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pmath.py
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pmath.py
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import numpy as np
import torch
def pair_wise_eud(x,y,c=1.0):
# input:
# x, m x d
# y, n x d
# output: m x n
m = x.size(0)
n = y.size(0)
d = x.size(1)
assert(x.size(1) == y.size(1))
xx = x.pow(2).sum(-1, keepdim = True)
yy = y.pow(2).sum(-1, keepdim = True)
xy = torch.einsum('ij,kj->ik', (x, y))
result = xx - 2*xy + yy.permute(1, 0)
return result
def pair_wise_cos(x,y,c=1.0):
# input:
# x, m x d
# y, n x d
# output: m x n
x_norm = torch.norm(x, dim=1, keepdim = True)# m x 1
y_norm = torch.norm(y, dim=1, keepdim = True)# n x 1
denominator = torch.matmul(x_norm, y_norm.t()) # m x n
numerator = torch.matmul(x, y.t()) # m x n, each element is a in-prod
return -numerator / denominator # m x n
def pair_wise_hyp(x, y, c=1.0):
c = torch.as_tensor(c)
return _dist_matrix(x, y, c)
def tanh(x, clamp=15):
return x.clamp(-clamp, clamp).tanh()
def tensor_dot(x, y):
res = torch.einsum('ij,kj->ik', (x, y))
return res
def _mobius_addition_batch(x, y, c):
xy = tensor_dot(x, y) # B x C
x2 = x.pow(2).sum(-1, keepdim=True) # B x 1
y2 = y.pow(2).sum(-1, keepdim=True) # C x 1
num = (1 + 2 * c * xy + c * y2.permute(1, 0)) # B x C
num = num.unsqueeze(2) * x.unsqueeze(1) # B x C x 1 * B x 1 x D = B x C x D
num = num + (1 - c * x2).unsqueeze(2) * y # B x C x D + B x 1 x 1 = B x C x D
denom_part1 = 1 + 2 * c * xy # B x C
denom_part2 = c ** 2 * x2 * y2.permute(1, 0) # B x 1 * 1 x C = B x C
denom = denom_part1 + denom_part2
res = num / (denom.unsqueeze(2) + 1e-5)
return res
def _mobius_addition_same_size(x, y, c):
xy = torch.einsum('ij,ij -> i', (x, y)).unsqueeze(1) # n x1
x2 = x.pow(2).sum(-1, keepdim=True) # n x 1
y2 = y.pow(2).sum(-1, keepdim=True) # n x 1
num = 1 + 2 * c * xy + c * y2 # n x 1
num2 = num * x # n x D
num3 = num + (1 - c * x2) * y # n x D
denom_part1 = 1 + 2 * c * xy # n x 1
denom_part2 = c ** 2 * x2 * y2 # n x 1
denom = denom_part1 + denom_part2
res = num3 / (denom + 1e-5)
return res
def _dist_matrix(x, y, c):
sqrt_c = c ** 0.5
return 2 / sqrt_c * artanh(sqrt_c * torch.norm(_mobius_addition_batch(-x, y, c=c), dim=-1))
def dist_same_size(x,y, c=1.0):
c = torch.as_tensor(c)
sqrt_c = c ** 0.5
return 2 / sqrt_c * artanh(sqrt_c * torch.norm(_mobius_addition_same_size(-x, y, c=c), dim=-1))
def project(x, *, c=1.0):
r"""
Safe projection on the manifold for numerical stability. This was mentioned in [1]_
Parameters
"""
c = torch.as_tensor(c).type_as(x)
return _project(x, c)
def _project(x, c):
"""Parameters
----------
x : tensor
point on the Poincare ball
c : float|tensor
ball negative curvature
Returns
-------
tensor
projected vector on the manifold"""
c = torch.as_tensor(c).type_as(x)
norm = torch.clamp_min(x.norm(dim=-1, keepdim=True, p=2), 1e-5)
maxnorm = (1 - 1e-3) / (c ** 0.5)
cond = norm > maxnorm
projected = x / norm * maxnorm
return torch.where(cond, projected, x)
def mobius_matvec(m, x, c):
x_norm = torch.clamp_min(x.norm(dim=-1, keepdim=True, p=2), 1e-5)
sqrt_c = c ** 0.5
mx = x @ m.transpose(-1, -2)
mx_norm = mx.norm(dim=-1, keepdim=True, p=2)
res_c = tanh(mx_norm / x_norm * artanh(sqrt_c * x_norm)) * mx / (mx_norm * sqrt_c)
cond = (mx == 0).prod(-1, keepdim=True, dtype=torch.uint8)
res_0 = torch.zeros(1, dtype=res_c.dtype, device=res_c.device)
res = torch.where(cond, res_0, res_c)
return _project(res, c)
def euc2hyp(x, c):
new_x = _project(expmap0(x, c), c)
return new_x
### 以下为原始代码
def dist(x, y, *, c=1.0, keepdim=False):
r"""
Distance on the Poincare ball
.. math::
d_c(x, y) = \frac{2}{\sqrt{c}}\tanh^{-1}(\sqrt{c}\|(-x)\oplus_c y\|_2)
.. plot:: plots/extended/poincare/distance.py
Parameters
----------
x : tensor
point on poincare ball
y : tensor
point on poincare ball
c : float|tensor
ball negative curvature
keepdim : bool
retain the last dim? (default: false)
Returns
-------
tensor
geodesic distance between :math:`x` and :math:`y`
"""
c = torch.as_tensor(c).type_as(x)
return _dist(x, y, c, keepdim=keepdim)
def _dist(x, y, c, keepdim: bool = False):
sqrt_c = c ** 0.5
dist_c = artanh(sqrt_c * _mobius_add(-x, y, c).norm(dim=-1, p=2, keepdim=keepdim))
return dist_c * 2 / sqrt_c
def mobius_add(x, y, *, c=1.0):
r"""
Mobius addition is a special operation in a hyperbolic space.
.. math::
x \oplus_c y = \frac{
(1 + 2 c \langle x, y\rangle + c \|y\|^2_2) x + (1 - c \|x\|_2^2) y
}{
1 + 2 c \langle x, y\rangle + c^2 \|x\|^2_2 \|y\|^2_2
}
In general this operation is not commutative:
.. math::
x \oplus_c y \ne y \oplus_c x
But in some cases this property holds:
* zero vector case
.. math::
\mathbf{0} \oplus_c x = x \oplus_c \mathbf{0}
* zero negative curvature case that is same as Euclidean addition
.. math::
x \oplus_0 y = y \oplus_0 x
Another usefull property is so called left-cancellation law:
.. math::
(-x) \oplus_c (x \oplus_c y) = y
Parameters
----------
x : tensor
point on the Poincare ball
y : tensor
point on the Poincare ball
c : float|tensor
ball negative curvature
Returns
-------
tensor
the result of mobius addition
"""
c = torch.as_tensor(c).type_as(x)
return _mobius_add(x, y, c)
def _mobius_add(x, y, c):
x2 = x.pow(2).sum(dim=-1, keepdim=True)
y2 = y.pow(2).sum(dim=-1, keepdim=True)
xy = (x * y).sum(dim=-1, keepdim=True)
num = (1 + 2 * c * xy + c * y2) * x + (1 - c * x2) * y
denom = 1 + 2 * c * xy + c ** 2 * x2 * y2
return num / (denom + 1e-5)
def expmap0(u, *, c=1.0):
r"""
Exponential map for Poincare ball model from :math:`0`.
.. math::
\operatorname{Exp}^c_0(u) = \tanh(\sqrt{c}/2 \|u\|_2) \frac{u}{\sqrt{c}\|u\|_2}
Parameters
----------
u : tensor
speed vector on poincare ball
c : float|tensor
ball negative curvature
Returns
-------
tensor
:math:`\gamma_{0, u}(1)` end point
"""
c = torch.as_tensor(c).type_as(u)
return _expmap0(u, c)
def _expmap0(u, c):
sqrt_c = c ** 0.5
u_norm = torch.clamp_min(u.norm(dim=-1, p=2, keepdim=True), 1e-5)
gamma_1 = tanh(sqrt_c * u_norm) * u / (sqrt_c * u_norm)
return gamma_1
def _hyperbolic_softmax(X, A, P, c):
lambda_pkc = 2 / (1 - c * P.pow(2).sum(dim=1))
k = lambda_pkc * torch.norm(A, dim=1) / torch.sqrt(c)
mob_add = _mobius_addition_batch(-P, X, c)
num = 2 * torch.sqrt(c) * torch.sum(mob_add * A.unsqueeze(1), dim=-1)
denom = torch.norm(A, dim=1, keepdim=True) * (1 - c * mob_add.pow(2).sum(dim=2))
logit = k.unsqueeze(1) * arsinh(num / denom)
return logit.permute(1, 0)
class Arsinh(torch.autograd.Function):
@staticmethod
def forward(ctx, x):
ctx.save_for_backward(x)
return (x + torch.sqrt_(1 + x.pow(2))).clamp_min_(1e-5).log_()
@staticmethod
def backward(ctx, grad_output):
input, = ctx.saved_tensors
return grad_output / (1 + input ** 2) ** 0.5
class Artanh(torch.autograd.Function):
@staticmethod
def forward(ctx, x):
x = x.clamp(-1 + 1e-5, 1 - 1e-5)
ctx.save_for_backward(x)
res = (torch.log_(1 + x).sub_(torch.log_(1 - x))).mul_(0.5)
return res
@staticmethod
def backward(ctx, grad_output):
input, = ctx.saved_tensors
return grad_output / (1 - input ** 2)
def artanh(x):
return Artanh.apply(x)
def arsinh(x):
return Arsinh.apply(x)