def test_multiply_by_i(self):
x = get_random_symmetric_matrices(10, 4)
result = sm.multiply_by_i(x)
expected_real = sm.real(result)
expected_imag = sm.imag(result)
self.assertAllEqual(expected_real, -sm.imag(x))
self.assertAllEqual(expected_imag, sm.real(x))
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 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_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_to_symmetric(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)
self.assertAllEqual(x_real, x_real_transpose)
self.assertAllEqual(x_imag, x_imag_transpose)
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_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_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_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 test_to_hermitian(self):
m = torch.rand(4, 2, 3, 3)
h = sm.to_hermitian(m)
h_real, h_imag = sm.real(h), sm.imag(h)
# 1 - real part is symmetric
self.assertAllEqual(h_real, h_real.transpose(-1, -2))
# 2 - Imaginary diagonal must be 0
imag_diag = torch.diagonal(h_imag, dim1=-2, dim2=-1)
self.assertAllEqual(imag_diag, torch.zeros_like(imag_diag))
# 3 - imaginary elements in the upper triangular part of the matrix must be of opposite sign than the
# elements in the lower triangular part
imag_triu = torch.triu(h_imag, diagonal=1)
imag_tril = torch.tril(h_imag, diagonal=-1)
self.assertAllEqual(imag_triu, imag_tril.transpose(-1, -2) * -1)
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
