# Source code for torch_geometric.nn.dense.mincut_pool

import torch

EPS = 1e-15

[docs]def dense_mincut_pool(x, adj, s, mask=None):
r"""MinCUt pooling operator from the "Mincut Pooling in Graph Neural
Networks" <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 pooled node feature matrix, coarsened 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 (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 (Tensor): Symmetrically normalized adjacency tensor
:math:\mathbf{A} \in \mathbb{R}^{B \times N \times N}.
s (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 beforehand, since it is executed
within this method.
mask (BoolTensor, optional): Mask matrix
:math:\mathbf{M} \in {\{ 0, 1 \}}^{B \times N} indicating
the valid nodes for each graph. (default: :obj:None)

:rtype: (:class:Tensor, :class:Tensor, :class:Tensor,
:class: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, 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)

# 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):
return torch.einsum('ijj->i', x)

def _rank3_diag(x):
eye = torch.eye(x.size(1)).type_as(x)
out = eye * x.unsqueeze(2).expand(*x.size(), x.size(1))
return out