Message Passing

Message passing module.

class topomodelx.base.message_passing.MessagePassing(aggr_func: Literal['sum', 'mean', 'add'] = 'sum', att: bool = False, initialization: Literal['uniform', 'xavier_uniform', 'xavier_normal'] = 'xavier_uniform', initialization_gain: float = 1.414)

Define message passing.

This class defines message passing through a single neighborhood N, by decomposing it into 2 steps:

1. 🟥 Create messages going from source cells to target cells through N.

2. 🟧 Aggregate messages coming from different sources cells onto each target cell.

This class should not be instantiated directly, but rather inherited through subclasses that effectively define a message passing function.

This class does not have trainable weights, but its subclasses should define these weights.

Parameters:

aggr_funcLiteral[“sum”, “mean”, “add”], default=”sum”

Aggregation function to use.

attbool, default=False

Whether to use attention.

initializationLiteral[“uniform”, “xavier_uniform”, “xavier_normal”], default=”xavier_uniform”

Initialization method for the weights of the layer.

initialization_gainfloat, default=1.414

Gain for the weight initialization.

Methods

add_module(name, module)	Add a child module to the current module.
aggregate(x_message)	Aggregate messages on each target cell.
apply(fn)	Apply fn recursively to every submodule (as returned by .children()) as well as self.
attention(x_source[, x_target])	Compute attention weights for messages.
bfloat16()	Casts all floating point parameters and buffers to bfloat16 datatype.
buffers([recurse])	Return an iterator over module buffers.
children()	Return an iterator over immediate children modules.
compile(*args, **kwargs)	Compile this Module's forward using torch.compile().
cpu()	Move all model parameters and buffers to the CPU.
cuda([device])	Move all model parameters and buffers to the GPU.
double()	Casts all floating point parameters and buffers to double datatype.
eval()	Set the module in evaluation mode.
extra_repr()	Return the extra representation of the module.
float()	Casts all floating point parameters and buffers to float datatype.
forward(x_source, neighborhood[, x_target])	Forward pass.
get_buffer(target)	Return the buffer given by target if it exists, otherwise throw an error.
get_extra_state()	Return any extra state to include in the module's state_dict.
get_parameter(target)	Return the parameter given by target if it exists, otherwise throw an error.
get_submodule(target)	Return the submodule given by target if it exists, otherwise throw an error.
half()	Casts all floating point parameters and buffers to half datatype.
ipu([device])	Move all model parameters and buffers to the IPU.
load_state_dict(state_dict[, strict, assign])	Copy parameters and buffers from state_dict into this module and its descendants.
message(x_source[, x_target])	Construct message from source cells to target cells.
modules([remove_duplicate])	Return an iterator over all modules in the network.
mtia([device])	Move all model parameters and buffers to the MTIA.
named_buffers([prefix, recurse, ...])	Return an iterator over module buffers, yielding both the name of the buffer as well as the buffer itself.
named_children()	Return an iterator over immediate children modules, yielding both the name of the module as well as the module itself.
named_modules([memo, prefix, remove_duplicate])	Return an iterator over all modules in the network, yielding both the name of the module as well as the module itself.
named_parameters([prefix, recurse, ...])	Return an iterator over module parameters, yielding both the name of the parameter as well as the parameter itself.
parameters([recurse])	Return an iterator over module parameters.
register_backward_hook(hook)	Register a backward hook on the module.
register_buffer(name, tensor[, persistent])	Add a buffer to the module.
register_forward_hook(hook, *[, prepend, ...])	Register a forward hook on the module.
register_forward_pre_hook(hook, *[, ...])	Register a forward pre-hook on the module.
register_full_backward_hook(hook[, prepend])	Register a backward hook on the module.
register_full_backward_pre_hook(hook[, prepend])	Register a backward pre-hook on the module.
register_load_state_dict_post_hook(hook)	Register a post-hook to be run after module's load_state_dict() is called.
register_load_state_dict_pre_hook(hook)	Register a pre-hook to be run before module's load_state_dict() is called.
register_module(name, module)	Alias for add_module().
register_parameter(name, param)	Add a parameter to the module.
register_state_dict_post_hook(hook)	Register a post-hook for the state_dict() method.
register_state_dict_pre_hook(hook)	Register a pre-hook for the state_dict() method.
requires_grad_([requires_grad])	Change if autograd should record operations on parameters in this module.
reset_parameters()	Reset learnable parameters.
set_extra_state(state)	Set extra state contained in the loaded state_dict.
set_submodule(target, module[, strict])	Set the submodule given by target if it exists, otherwise throw an error.
share_memory()	See torch.Tensor.share_memory_().
state_dict(*args[, destination, prefix, ...])	Return a dictionary containing references to the whole state of the module.
to(*args, **kwargs)	Move and/or cast the parameters and buffers.
to_empty(*, device[, recurse])	Move the parameters and buffers to the specified device without copying storage.
train([mode])	Set the module in training mode.
type(dst_type)	Casts all parameters and buffers to dst_type.
xpu([device])	Move all model parameters and buffers to the XPU.
zero_grad([set_to_none])	Reset gradients of all model parameters.

__call__

References

[1]

Hajij, Zamzmi, Papamarkou, Miolane, Guzmán-Sáenz, Ramamurthy, Birdal, Dey, Mukherjee, Samaga, Livesay, Walters, Rosen, Schaub. Topological deep learning: going beyond graph data (2023). https://arxiv.org/abs/2206.00606.

[2]

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

aggregate(x_message)

Aggregate messages on each target cell.

A target cell receives messages from several source cells. This function aggregates these messages into a single output feature per target cell.

🟧 This function corresponds to the within-neighborhood aggregation defined in [1]_ and [2]_.

Parameters:

x_messagetorch.Tensor, shape = (…, n_messages, out_channels)

Features associated with each message. One message is sent from a source cell to a target cell.

Returns:

Tensor, shape = (…, n_target_cells, out_channels)

Output features on target cells. Each target cell aggregates messages from several source cells. Target cells with an empty neighborhood receive zero features. Assumes that all target cells have the same rank s.

attention(x_source, x_target=None)

Compute attention weights for messages.

This provides a default attention function to the message-passing scheme.

Alternatively, users can subclass MessagePassing and overwrite the attention method in order to replace it with their own attention mechanism.

The implementation follows [1]_.

Parameters:

x_sourcetorch.Tensor, shape = (n_source_cells, in_channels)

Input features on source cells. Assumes that all source cells have the same rank r.

x_targettorch.Tensor, shape = (n_target_cells, in_channels)

Input features on source cells. Assumes that all source cells have the same rank r.

Returns:

torch.Tensor, shape = (n_messages, 1)

Attention weights: one scalar per message between a source and a target cell.

forward(x_source, neighborhood, x_target=None)

Forward pass.

This implements message passing for a given neighborhood:

• from source cells with input features x_source,

• via neighborhood defining where messages can pass,

• to target cells with input features x_target.

In practice, this will update the features on the target cells.

If not provided, x_target is assumed to be x_source, i.e. source cells send messages to themselves.

The message passing is decomposed into two steps:

1. 🟥 Message: A message \(m_{y \rightarrow x}^{\left(r \rightarrow s\right)}\) travels from a source cell \(y\) of rank r to a target cell \(x\) of rank s through a neighborhood of \(x\), denoted \(\mathcal{N} (x)\), via the message function \(M_\mathcal{N}\):

\[m_{y \rightarrow x}^{\left(r \rightarrow s\right)} = M_{\mathcal{N}}\left(\mathbf{h}_x^{(s)}, \mathbf{h}_y^{(r)}, \Theta \right),\]

where:

• \(\mathbf{h}_y^{(r)}\) are input features on the source cells, called x_source,

• \(\mathbf{h}_x^{(s)}\) are input features on the target cells, called x_target,

• \(\Theta\) are optional parameters (weights) of the message passing function.

Optionally, attention can be applied to the message, such that:

\[m_{y \rightarrow x}^{\left(r \rightarrow s\right)} \leftarrow att(\mathbf{h}_y^{(r)}, \mathbf{h}_x^{(s)}) . m_{y \rightarrow x}^{\left(r \rightarrow s\right)}\]

2. 🟧 Aggregation: Messages are aggregated across source cells \(y\) belonging to the neighborhood \(\mathcal{N}(x)\):

\[m_x^{\left(r \rightarrow s\right)} = \text{AGG}_{y \in \mathcal{N}(x)} m_{y \rightarrow x}^{\left(r\rightarrow s\right)},\]

resulting in the within-neighborhood aggregated message \(m_x^{\left(r \rightarrow s\right)}\).

Details can be found in [1]_ and [2]_.

Parameters:

x_sourceTensor, shape = (…, n_source_cells, in_channels)

Input features on source cells. Assumes that all source cells have the same rank r.

neighborhoodtorch.sparse, shape = (n_target_cells, n_source_cells)

Neighborhood matrix.

x_targetTensor, shape = (…, n_target_cells, in_channels)

Input features on target cells. Assumes that all target cells have the same rank s. Optional. If not provided, x_target is assumed to be x_source, i.e. source cells send messages to themselves.

Returns:

torch.Tensor, shape = (…, n_target_cells, out_channels)

Output features on target cells. Assumes that all target cells have the same rank s.

message(x_source, x_target=None)

Construct message from source cells to target cells.

🟥 This provides a default message function to the message passing scheme.

Alternatively, users can subclass MessagePassing and overwrite the message method in order to replace it with their own message mechanism.

Parameters:

x_sourceTensor, shape = (…, n_source_cells, in_channels)

Input features on source cells. Assumes that all source cells have the same rank r.

x_targetTensor, shape = (…, n_target_cells, in_channels)

Input features on target cells. Assumes that all target cells have the same rank s. Optional. If not provided, x_target is assumed to be x_source, i.e. source cells send messages to themselves.

Returns:

torch.Tensor, shape = (…, n_source_cells, in_channels)

Messages on source cells.

reset_parameters()

Reset learnable parameters.

Notes

This function will be called by subclasses of MessagePassing that have trainable weights.