def get_id_minus_conjugate_z_times_z(z: torch.Tensor): """ :param z: b x 2 x n x n :return: Id - \overline(z)z """ identity = sm.identity_like(z) conj_z_z = sm.bmm(sm.conjugate(z), z) return sm.subtract(identity, conj_z_z)
def test_takagi_factorization_imag_identity(self): a = sm.identity_like(get_random_symmetric_matrices(3, 3)) a = sm.stick(sm.imag(a), sm.real(a)) eigenvalues, s = TakagiFactorization(3).factorize(a) diagonal = torch.diag_embed(eigenvalues) diagonal = sm.stick(diagonal, torch.zeros_like(diagonal)) self.assertAllClose( a, sm.bmm3(sm.conjugate(s), diagonal, sm.conj_trans(s)))
def inverse_cayley_transform(z: torch.Tensor) -> torch.Tensor: """ T^-1(Z): Bounded Domain Model -> Upper Half Space model T^-1(Z) = i (Id + Z)(Id - Z)^-1 :param z: b x 2 x n x n: PRE: z \in Bounded Domain Manifold :return: y \in Upper Half Space Manifold """ identity = sm.identity_like(z) i_z_plus_id = sm.multiply_by_i(sm.add(identity, z)) inv_z_minus_id = sm.inverse(sm.subtract(identity, z)) return sm.bmm(i_z_plus_id, inv_z_minus_id)
def cayley_transform(z: torch.Tensor) -> torch.Tensor: """ T(Z): Upper Half Space model -> Bounded Domain Model T(Z) = (Z - i Id)(Z + i Id)^-1 :param z: b x 2 x n x n: PRE: z \in Upper Half Space Manifold :return: y \in Bounded Domain Manifold """ identity = sm.identity_like(z) i_identity = sm.stick(sm.imag(identity), sm.real(identity)) z_minus_id = sm.subtract(z, i_identity) inv_z_plus_id = sm.inverse(sm.add(z, i_identity)) return sm.bmm(z_minus_id, inv_z_plus_id)
def test_takagi_factorization_real_diagonal(self): a = get_random_symmetric_matrices(3, 3) * 10 a = torch.where(sm.identity_like(a) == 1, a, torch.zeros_like(a)) eigenvalues, s = TakagiFactorization(3).factorize(a) diagonal = torch.diag_embed(eigenvalues) diagonal = sm.stick(diagonal, torch.zeros_like(diagonal)) self.assertAllClose( a, sm.bmm3(sm.conjugate(s), diagonal, sm.conj_trans(s))) # real part of eigenvectors is made of vectors with one 1 and all zeros real_part = torch.sum(torch.abs(sm.real(s)), dim=-1) self.assertAllClose(torch.ones_like(real_part), real_part) # imaginary part of eigenvectors is all zeros self.assertAllClose(torch.zeros(1), torch.sum(sm.imag(s)))
def egrad2rgrad(self, z: torch.Tensor, u: torch.Tensor) -> torch.Tensor: """ Transform gradient computed using autodiff to the correct Riemannian gradient for the point :math:`x`. If you have a function f(z) on Mn, then the Riemannian gradient is grad_R(f(z)) = (Id + ẑz) * grad_E(f(z)) * (Id + zẑ) :param z: point on the manifold. Shape: (b, 2, n, n) :param u: gradient to be projected: Shape: same than z :return grad vector in the Riemannian manifold. Shape: same than z """ id = sm.identity_like(z) conjz = sm.conjugate(z) id_plus_conjz_z = id + sm.bmm(conjz, z) id_plus_z_conjz = id + sm.bmm(z, conjz) riem_grad = sm.bmm3(id_plus_conjz_z, u, id_plus_z_conjz) return riem_grad
def inner(self, z: torch.Tensor, u: torch.Tensor, v=None, *, keepdim=False) -> torch.Tensor: """ Inner product for tangent vectors at point :math:`z`. For the bounded domain model, the inner product at point z of the vectors u, v it is (z, u, v are complex symmetric matrices): g_{z}(u, v) = tr[ (Id - ẑz)^-1 u (Id - zẑ)^-1 \ov{v} ] :param z: torch.Tensor point on the manifold: b x 2 x n x n :param u: torch.Tensor tangent vector at point :math:`z`: b x 2 x n x n :param v: Optional[torch.Tensor] tangent vector at point :math:`z`: b x 2 x n x n :param keepdim: bool keep the last dim? :return: torch.Tensor inner product (broadcastable): b x 2 x 1 x 1 """ if v is None: v = u identity = sm.identity_like(z) conj_z = sm.conjugate(z) conj_z_z = sm.bmm(conj_z, z) z_conj_z = sm.bmm(z, conj_z) inv_id_minus_conj_z_z = sm.subtract(identity, conj_z_z) inv_id_minus_z_conj_z = sm.subtract(identity, z_conj_z) inv_id_minus_conj_z_z = sm.inverse(inv_id_minus_conj_z_z) inv_id_minus_z_conj_z = sm.inverse(inv_id_minus_z_conj_z) res = sm.bmm3(inv_id_minus_conj_z_z, u, inv_id_minus_z_conj_z) res = sm.bmm(res, sm.conjugate(v)) real_part = sm.trace(sm.real(res), keepdim=True) real_part = torch.unsqueeze(real_part, -1) # b x 1 x 1 return sm.stick(real_part, real_part) # # b x 2 x 1 x 1