_back_log(vec2, U, lmbd)
Sresult = Su + Svec2
# eig decomposition of a new point from S
log_new_lmbd, new_U = tf.linalg.eigh(Sresult)
# new lmbd
new_lmbd = tf.exp(log_new_lmbd)
# transported vector
new_vec1 = _push_forward_log(_pull_back_log(vec1, U, lmbd), new_U,
new_lmbd)
return new_vec1
elif self._metric == 'log_cholesky':
v = self.retraction(u, vec2)
L = tf.linalg.cholesky(u)
inv_L = tf.linalg.inv(L)
inv_diag_L = tf.linalg.diag(1 / tf.linalg.diag_part(L))
X = _pull_back_chol(vec1, L, inv_L)
K = tf.linalg.cholesky(v)
L_transport = _lower(X) + tf.linalg.band_part(K, 0, 0) *\
inv_diag_L * tf.linalg.band_part(X, 0, 0)
return K @ adj(L_transport) + L_transport @ adj(K)
def retraction_transport(self, u, vec1, vec2):
"""Performs a retraction and a vector transport simultaneously.
Args:
u: complex valued tensor of shape (..., n, n),
a set of points from the manifold, starting points.
vec1: complex valued tensor of shape (..., n, n),
a set of vectors to be transported.
vec2: complex valued tensor of shape (..., n, n),
a set of direction vectors.
Returns:
two complex valued tensors of shape (..., n, n),
a set of transported points and vectors."""
if self.metric == 'log_euclidean':
lmbd, U = tf.linalg.eigh(u)
# geoidesic in S
Su = U @ tf.linalg.diag(tf.math.log(lmbd)) @ adj(U)
Svec2 = _pull_back_log(vec2, U, lmbd)
Sresult = Su + Svec2
# eig decomposition of new point from S
log_new_lmbd, new_U = tf.linalg.eigh(Sresult)
# new point from S++
new_point = new_U @ tf.linalg.diag(tf.exp(log_new_lmbd)) @\
adj(new_U)
# new lmbd
new_lmbd = tf.exp(log_new_lmbd)
# transported vector
new_vec1 = _push_forward_log(_pull_back_log(vec1, U, lmbd),
new_U, new_lmbd)
return new_point, new_vec1
elif self.metric == 'log_cholesky':
v = self.retraction(u, vec2)
L = tf.linalg.cholesky(u)
inv_L = tf.linalg.inv(L)
inv_diag_L = tf.linalg.diag(1 / tf.linalg.diag_part(L))
X = _pull_back_chol(vec1, L, inv_L)
K = tf.linalg.cholesky(v)
L_transport = _lower(X) + tf.linalg.band_part(K, 0, 0) *\
inv_diag_L * tf.linalg.band_part(X, 0, 0)
return v, K @ adj(L_transport) + L_transport @ adj(K)
def inner(self, u, vec1, vec2):
"""Returns manifold wise inner product of vectors from
a tangent space.
Args:
u: complex valued tensor of shape (..., n, n),
a set of points from the manifold.
vec1: complex valued tensor of shape (..., n, n),
a set of tangent vectors from the manifold.
vec2: complex valued tensor of shape (..., n, n),
a set of tangent vectors from the manifold.
Returns:
complex valued tensor of shape (..., 1, 1),
manifold wise inner product.
Note:
The complexity O(n^3) for both inner products."""
if self._metric == 'log_euclidean':
lmbd, U = tf.linalg.eigh(u)
W = _pull_back_log(vec1, U, lmbd)
V = _pull_back_log(vec2, U, lmbd)
prod = tf.math.real(
tf.reduce_sum(tf.math.conj(W) * V,
axis=(-2, -1),
keepdims=True))
prod = tf.cast(prod, dtype=u.dtype)
return prod
elif self._metric == 'log_cholesky':
u_shape = tf.shape(u)
L = tf.linalg.cholesky(u)
inv_L = tf.linalg.inv(L)
W = _pull_back_chol(vec1, L, inv_L)
V = _pull_back_chol(vec2, L, inv_L)
mask = tf.ones(u_shape[-2:], dtype=u.dtype)
mask = _lower(mask)
G = mask + tf.linalg.diag(1 / (tf.linalg.diag_part(L)**2))
prod = tf.reduce_sum(tf.math.conj(W) * G * V, axis=(-2, -1))
prod = tf.math.real(prod)
prod = prod[..., tf.newaxis, tf.newaxis]
prod = tf.cast(prod, dtype=u.dtype)
return prod
def retraction(self, u, vec):
"""Transports a set of points from the manifold via a
retraction map.
Args:
u: complex valued tensor of shape (..., n, n), a set
of points to be transported.
vec: complex valued tensor of shape (..., n, n),
a set of direction vectors.
Returns:
complex valued tensor of shape (..., n, n),
a set of transported points.
Note:
The complexity O(n^3)."""
if self._metric == 'log_euclidean':
lmbd, U = tf.linalg.eigh(u)
# geodesic in S
Su = U @ tf.linalg.diag(tf.math.log(lmbd)) @ adj(U)
Svec = _pull_back_log(vec, U, lmbd)
Sresult = Su + Svec
return tf.linalg.expm(Sresult)
elif self._metric == 'log_cholesky':
L = tf.linalg.cholesky(u)
inv_L = tf.linalg.inv(L)
X = _pull_back_chol(vec, L, inv_L)
inv_diag_L = tf.linalg.diag(1 / tf.linalg.diag_part(L))
cholesky_retraction = _lower(L) + _lower(X) +\
tf.linalg.band_part(L, 0, 0) *\
tf.exp(tf.linalg.band_part(X, 0, 0) * inv_diag_L)
return cholesky_retraction @ adj(cholesky_retraction)