GMMConv

class dgl.nn.mxnet.conv.GMMConv(in_feats, out_feats, dim, n_kernels, aggregator_type='sum', residual=False, bias=True, allow_zero_in_degree=False)[source]

Bases: mxnet.gluon.block.Block

Gaussian Mixture Model Convolution layer from Geometric Deep Learning on Graphs and Manifolds using Mixture Model CNNs

\[ \begin{align}\begin{aligned}u_{ij} &= f(x_i, x_j), x_j \in \mathcal{N}(i)\\w_k(u) &= \exp\left(-\frac{1}{2}(u-\mu_k)^T \Sigma_k^{-1} (u - \mu_k)\right)\\h_i^{l+1} &= \mathrm{aggregate}\left(\left\{\frac{1}{K} \sum_{k}^{K} w_k(u_{ij}), \forall j\in \mathcal{N}(i)\right\}\right)\end{aligned}\end{align} \]

where \(u\) denotes the pseudo-coordinates between a vertex and one of its neighbor, computed using function \(f\), \(\Sigma_k^{-1}\) and \(\mu_k\) 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 \(x_i\).

  • out_feats (int) – Number of output features; i.e., the number of dimensions of \(h_i^{(l+1)}\).

  • dim (int) – Dimensionality of pseudo-coordinte; i.e, the number of dimensions of \(u_{ij}\).

  • n_kernels (int) – Number of kernels \(K\).

  • 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.heterograph({('_N', '_E', '_N'):(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)>
forward(graph, feat, pseudo)[source]

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 \((N, D_{in})\) where \(D_{in}\) is size of input feature, \(N\) is the number of nodes. If a pair of tensors are given, the pair must contain two tensors of shape \((N_{in}, D_{in_{src}})\) and \((N_{out}, D_{in_{dst}})\).

  • pseudo (mxnet.NDArray) – The pseudo coordinate tensor of shape \((E, D_{u})\) where \(E\) is the number of edges of the graph and \(D_{u}\) is the dimensionality of pseudo coordinate.

Returns

The output feature of shape \((N, D_{out})\) where \(D_{out}\) is the output feature size.

Return type

mxnet.NDArray

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.