CCXN_Layer

Implementation of a simplified, convolutional version of CCXN layer from Hajij et. al: Cell Complex Neural Networks.

class topomodelx.nn.cell.ccxn_layer.CCXNLayer(in_channels_0, in_channels_1, in_channels_2, att: bool = False, **kwargs)

Layer of a Convolutional Cell Complex Network (CCXN).

Implementation of a simplified version of the CCXN layer proposed in [1].

This layer is composed of two convolutional layers: 1. A convolutional layer sending messages from nodes to nodes. 2. A convolutional layer sending messages from edges to faces. Optionally, attention mechanisms can be used.

Parameters:

in_channels_0int

Dimension of input features on nodes (0-cells).

in_channels_1int

Dimension of input features on edges (1-cells).

in_channels_2int

Dimension of input features on faces (2-cells).

attbool, default=False

Whether to use attention.

**kwargsoptional

Additional arguments for the modules of the CCXN layer.

References

[1]

Hajij, Istvan, Zamzmi. Cell complex neural networks. Topological data analysis and beyond workshop at NeurIPS 2020. https://arxiv.org/pdf/2010.00743.pdf

[2]

Papillon, Sanborn, Hajij, Miolane. Equations of topological neural networks (2023). awesome-tnns/awesome-tnns

[3]

Papillon, Sanborn, Hajij, Miolane. Architectures of topological deep learning: a survey on topological neural networks (2023). https://arxiv.org/abs/2304.10031.

forward(x_0, x_1, adjacency_0, incidence_2_t, x_2=None)

Forward pass.

The forward pass was initially proposed in [1]_. Its equations are given in [2]_ and graphically illustrated in [3]_.

The forward pass of this layer is composed of two steps.

1. The convolution from nodes to nodes is given by an adjacency message passing scheme (AMPS):

\[\begin{split}\begin{align*} &🟥 \quad m_{y \rightarrow \{z\} \rightarrow x}^{(0 \rightarrow 1 \rightarrow 0)} = M_{\mathcal{L}_\uparrow}(h_x^{(0)}, h_y^{(0)}, \Theta^{(y \rightarrow x)})\\ &🟧 \quad m_x^{(0 \rightarrow 1 \rightarrow 0)} = \text{AGG}_{y \in \mathcal{L}_\uparrow(x)}(m_{y \rightarrow \{z\} \rightarrow x}^{0 \rightarrow 1 \rightarrow 0})\\ &🟩 \quad m_x^{(0)} = m_x^{(0 \rightarrow 1 \rightarrow 0)}\\ &🟦 \quad h_x^{t+1,(0)} = U^{t}(h_x^{(0)}, m_x^{(0)}) \end{align*}\end{split}\]

2. The convolution from edges to faces is given by cohomology message passing scheme, using the coboundary neighborhood:

\[\begin{split}\begin{align*} &🟥 \quad m_{y \rightarrow x}^{(r' \rightarrow r)} = M^t_{\mathcal{C}}(h_{x}^{t,(r)}, h_y^{t,(r')}, x, y)\\ &🟧 \quad m_x^{(r' \rightarrow r)} = \text{AGG}_{y \in \mathcal{C}(x)} m_{y \rightarrow x}^{(r' \rightarrow r)}\\ &🟩 \quad m_x^{(r)} = m_x^{(r' \rightarrow r)}\\ &🟦 \quad h_{x}^{t+1,(r)} = U^{t,(r)}(h_{x}^{t,(r)}, m_{x}^{(r)}) \end{align*}\end{split}\]

Parameters:

x_0torch.Tensor, shape = (n_0_cells, channels)

Input features on the nodes of the cell complex.

x_1torch.Tensor, shape = (n_1_cells, channels)

Input features on the edges of the cell complex.

adjacency_0torch.sparse, shape = (n_0_cells, n_0_cells)

Neighborhood matrix mapping nodes to nodes (A_0_up).

incidence_2_ttorch.sparse, shape = (n_2_cells, n_1_cells)

Neighborhood matrix mapping edges to faces (B_2^T).

x_2torch.Tensor, shape = (n_2_cells, channels)

Input features on the faces of the cell complex. Optional, only required if attention is used between edges and faces.

Returns:

torch.Tensor, shape = (1, num_classes)

Output prediction on the entire cell complex.