from typing import Optional, Tuple
import torch
from torch import Tensor
[docs]def dense_mincut_pool(
x: Tensor,
adj: Tensor,
s: Tensor,
mask: Optional[Tensor] = None,
temp: float = 1.0,
) -> Tuple[Tensor, Tensor, Tensor, Tensor]:
r"""The MinCut pooling operator from the `"Spectral Clustering in Graph
Neural Networks for Graph Pooling" <https://arxiv.org/abs/1907.00481>`_
paper.
.. math::
\mathbf{X}^{\prime} &= {\mathrm{softmax}(\mathbf{S})}^{\top} \cdot
\mathbf{X}
\mathbf{A}^{\prime} &= {\mathrm{softmax}(\mathbf{S})}^{\top} \cdot
\mathbf{A} \cdot \mathrm{softmax}(\mathbf{S})
based on dense learned assignments :math:`\mathbf{S} \in \mathbb{R}^{B
\times N \times C}`.
Returns the pooled node feature matrix, the coarsened and symmetrically
normalized adjacency matrix and two auxiliary objectives: (1) The MinCut
loss
.. math::
\mathcal{L}_c = - \frac{\mathrm{Tr}(\mathbf{S}^{\top} \mathbf{A}
\mathbf{S})} {\mathrm{Tr}(\mathbf{S}^{\top} \mathbf{D}
\mathbf{S})}
where :math:`\mathbf{D}` is the degree matrix, and (2) the orthogonality
loss
.. math::
\mathcal{L}_o = {\left\| \frac{\mathbf{S}^{\top} \mathbf{S}}
{{\|\mathbf{S}^{\top} \mathbf{S}\|}_F} -\frac{\mathbf{I}_C}{\sqrt{C}}
\right\|}_F.
Args:
x (torch.Tensor): Node feature tensor
:math:`\mathbf{X} \in \mathbb{R}^{B \times N \times F}`, with
batch-size :math:`B`, (maximum) number of nodes :math:`N` for
each graph, and feature dimension :math:`F`.
adj (torch.Tensor): Adjacency tensor
:math:`\mathbf{A} \in \mathbb{R}^{B \times N \times N}`.
s (torch.Tensor): Assignment tensor
:math:`\mathbf{S} \in \mathbb{R}^{B \times N \times C}`
with number of clusters :math:`C`.
The softmax does not have to be applied before-hand, since it is
executed within this method.
mask (torch.Tensor, optional): Mask matrix
:math:`\mathbf{M} \in {\{ 0, 1 \}}^{B \times N}` indicating
the valid nodes for each graph. (default: :obj:`None`)
temp (float, optional): Temperature parameter for softmax function.
(default: :obj:`1.0`)
:rtype: (:class:`torch.Tensor`, :class:`torch.Tensor`,
:class:`torch.Tensor`, :class:`torch.Tensor`)
"""
x = x.unsqueeze(0) if x.dim() == 2 else x
adj = adj.unsqueeze(0) if adj.dim() == 2 else adj
s = s.unsqueeze(0) if s.dim() == 2 else s
(batch_size, num_nodes, _), k = x.size(), s.size(-1)
s = torch.softmax(s / temp if temp != 1.0 else s, dim=-1)
if mask is not None:
mask = mask.view(batch_size, num_nodes, 1).to(x.dtype)
x, s = x * mask, s * mask
out = torch.matmul(s.transpose(1, 2), x)
out_adj = torch.matmul(torch.matmul(s.transpose(1, 2), adj), s)
# MinCut regularization.
mincut_num = _rank3_trace(out_adj)
d_flat = torch.einsum('ijk->ij', adj)
d = _rank3_diag(d_flat)
mincut_den = _rank3_trace(
torch.matmul(torch.matmul(s.transpose(1, 2), d), s))
mincut_loss = -(mincut_num / mincut_den)
mincut_loss = torch.mean(mincut_loss)
# Orthogonality regularization.
ss = torch.matmul(s.transpose(1, 2), s)
i_s = torch.eye(k).type_as(ss)
ortho_loss = torch.norm(
ss / torch.norm(ss, dim=(-1, -2), keepdim=True) -
i_s / torch.norm(i_s), dim=(-1, -2))
ortho_loss = torch.mean(ortho_loss)
EPS = 1e-15
# Fix and normalize coarsened adjacency matrix.
ind = torch.arange(k, device=out_adj.device)
out_adj[:, ind, ind] = 0
d = torch.einsum('ijk->ij', out_adj)
d = torch.sqrt(d)[:, None] + EPS
out_adj = (out_adj / d) / d.transpose(1, 2)
return out, out_adj, mincut_loss, ortho_loss
def _rank3_trace(x: Tensor) -> Tensor:
return torch.einsum('ijj->i', x)
def _rank3_diag(x: Tensor) -> Tensor:
eye = torch.eye(x.size(1)).type_as(x)
out = eye * x.unsqueeze(2).expand(x.size(0), x.size(1), x.size(1))
return out