"""Spatial Encoder"""
import torch as th
import torch.nn as nn
import torch.nn.functional as F
def gaussian(x, mean, std):
"""compute gaussian basis kernel function"""
const_pi = 3.14159
a = (2 * const_pi) ** 0.5
return th.exp(-0.5 * (((x - mean) / std) ** 2)) / (a * std)
[docs]class SpatialEncoder(nn.Module):
r"""Spatial Encoder, as introduced in
`Do Transformers Really Perform Bad for Graph Representation?
<https://proceedings.neurips.cc/paper/2021/file/f1c1592588411002af340cbaedd6fc33-Paper.pdf>`__
This module is a learnable spatial embedding module, which encodes
the shortest distance between each node pair for attention bias.
Parameters
----------
max_dist : int
Upper bound of the shortest path distance
between each node pair to be encoded.
All distance will be clamped into the range `[0, max_dist]`.
num_heads : int, optional
Number of attention heads if multi-head attention mechanism is applied.
Default : 1.
Examples
--------
>>> import torch as th
>>> import dgl
>>> from dgl.nn import SpatialEncoder
>>> from dgl import shortest_dist
>>> g1 = dgl.graph(([0,0,0,1,1,2,3,3], [1,2,3,0,3,0,0,1]))
>>> g2 = dgl.graph(([0,1], [1,0]))
>>> n1, n2 = g1.num_nodes(), g2.num_nodes()
>>> # use -1 padding since shortest_dist returns -1 for unreachable node pairs
>>> dist = -th.ones((2, 4, 4), dtype=th.long)
>>> dist[0, :n1, :n1] = shortest_dist(g1, root=None, return_paths=False)
>>> dist[1, :n2, :n2] = shortest_dist(g2, root=None, return_paths=False)
>>> spatial_encoder = SpatialEncoder(max_dist=2, num_heads=8)
>>> out = spatial_encoder(dist)
>>> print(out.shape)
torch.Size([2, 4, 4, 8])
"""
def __init__(self, max_dist, num_heads=1):
super().__init__()
self.max_dist = max_dist
self.num_heads = num_heads
# deactivate node pair between which the distance is -1
self.embedding_table = nn.Embedding(
max_dist + 2, num_heads, padding_idx=0
)
[docs] def forward(self, dist):
"""
Parameters
----------
dist : Tensor
Shortest path distance of the batched graph with -1 padding, a tensor
of shape :math:`(B, N, N)`, where :math:`B` is the batch size of
the batched graph, and :math:`N` is the maximum number of nodes.
Returns
-------
torch.Tensor
Return attention bias as spatial encoding of shape
:math:`(B, N, N, H)`, where :math:`H` is :attr:`num_heads`.
"""
spatial_encoding = self.embedding_table(
th.clamp(
dist,
min=-1,
max=self.max_dist,
)
+ 1
)
return spatial_encoding
[docs]class SpatialEncoder3d(nn.Module):
r"""3D Spatial Encoder, as introduced in
`One Transformer Can Understand Both 2D & 3D Molecular Data
<https://arxiv.org/pdf/2210.01765.pdf>`__
This module encodes pair-wise relation between node pair :math:`(i,j)` in
the 3D geometric space, according to the Gaussian Basis Kernel function:
:math:`\psi _{(i,j)} ^k = \frac{1}{\sqrt{2\pi} \lvert \sigma^k \rvert}
\exp{\left ( -\frac{1}{2} \left( \frac{\gamma_{(i,j)} \lvert \lvert r_i -
r_j \rvert \rvert + \beta_{(i,j)} - \mu^k}{\lvert \sigma^k \rvert} \right)
^2 \right)}，k=1,...,K,`
where :math:`K` is the number of Gaussian Basis kernels. :math:`r_i` is the
Cartesian coordinate of node :math:`i`.
:math:`\gamma_{(i,j)}, \beta_{(i,j)}` are learnable scaling factors and
biases determined by node types. :math:`\mu^k, \sigma^k` are learnable
centers and standard deviations of the Gaussian Basis kernels.
Parameters
----------
num_kernels : int
Number of Gaussian Basis Kernels to be applied. Each Gaussian Basis
Kernel contains a learnable kernel center and a learnable standard
deviation.
num_heads : int, optional
Number of attention heads if multi-head attention mechanism is applied.
Default : 1.
max_node_type : int, optional
Maximum number of node types. Each node type has a corresponding
learnable scaling factor and a bias. Default : 100.
Examples
--------
>>> import torch as th
>>> import dgl
>>> from dgl.nn import SpatialEncoder3d
>>> coordinate = th.rand(1, 4, 3)
>>> node_type = th.tensor([[1, 0, 2, 1]])
>>> spatial_encoder = SpatialEncoder3d(num_kernels=4,
... num_heads=8,
... max_node_type=3)
>>> out = spatial_encoder(coordinate, node_type=node_type)
>>> print(out.shape)
torch.Size([1, 4, 4, 8])
"""
def __init__(self, num_kernels, num_heads=1, max_node_type=100):
super().__init__()
self.num_kernels = num_kernels
self.num_heads = num_heads
self.max_node_type = max_node_type
self.means = nn.Parameter(th.empty(num_kernels))
self.stds = nn.Parameter(th.empty(num_kernels))
self.linear_layer_1 = nn.Linear(num_kernels, num_kernels)
self.linear_layer_2 = nn.Linear(num_kernels, num_heads)
# There are 2 * max_node_type + 3 pairs of gamma and beta parameters:
# 1. Parameters at position 0 are for default gamma/beta when no node
# type is given
# 2. Parameters at position 1 to max_node_type+1 are for src node types.
# (position 1 is for padded unexisting nodes)
# 3. Parameters at position max_node_type+2 to 2*max_node_type+2 are
# for tgt node types. (position max_node_type+2 is for padded)
# unexisting nodes)
self.gamma = nn.Embedding(2 * max_node_type + 3, 1, padding_idx=0)
self.beta = nn.Embedding(2 * max_node_type + 3, 1, padding_idx=0)
nn.init.uniform_(self.means, 0, 3)
nn.init.uniform_(self.stds, 0, 3)
nn.init.constant_(self.gamma.weight, 1)
nn.init.constant_(self.beta.weight, 0)
[docs] def forward(self, coord, node_type=None):
"""
Parameters
----------
coord : torch.Tensor
3D coordinates of nodes in shape :math:`(B, N, 3)`, where :math:`B`
is the batch size, :math:`N`: is the maximum number of nodes.
node_type : torch.Tensor, optional
Node type ids of nodes. Default : None.
* If specified, :attr:`node_type` should be a tensor in shape
:math:`(B, N,)`. The scaling factors in gaussian kernels of each
pair of nodes are determined by their node types.
* Otherwise, :attr:`node_type` will be set to zeros of the same
shape by default.
Returns
-------
torch.Tensor
Return attention bias as 3D spatial encoding of shape
:math:`(B, N, N, H)`, where :math:`H` is :attr:`num_heads`.
"""
bsz, N = coord.shape[:2]
euc_dist = th.cdist(coord, coord, p=2.0) # shape: [B, n, n]
if node_type is None:
node_type = th.zeros([bsz, N, N, 2], device=coord.device).long()
else:
src_node_type = node_type.unsqueeze(-1).repeat(1, 1, N)
tgt_node_type = node_type.unsqueeze(1).repeat(1, N, 1)
node_type = th.stack(
[src_node_type + 2, tgt_node_type + self.max_node_type + 3],
dim=-1,
) # shape: [B, n, n, 2]
# scaled euclidean distance
gamma = self.gamma(node_type).sum(dim=-2) # shape: [B, n, n, 1]
beta = self.beta(node_type).sum(dim=-2) # shape: [B, n, n, 1]
euc_dist = gamma * euc_dist.unsqueeze(-1) + beta # shape: [B, n, n, 1]
# gaussian basis kernel
euc_dist = euc_dist.expand(-1, -1, -1, self.num_kernels)
gaussian_kernel = gaussian(
euc_dist, self.means, self.stds.abs() + 1e-2
) # shape: [B, n, n, K]
# linear projection
encoding = self.linear_layer_1(gaussian_kernel)
encoding = F.gelu(encoding)
encoding = self.linear_layer_2(encoding) # shape: [B, n, n, H]
return encoding