# Source code for dgl.nn.mxnet.conv.gmmconv

"""Torch Module for GMM Conv"""
# pylint: disable= no-member, arguments-differ, invalid-name
import math
import mxnet as mx
from mxnet import nd
from mxnet.gluon import nn
from mxnet.gluon.contrib.nn import Identity

from .... import function as fn
from ....base import DGLError
from ....utils import expand_as_pair

[docs]class GMMConv(nn.Block):
r"""

Description
-----------
The Gaussian Mixture Model Convolution layer from Geometric Deep
Learning on Graphs and Manifolds using Mixture Model CNNs
<http://openaccess.thecvf.com/content_cvpr_2017/papers/Monti_Geometric_Deep_Learning_CVPR_2017_paper.pdf>__.

.. math::
u_{ij} &= f(x_i, x_j), x_j \in \mathcal{N}(i)

w_k(u) &= \exp\left(-\frac{1}{2}(u-\mu_k)^T \Sigma_k^{-1} (u - \mu_k)\right)

h_i^{l+1} &= \mathrm{aggregate}\left(\left\{\frac{1}{K}
\sum_{k}^{K} w_k(u_{ij}), \forall j\in \mathcal{N}(i)\right\}\right)

where :math:u denotes the pseudo-coordinates between a vertex and one of its neighbor,
computed using function :math:f, :math:\Sigma_k^{-1} and :math:\mu_k are
learnable parameters representing the covariance matrix and mean vector of a Gaussian kernel.

Parameters
----------
in_feats : int
Number of input features; i.e., the number of dimensions of :math:x_i.
out_feats : int
Number of output features; i.e., the number of dimensions of :math:h_i^{(l+1)}.
dim : int
Dimensionality of pseudo-coordinte; i.e, the number of dimensions of :math:u_{ij}.
n_kernels : int
Number of kernels :math:K.
aggregator_type : str
Aggregator type (sum, mean, max). Default: sum.
residual : bool
If True, use residual connection inside this layer. Default: False.
bias : bool
If True, adds a learnable bias to the output. Default: True.
allow_zero_in_degree : bool, optional
If there are 0-in-degree nodes in the graph, output for those nodes will be invalid
since no message will be passed to those nodes. This is harmful for some applications
causing silent performance regression. This module will raise a DGLError if it detects
0-in-degree nodes in input graph. By setting True, it will suppress the check
and let the users handle it by themselves. Default: False.

Note
----
Zero in-degree nodes will lead to invalid output value. This is because no message
will be passed to those nodes, the aggregation function will be appied on empty input.
A common practice to avoid this is to add a self-loop for each node in the graph if
it is homogeneous, which can be achieved by:

>>> g = ... # a DGLGraph
>>> g = dgl.add_self_loop(g)

Calling add_self_loop will not work for some graphs, for example, heterogeneous graph
since the edge type can not be decided for self_loop edges. Set allow_zero_in_degree
to True for those cases to unblock the code and handle zero-in-degree nodes manually.
A common practise to handle this is to filter out the nodes with zero-in-degree when use
after conv.

Examples
--------
>>> import dgl
>>> import numpy as np
>>> import mxnet as mx
>>> from dgl.nn import GMMConv
>>>
>>> # Case 1: Homogeneous graph
>>> g = dgl.graph(([0,1,2,3,2,5], [1,2,3,4,0,3]))
>>> g = dgl.add_self_loop(g)
>>> feat = mx.nd.ones((6, 10))
>>> conv = GMMConv(10, 2, 3, 2, 'mean')
>>> conv.initialize(ctx=mx.cpu(0))
>>> pseudo = mx.nd.ones((12, 3))
>>> res = conv(g, feat, pseudo)
>>> res
[[-0.05083769 -0.1567954 ]
[-0.05083769 -0.1567954 ]
[-0.05083769 -0.1567954 ]
[-0.05083769 -0.1567954 ]
[-0.05083769 -0.1567954 ]
[-0.05083769 -0.1567954 ]]
<NDArray 6x2 @cpu(0)>

>>> # Case 2: Unidirectional bipartite graph
>>> u = [0, 1, 0, 0, 1]
>>> v = [0, 1, 2, 3, 2]
>>> g = dgl.bipartite((u, v))
>>> u_fea = mx.nd.random.randn(2, 5)
>>> v_fea = mx.nd.random.randn(4, 10)
>>> pseudo = mx.nd.ones((5, 3))
>>> conv = GMMConv((5, 10), 2, 3, 2, 'mean')
>>> conv.initialize(ctx=mx.cpu(0))
>>> res = conv(g, (u_fea, v_fea), pseudo)
>>> res
[[-0.1005067  -0.09494358]
[-0.0023314  -0.07597432]
[-0.05141905 -0.08545895]
[-0.1005067  -0.09494358]]
<NDArray 4x2 @cpu(0)>
"""
def __init__(self,
in_feats,
out_feats,
dim,
n_kernels,
aggregator_type='sum',
residual=False,
bias=True,
allow_zero_in_degree=False):
super(GMMConv, self).__init__()

self._in_src_feats, self._in_dst_feats = expand_as_pair(in_feats)
self._out_feats = out_feats
self._dim = dim
self._n_kernels = n_kernels
self._allow_zero_in_degree = allow_zero_in_degree
if aggregator_type == 'sum':
self._reducer = fn.sum
elif aggregator_type == 'mean':
self._reducer = fn.mean
elif aggregator_type == 'max':
self._reducer = fn.max
else:
raise KeyError("Aggregator type {} not recognized.".format(aggregator_type))

with self.name_scope():
self.mu = self.params.get('mu',
shape=(n_kernels, dim),
init=mx.init.Normal(0.1))
self.inv_sigma = self.params.get('inv_sigma',
shape=(n_kernels, dim),
init=mx.init.Constant(1))
self.fc = nn.Dense(n_kernels * out_feats,
in_units=self._in_src_feats,
use_bias=False,
weight_initializer=mx.init.Xavier(magnitude=math.sqrt(2.0)))
if residual:
if self._in_dst_feats != out_feats:
self.res_fc = nn.Dense(out_feats, in_units=self._in_dst_feats, use_bias=False)
else:
self.res_fc = Identity()
else:
self.res_fc = None

if bias:
self.bias = self.params.get('bias',
shape=(out_feats,),
init=mx.init.Zero())
else:
self.bias = None

def set_allow_zero_in_degree(self, set_value):
r"""

Description
-----------
Set allow_zero_in_degree flag.

Parameters
----------
set_value : bool
The value to be set to the flag.
"""
self._allow_zero_in_degree = set_value

[docs]    def forward(self, graph, feat, pseudo):
"""

Description
-----------
Compute Gaussian Mixture Model Convolution layer.

Parameters
----------
graph : DGLGraph
The graph.
feat : mxnet.NDArray
If a single tensor is given, the input feature of shape :math:(N, D_{in}) where
:math:D_{in} is size of input feature, :math:N is the number of nodes.
If a pair of tensors are given, the pair must contain two tensors of shape
:math:(N_{in}, D_{in_{src}}) and :math:(N_{out}, D_{in_{dst}}).
pseudo : mxnet.NDArray
The pseudo coordinate tensor of shape :math:(E, D_{u}) where
:math:E is the number of edges of the graph and :math:D_{u}
is the dimensionality of pseudo coordinate.

Returns
-------
mxnet.NDArray
The output feature of shape :math:(N, D_{out}) where :math:D_{out}
is the output feature size.

Raises
------
DGLError
If there are 0-in-degree nodes in the input graph, it will raise DGLError
since no message will be passed to those nodes. This will cause invalid output.
The error can be ignored by setting allow_zero_in_degree parameter to True.
"""
if not self._allow_zero_in_degree:
if graph.in_degrees().min() == 0:
raise DGLError('There are 0-in-degree nodes in the graph, '
'output for those nodes will be invalid. '
'This is harmful for some applications, '
'causing silent performance regression. '
'Adding self-loop on the input graph by '
'calling g = dgl.add_self_loop(g) will resolve '
'the issue. Setting allow_zero_in_degree '
'to be True when constructing this module will '
'suppress the check and let the code run.')

feat_src, feat_dst = expand_as_pair(feat, graph)
with graph.local_scope():
graph.srcdata['h'] = self.fc(feat_src).reshape(
-1, self._n_kernels, self._out_feats)
E = graph.number_of_edges()
# compute gaussian weight
gaussian = -0.5 * ((pseudo.reshape(E, 1, self._dim) -
self.mu.data(feat_src.context)
.reshape(1, self._n_kernels, self._dim)) ** 2)
gaussian = gaussian *\
(self.inv_sigma.data(feat_src.context)
.reshape(1, self._n_kernels, self._dim) ** 2)
gaussian = nd.exp(gaussian.sum(axis=-1, keepdims=True)) # (E, K, 1)
graph.edata['w'] = gaussian
graph.update_all(fn.u_mul_e('h', 'w', 'm'), self._reducer('m', 'h'))
rst = graph.dstdata['h'].sum(1)
# residual connection
if self.res_fc is not None:
rst = rst + self.res_fc(feat_dst)
# bias
if self.bias is not None:
rst = rst + self.bias.data(feat_dst.context)
return rst