# tfg.geometry.convolution.graph_convolution.edge_convolution_template

A template for edge convolutions.

This function implements a general edge convolution for graphs of the form $$y_i = \sum_{j \in \mathcal{N}(i)} w_{ij} f(x_i, x_j)$$, where $$\mathcal{N}(i)$$ is the set of vertices in the neighborhood of vertex $$i$$, $$x_i \in \mathbb{R}^C$$ are the features at vertex $$i$$, $$w_{ij} \in \mathbb{R}$$ is the weight for the edge between vertex $$i$$ and vertex $$j$$, and finally $$f(x_i, x_j): \mathbb{R}^{C} \times \mathbb{R}^{C} \to \mathbb{R}^{D}$$ is a user-supplied function.

This template also implements the same general edge convolution described above with a max-reduction instead of a weighted sum.

An example of how this template can be used is for Laplacian smoothing, which is defined as

$$y_i = \frac{1}{|\mathcal{N(i)}|} \sum_{j \in \mathcal{N(i)} } x_j$$

. edge_convolution_template can be used to perform Laplacian smoothing by setting

$$w_{ij} = \frac{1}{|\mathcal{N(i)}|}$$

, edge_function=lambda x, y: y, and reduction='weighted'.

The shorthands used below are V: The number of vertices. C: The number of channels in the input data.

#### Note:

In the following, A1 to An are optional batch dimensions.

data A float tensor with shape [A1, ..., An, V, C].
neighbors A SparseTensor with the same type as data and with shape [A1, ..., An, V, V] representing vertex neighborhoods. The neighborhood of a vertex defines the support region for convolution. The value at neighbors[A1, ..., An, i, j] corresponds to the weight $$w_{ij}$$ above. Each vertex must have at least one neighbor.
sizes An int tensor of shape [A1, ..., An] indicating the true input sizes in case of padding (sizes=None indicates no padding). Note that sizes[A1, ..., An] <= V. If data and neighbors are 2-D, sizes will be ignored. As an example, consider an input consisting of three graphs G0, G1, and G2 with V0, V1, and V2 vertices respectively. The padded input would have the shapes [3, V, C], and [3, V, V] for data and neighbors respectively, where V = max([V0, V1, V2]). The true sizes of each graph will be specified by sizes=[V0, V1, V2] and data[i, :Vi, :] and neighbors[i, :Vi, :Vi] will be the vertex and neighborhood data of graph Gi. The SparseTensor neighbors should have no nonzero entries in the padded regions.
edge_function A callable that takes at least two arguments of vertex features and returns a tensor of vertex features. Y = f(X1, X2, **kwargs), where X1 and X2 have shape [V3, C] and Y must have shape [V3, D], D >= 1.
reduction Either 'weighted' or 'max'. Specifies the reduction over the neighborhood. For 'weighted', the reduction is a weighted sum as shown in the equation above. For 'max' the reduction is a max over features in which case the weights

$$w_{ij}$$

are ignored.

edge_function_kwargs A dict containing any additional keyword arguments to be passed to edge_function.
name A name for this op. Defaults to graph_convolution_edge_convolution_template.

Tensor with shape [A1, ..., An, V, D].

TypeError if the input types are invalid.
ValueError if the input dimensions are invalid.