def _check_point_on_manifold(self, x, atol, rtol):
x_t = utils.transposem(x)
eigvals, _ = tf.linalg.eigh(x)
is_symmetric = utils.allclose(x, x_t, atol, rtol)
is_pos_vals = utils.allclose(eigvals, tf.abs(eigvals), atol, rtol)
is_zero_vals = utils.allclose(eigvals, tf.zeros_like(eigvals), atol,
rtol)
return is_symmetric & is_pos_vals & tf.logical_not(is_zero_vals)
def _check_point_on_manifold(self, x, atol, rtol):
xtx = utils.transposem(x) @ x
eye = tf.eye(
tf.shape(xtx)[-1], batch_shape=tf.shape(xtx)[:-2], dtype=x.dtype
)
is_orth = utils.allclose(xtx, eye, atol, rtol)
det = tf.linalg.det(x)
is_unit_det = utils.allclose(det, tf.ones_like(det), atol, rtol)
return is_orth & is_unit_det
def _check_point_on_manifold(self, x, atol, rtol):
xtx = utils.transposem(x) @ x
shape = xtx.shape.as_list()
eye = tf.eye(shape[-1], batch_shape=shape[:-2])
is_idempotent = utils.allclose(xtx, tf.cast(eye, x.dtype), atol, rtol)
s = tf.linalg.svd(x, compute_uv=False)
rank = tf.math.count_nonzero(s, axis=-1, dtype=tf.float32)
k = tf.ones_like(rank) * int(x.shape[-1])
is_col_rank = utils.allclose(rank, k, atol, rtol)
return is_idempotent & is_col_rank
def _check_vector_on_tangent(self, x, u, atol, rtol):
xtu = utils.transposem(x) @ u
return utils.allclose(xtu, tf.zeros_like(xtu), atol, rtol)
def _check_vector_on_tangent(self, x, u, atol, rtol):
u_t = utils.transposem(u)
return utils.allclose(u, u_t, atol, rtol)
def _check_vector_on_tangent(self, x, u, atol, rtol):
lower_triang = tf.linalg.band_part(x, -1, 0)
return utils.allclose(x, lower_triang, atol, rtol)
def _check_point_on_manifold(self, x, atol, rtol):
lower_triang = tf.linalg.band_part(x, -1, 0)
is_lower_triang = utils.allclose(x, lower_triang, atol, rtol)
diag = tf.linalg.diag_part(x)
is_pos_diag = utils.allclose(diag, tf.abs(diag), atol, rtol)
return is_lower_triang & is_pos_diag
def _check_vector_on_tangent(self, x, u, atol, rtol):
inner = self.inner(x, x, u)
rtol = 100 * utils.get_eps(x) if rtol is None else rtol
return utils.allclose(inner, tf.zeros_like(inner), atol, rtol)
def _check_point_on_manifold(self, x, atol, rtol):
x_sq = tf.square(x)
quad_form = -x_sq[..., :1] + tf.reduce_sum(
x_sq[..., 1:], axis=-1, keepdims=True)
return utils.allclose(quad_form,
tf.ones_like(quad_form) * -self.k, atol, rtol)
def _check_vector_on_tangent(self, x, u, atol, rtol):
diff = utils.transposem(u) + u
return utils.allclose(diff, tf.zeros_like(diff), atol, rtol)
def _check_vector_on_tangent(self, x, u, atol, rtol):
inner = self.inner(x, x, u, keepdims=True)
return utils.allclose(inner, tf.zeros_like(inner), atol, rtol)
def _check_point_on_manifold(self, x, atol, rtol):
norm = tf.linalg.norm(x, axis=-1)
return utils.allclose(norm, tf.ones_like(norm), atol, rtol)
def _check_point_on_manifold(self, x, atol, rtol):
xtx = utils.transposem(x) @ x
eye = tf.eye(tf.shape(xtx)[-1],
batch_shape=tf.shape(xtx)[:-2],
dtype=xtx.dtype)
return utils.allclose(xtx, eye, atol, rtol)