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 upper half space model, the inner product at point z = x + iy of the vectors u, v it is (z, u, v are complex symmetric matrices): g_{z}(u, v) = tr[ y^-1 u y^-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 inv_imag_z = torch.inverse(sm.imag(z)) inv_imag_z = sm.stick(inv_imag_z, torch.zeros_like(inv_imag_z)) res = sm.bmm3(inv_imag_z, u, inv_imag_z) res = sm.bmm(res, sm.conjugate(v)) real_part = sm.trace(sm.real(res), keepdim=True) # b x 1 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
def test_takagi_factorization_real_neg_imag_neg(self): a = get_random_symmetric_matrices(3, 3) a = sm.stick(sm.real(a) * -1, sm.imag(a) * -1) 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 test_distance_is_symmetric_only_imaginary_matrices(self): x = self.manifold.random(10) y = self.manifold.random(10) zeros = torch.zeros_like(sm.real(x)) x = sm.stick(zeros, sm.imag(x)) y = sm.stick(zeros, sm.imag(y)) dist_xy = self.manifold.dist(x, y) dist_yx = self.manifold.dist(y, x) self.assertAllClose(dist_xy, dist_yx)
def test_pow_square(self): x_real = torch.Tensor([[[1, -3], [5, -7]]]) x_imag = torch.Tensor([[[9, -11], [-14, 15]]]) x = sm.stick(x_real, x_imag) expected_real = torch.Tensor([[[-80, -112], [-171, -176]]]) expected_imag = torch.Tensor([[[18, 66], [-140, -210]]]) expected = sm.stick(expected_real, expected_imag) result = sm.pow(x, 2) self.assertAllClose(expected, result)
def test_pow_cube(self): x_real = torch.Tensor([[[1, -3], [5, -7]]]) x_imag = torch.Tensor([[[9, -11], [-14, 15]]]) x = sm.stick(x_real, x_imag) expected_real = torch.Tensor([[[-242, 1062], [-2815, 4382]]]) expected_imag = torch.Tensor([[[-702, 1034], [1694, -1170]]]) expected = sm.stick(expected_real, expected_imag) result = sm.pow(x, 3) self.assertAllClose(expected, result)
def test_pow_square_root(self): x_real = torch.Tensor([[[1, -3], [5, -7]]]) x_imag = torch.Tensor([[[9, -11], [-14, 15]]]) x = sm.stick(x_real, x_imag) expected_real = torch.Tensor([[[2.24225167, 2.04960413], [3.15167167, 2.18551428]]]) expected_imag = torch.Tensor([[[2.00691120, -2.68344501], [-2.22104353, 3.43168656]]]) expected = sm.stick(expected_real, expected_imag) result = sm.pow(x, 0.5) self.assertAllClose(expected, result)
def test_pow_inverse(self): x_real = torch.Tensor([[[1, -3], [5, -7]]]) x_imag = torch.Tensor([[[9, -11], [-14, 15]]]) x = sm.stick(x_real, x_imag) expected_real = torch.Tensor([[[0.01219512, -0.02307692], [0.02262443, -0.02554744]]]) expected_imag = torch.Tensor([[[-0.10975609, 0.08461538], [0.06334841, -0.05474452]]]) expected = sm.stick(expected_real, expected_imag) result = sm.pow(x, -1) self.assertAllClose(expected, result)
def test_hermitian_matrix_sqrt(self): # expected result from https://www.wolframalpha.com/ x_real = torch.Tensor([[[0.9408, 0.1332], [0.1332, 0.5936]]]) x_imag = torch.Tensor([[[0.0000, 0.5677], [-0.5677, 0.0000]]]) x = sm.stick(x_real, x_imag) expected_real = torch.Tensor([[[0.896153, 0.0847679], [0.0847679, 0.675196]]]) expected_imag = torch.Tensor([[[0, 0.361282], [-0.361282, 0]]]) expected = sm.stick(expected_real, expected_imag) result = sm.hermitian_matrix_sqrt(x) self.assertAllClose(expected, result, rtol=1e-05, atol=1e-06)
def test_inverse_symmetric_2d(self): # expected result from https://adrianstoll.com/linear-algebra/matrix-inversion.html x_real = torch.Tensor([[[1, -3], [-3, 7]]]) x_imag = torch.Tensor([[[-9, 11], [11, 15]]]) x = sm.stick(x_real, x_imag) expected_real = torch.Tensor([[[256 / 8105, 141 / 16210], [141 / 16210, 23 / 16210]]]) expected_imag = torch.Tensor([[[921 / 16210, -356 / 8105], [-356 / 8105, -288 / 8105]]]) expected = sm.stick(expected_real, expected_imag) result = sm.inverse(x) self.assertAllClose(expected, result)
def projx(self, z: torch.Tensor) -> torch.Tensor: """ Project point :math:`z` on the manifold. In this space, we need to ensure that Y = Im(X) is positive definite. Since the matrix Y is symmetric, it is possible to diagonalize it. For a diagonal matrix the condition is just that all diagonal entries are positive, so we clamp the values that are <=0 in the diagonal to an epsilon, and then restore the matrix back into non-diagonal form using the base change matrix that was obtained from the diagonalization. Steps to project: Y = Im(z) 1) Y = SDS^-1 2) D_tilde = clamp(D, min=epsilon) 3) Y_tilde = SD_tildeS^-1 :param z: points to be projected: (b, 2, n, n) """ z = super().projx(z) y = sm.imag(z) y_tilde, batchwise_mask = sm.positive_conjugate_projection(y) self.projected_points += len(z) - sum(batchwise_mask).item() return sm.stick(sm.real(z), y_tilde)
def test_bmm(self): x_real = torch.Tensor([[[1, -3], [5, -7]]]) x_imag = torch.Tensor([[[9, -11], [-14, 15]]]) x = sm.stick(x_real, x_imag) y_real = torch.Tensor([[[9, -11], [-14, 15]]]) y_imag = torch.Tensor([[[1, -3], [5, -7]]]) y = sm.stick(y_real, y_imag) expected_real = torch.Tensor([[[97, -106], [82, -97]]]) expected_imag = torch.Tensor([[[221, -246], [-366, 413]]]) expected = sm.stick(expected_real, expected_imag) result = sm.bmm(x, y) self.assertAllClose(expected, result)
def symmetric_svd_with_eigenvectors(self, z: torch.Tensor): """ :param z: complex symmetric matrix :return: """ compound_z = sm.to_compound_symmetric( z) # b x 2 x 2n x 2n. Z = A + iB, then [(A, B),(B, -A)] evalues, q = self.symeig( compound_z, eigenvectors=True) # b x n in ascending order, b x 2n x 2n # I can think of Q as 4 n x n matrices. # Q = [(X, Re(U)), # (Y, -Im(U))] where X, Y are irrelevant and I need to build U real_u_on_top_of_minus_imag_u = torch.chunk(q, 2, dim=-1)[-1] real_u, minus_imag_u = torch.chunk(real_u_on_top_of_minus_imag_u, 2, dim=-2) u = sm.stick(real_u, -minus_imag_u) # b x 2 x n x n sing_values = evalues[:, z.shape[-1]:] # sing_values_matrix = sm.bmm3(sm.transpose(u), z, u) # b x 2 x n x n. imag part should be zero # sing_values = torch.diagonal(sm.real(sing_values_matrix), offset=0, dim1=-2, dim2=-1) return sing_values, u
def test_stick(self): x_real = torch.rand(10, 2, 4, 4) x_imag = torch.rand(10, 2, 4, 4) x = sm.stick(x_real, x_imag) self.assertAllEqual(x_real, sm.real(x)) self.assertAllEqual(x_imag, sm.imag(x))
def test_distance_is_symmetric_real_neg_imag_pos(self): x = self.manifold.random(10) x = sm.stick(sm.real(x) * -1, sm.imag(x)) y = self.manifold.random(10) dist_xy = self.manifold.dist(x, y) dist_yx = self.manifold.dist(y, x) self.assertAllClose(dist_xy, dist_yx)
def test_takagi_factorization_very_large_values(self): a = get_random_symmetric_matrices(3, 3) * 1000 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 test_inverse_symmetric_3d(self): # expected result from https://adrianstoll.com/linear-algebra/matrix-inversion.html x_real = torch.Tensor([[[-1, -3, 9], [-3, 5, 7], [9, 7, 11]]]) x_imag = torch.Tensor([[[9, 4, -6], [4, 7, 9], [-6, 9, -3]]]) x = sm.stick(x_real, x_imag) expected_real = torch.Tensor( [[[-36251 / 845665, -27631 / 845665, 188 / 9949], [-27631 / 845665, 251611 / 3382660, -689 / 39796], [188 / 9949, -689 / 39796, 1299 / 39796]]]) expected_imag = torch.Tensor( [[[-18757 / 845665, -35642 / 845665, 532 / 9949], [-35642 / 845665, -112703 / 3382660, -1103 / 39796], [532 / 9949, -1103 / 39796, 289 / 39796]]]) expected = sm.stick(expected_real, expected_imag) result = sm.inverse(x) self.assertAllClose(expected, result)
def test_inverse_nonsymmetric_3d(self): # expected result from https://adrianstoll.com/linear-algebra/matrix-inversion.html x_real = torch.Tensor([[[-1, -3, 9], [3, 5, 7], [2, 9, 11]]]) x_imag = torch.Tensor([[[9, 4, -6], [-4, 7, 9], [-2, 7, -3]]]) x = sm.stick(x_real, x_imag) expected_real = torch.Tensor( [[[951 / 16589, 3223 / 16589, -4496 / 16589], [5029 / 66356, 2486 / 16589, 1387 / 33178], [2137 / 66356, 1030 / 16589, -1828 / 16589]]]) expected_imag = torch.Tensor( [[[-3143 / 16589, -3622 / 16589, 1532 / 16589], [5533 / 66356, 6925 / 33178, -17977 / 66356], [-4167 / 66356, -5779 / 33178, 6565 / 66356]]]) expected = sm.stick(expected_real, expected_imag) result = sm.inverse(x) self.assertAllClose(expected, result)
def test_to_hermitian_from_compound_real_symmetric_2d(self): x = torch.Tensor([[[1, -3, 0, -5], [-3, 7, 5, 0], [0, 5, 1, -3], [-5, 0, -3, 7]]]) expected_real = torch.Tensor([[[1, -3], [-3, 7]]]) expected_imag = torch.Tensor([[[0, 5], [-5, 0]]]) expected = sm.stick(expected_real, expected_imag) result = sm.to_hermitian_from_compound_real_symmetric(x) self.assertAllEqual(expected, result)
def test_bmm3(self): x_real = torch.Tensor([[[1, -3], [5, -7]]]) x_imag = torch.Tensor([[[9, -11], [-14, 15]]]) x = sm.stick(x_real, x_imag) y_real = torch.Tensor([[[9, -11], [-14, 15]]]) y_imag = torch.Tensor([[[1, -3], [5, -7]]]) y = sm.stick(y_real, y_imag) z_real = torch.Tensor([[[-3, -1], [-2, 5]]]) z_imag = torch.Tensor([[[-1, 3], [0, -2]]]) z = sm.stick(z_real, z_imag) expected_real = torch.Tensor([[[142, -1782], [-418, 1357]]]) expected_imag = torch.Tensor([[[-268, -948], [190, 2871]]]) expected = sm.stick(expected_real, expected_imag) result = sm.bmm3(x, y, z) self.assertAllClose(expected, result)
def test_hermitian_eig(self): # expected result from https://www.arndt-bruenner.de/mathe/scripts/engl_eigenwert2.htm x_real = torch.Tensor([[[1, -3], [-3, 7]]]) x_imag = torch.Tensor([[[0, 5], [-5, 0]]]) x = sm.stick(x_real, x_imag) expected = torch.Tensor([[[-2.55743852, 10.55743852]]]) _, _, result = sm.hermitian_eig(x) self.assertAllClose(expected, result)
def test_takagi_factorization_real_identity(self): a = sm.identity_like(get_random_symmetric_matrices(3, 3)) 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))) self.assertAllClose(a, s) self.assertAllClose(torch.ones_like(eigenvalues), eigenvalues)
def test_proj_x_real_neg_imag_neg(self): x = get_random_symmetric_matrices(10, self.dims) x = sm.stick(sm.real(x) * -1, sm.imag(x) * -1) proj_x = self.manifold.projx(x) # assert symmetry self.assertAllClose(proj_x, sm.transpose(proj_x)) # assert all points belong to the manifold for point in proj_x: self.assertTrue(self.manifold.check_point_on_manifold(point))
def test_distance_is_symmetric_with_diagonal_matrices(self): x = self.manifold.random(10) y = self.manifold.random(10) diagonal_mask = torch.eye(self.dims).unsqueeze(0).repeat(10, 1, 1).bool() diagonal_mask = sm.stick(diagonal_mask, diagonal_mask) x = torch.where(diagonal_mask, x, torch.zeros_like(x)) y = torch.where(diagonal_mask, y, torch.zeros_like(y)) dist_xy = self.manifold.dist(x, y) dist_yx = self.manifold.dist(y, x) self.assertAllClose(dist_xy, dist_yx)
def test_conj_transpose(self): x = get_random_symmetric_matrices(10, 4) x_real = sm.real(x) x_imag = sm.imag(x) x_real_transpose = x_real.transpose(-1, -2) x_imag_transpose = x_imag.transpose(-1, -2) x_expected_conj_transpose = sm.stick(x_real_transpose, x_imag_transpose * -1) x_result_conj_transpose = sm.conj_trans(x) self.assertAllEqual(x_expected_conj_transpose, x_result_conj_transpose)
def test_transpose(self): x = get_random_symmetric_matrices(10, 4) x_real = sm.real(x) x_imag = sm.imag(x) x_real_transpose = x_real.transpose(-1, -2) x_imag_transpose = x_imag.transpose(-1, -2) x_expected_transpose = sm.stick(x_real_transpose, x_imag_transpose) x_result_transpose = sm.transpose(x) self.assertAllEqual(x_expected_transpose, x_result_transpose) self.assertAllEqual( x, x_result_transpose) # because they are symmetric matrices
def test_cayley_transform_real_neg_imag_pos(self): x = self.upper_half_manifold.random(10) x = sm.stick(sm.real(x) * -1, sm.imag(x)) tran_x = cayley_transform(x) result = inverse_cayley_transform(tran_x) self.assertAllClose(x, result) # the intermediate result belongs to the Bounded domain manifold for point in tran_x: self.assertTrue(self.bounded_manifold.check_point_on_manifold(point)) # the final result belongs to the Upper Half Space manifold for point in result: self.assertTrue(self.upper_half_manifold.check_point_on_manifold(point))
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 Hn, then the gradient is the y * grad_eucl(f(z)) * y, where y is the imaginary part of z, and multiplication is just matrix multiplication. :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 """ real_grad, imag_grad = sm.real(u), sm.imag(u) y = sm.imag(z) real_grad = y.bmm(real_grad).bmm(y) imag_grad = y.bmm(imag_grad).bmm(y) return sm.stick(real_grad, imag_grad)
def random(self, *size, dtype=None, device=None, **kwargs) -> torch.Tensor: """ Random sampling on the manifold. The exact implementation depends on manifold and usually does not follow all assumptions about uniform measure, etc. """ from_ = kwargs.get("from_", -0.001) to = kwargs.get("to", 0.001) dims = self.dims perturbation = sm.squared_to_symmetric( torch.Tensor(size[0], dims, dims).uniform_(from_, to)) identity = torch.eye(dims).unsqueeze(0).repeat(size[0], 1, 1) imag_part = identity + perturbation real_part = sm.squared_to_symmetric( torch.Tensor(size[0], dims, dims).uniform_(from_, to)) return sm.stick(real_part, imag_part).to(device=device, dtype=dtype)