Simplicial Attention Network (SAN) Layer.

class topomodelx.nn.simplicial.san_layer.SANConv(in_channels, out_channels, n_filters, initialization: Literal['xavier_uniform', 'xavier_normal'] = 'xavier_uniform')

Simplicial Attention Network (SAN) Convolution from [1].

Parameters:

in_channelsint

Number of input channels.

out_channelsint

Number of output channels.

n_filtersint

Number of simplicial filters.

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

Weight initialization method.

Methods

add_module(name, module)	Adds a child module to the current module.
aggregate(x_message)	Aggregate messages on each target cell.
apply(fn)	Applies 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])	Returns an iterator over module buffers.
children()	Returns an iterator over immediate children modules.
cpu()	Moves all model parameters and buffers to the CPU.
cuda([device])	Moves all model parameters and buffers to the GPU.
double()	Casts all floating point parameters and buffers to double datatype.
eval()	Sets the module in evaluation mode.
extra_repr()	Set the extra representation of the module
float()	Casts all floating point parameters and buffers to float datatype.
forward(x_source, neighborhood)	Forward pass.
get_buffer(target)	Returns the buffer given by target if it exists, otherwise throws an error.
get_extra_state()	Returns any extra state to include in the module's state_dict.
get_parameter(target)	Returns the parameter given by target if it exists, otherwise throws an error.
get_submodule(target)	Returns the submodule given by target if it exists, otherwise throws an error.
half()	Casts all floating point parameters and buffers to half datatype.
ipu([device])	Moves all model parameters and buffers to the IPU.
load_state_dict(state_dict[, strict])	Copies 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()	Returns an iterator over all modules in the network.
named_buffers([prefix, recurse, ...])	Returns an iterator over module buffers, yielding both the name of the buffer as well as the buffer itself.
named_children()	Returns 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])	Returns 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, ...])	Returns an iterator over module parameters, yielding both the name of the parameter as well as the parameter itself.
parameters([recurse])	Returns an iterator over module parameters.
register_backward_hook(hook)	Registers a backward hook on the module.
register_buffer(name, tensor[, persistent])	Adds a buffer to the module.
register_forward_hook(hook, *[, prepend, ...])	Registers a forward hook on the module.
register_forward_pre_hook(hook, *[, ...])	Registers a forward pre-hook on the module.
register_full_backward_hook(hook[, prepend])	Registers a backward hook on the module.
register_full_backward_pre_hook(hook[, prepend])	Registers a backward pre-hook on the module.
register_load_state_dict_post_hook(hook)	Registers a post hook to be run after module's load_state_dict is called.
register_module(name, module)	Alias for add_module().
register_parameter(name, param)	Adds a parameter to the module.
register_state_dict_pre_hook(hook)	These hooks will be called with arguments: self, prefix, and keep_vars before calling state_dict on self.
requires_grad_([requires_grad])	Change if autograd should record operations on parameters in this module.
reset_parameters()	Reset learnable parameters.
set_extra_state(state)	This function is called from load_state_dict() to handle any extra state found within the state_dict.
share_memory()	See torch.Tensor.share_memory_()
state_dict(*args[, destination, prefix, ...])	Returns a dictionary containing references to the whole state of the module.
to(*args, **kwargs)	Moves and/or casts the parameters and buffers.
to_empty(*, device)	Moves the parameters and buffers to the specified device without copying storage.
train([mode])	Sets the module in training mode.
type(dst_type)	Casts all parameters and buffers to dst_type.
update(x_message_on_target)	Update embeddings on each cell (step 4).
xpu([device])	Moves all model parameters and buffers to the XPU.
zero_grad([set_to_none])	Sets gradients of all model parameters to zero.

__call__

References

[1]

Giusti, Battiloro, Di Lorenzo, Sardellitti and Barbarossa. Simplicial attention neural networks (2022). https://arxiv.org/abs/2203.07485.

[2]

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

forward(x_source, neighborhood)

Forward pass.

This implements message passing: - from source cells with input features x_source, - via neighborhood defining where messages can pass, - to target cells, which are the same source cells.

In practice, this will update the features on the target cells [2]_.

\[\begin{split}\begin{align*} &🟥 \quad m_{y \rightarrow \{z\} \rightarrow x}^{u,(1 \rightarrow 2 \rightarrow 1)} = ((L_{\uparrow,1} \odot \operatorname{att}(h_z^{t,(2)}, h_y^{t,(1)}))^u)\_{xy} \cdot h_y^{t,(1)} \cdot \Theta^{t,u}\\ &🟥 \quad m_{y \rightarrow \{z\} \rightarrow x}^{d,(1 \rightarrow 0 \rightarrow 1)} = ((L_{\downarrow,1} \odot \operatorname{att}(h_z^{t,(0)}, h_y^{t,(1)}))^d)\_{xy} \cdot h_y^{t,(1)} \cdot \Theta^{t,d}\\ &🟥 \quad m^{p,(1 \rightarrow 1)}\_{y \rightarrow x} = ((1-wH_1)^p)\_{xy} \cdot h_y^{t,(1)} \cdot \Theta^{t,p}\\ &🟧 \quad m_{x}^{u,(1 \rightarrow 2 \rightarrow 1)} = \sum_{y \in \mathcal{L}\_\uparrow(x)} m_{y \rightarrow \{z\} \rightarrow x}^{u,(1 \rightarrow 2 \rightarrow 1)}\\ &🟧 \quad m_{x}^{d,(1 \rightarrow 0 \rightarrow 1)} = \sum_{y \in \mathcal{L}\downarrow(X)} m_{y \rightarrow \{z\} \rightarrow x}^{d,(1 \rightarrow 0 \rightarrow 1)}\\ &🟧 \quad m^{p,(1 \rightarrow 1)}\_{x} = m^{p,(1 \rightarrow 1)}\_{x \rightarrow x}\\ &🟩 \quad m_x^{(1)} = \sum_{p=1}^P m_x^{p,(1 \rightarrow 1)} + \sum_{u=1}^{U} m_{x}^{u,(1 \rightarrow 2 \rightarrow 1)} + \sum_{d=1}^{D} m_{x}^{d,(1 \rightarrow 0 \rightarrow 1)}\\ &🟦 \quad h_x^{t+1, (1)} = \sigma(m_x^{(1)}) \end{align*}\end{split}\]

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.

Returns:

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

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

class topomodelx.nn.simplicial.san_layer.SANLayer(in_channels, out_channels, n_filters: int = 2)

Implementation of the Simplicial Attention Network (SAN) Layer proposed in [1]_.

Parameters:

in_channelsint

Number of input channels.

out_channelsint

Number of output channels.

n_filtersint, default = 2

Approximation order.

Methods

add_module(name, module)	Adds a child module to the current module.
apply(fn)	Applies fn recursively to every submodule (as returned by .children()) as well as self.
bfloat16()	Casts all floating point parameters and buffers to bfloat16 datatype.
buffers([recurse])	Returns an iterator over module buffers.
children()	Returns an iterator over immediate children modules.
cpu()	Moves all model parameters and buffers to the CPU.
cuda([device])	Moves all model parameters and buffers to the GPU.
double()	Casts all floating point parameters and buffers to double datatype.
eval()	Sets the module in evaluation mode.
extra_repr()	Set the extra representation of the module
float()	Casts all floating point parameters and buffers to float datatype.
forward(x, laplacian_up, laplacian_down, ...)	Forward pass of the SAN Layer.
get_buffer(target)	Returns the buffer given by target if it exists, otherwise throws an error.
get_extra_state()	Returns any extra state to include in the module's state_dict.
get_parameter(target)	Returns the parameter given by target if it exists, otherwise throws an error.
get_submodule(target)	Returns the submodule given by target if it exists, otherwise throws an error.
half()	Casts all floating point parameters and buffers to half datatype.
ipu([device])	Moves all model parameters and buffers to the IPU.
load_state_dict(state_dict[, strict])	Copies parameters and buffers from state_dict into this module and its descendants.
modules()	Returns an iterator over all modules in the network.
named_buffers([prefix, recurse, ...])	Returns an iterator over module buffers, yielding both the name of the buffer as well as the buffer itself.
named_children()	Returns 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])	Returns 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, ...])	Returns an iterator over module parameters, yielding both the name of the parameter as well as the parameter itself.
parameters([recurse])	Returns an iterator over module parameters.
register_backward_hook(hook)	Registers a backward hook on the module.
register_buffer(name, tensor[, persistent])	Adds a buffer to the module.
register_forward_hook(hook, *[, prepend, ...])	Registers a forward hook on the module.
register_forward_pre_hook(hook, *[, ...])	Registers a forward pre-hook on the module.
register_full_backward_hook(hook[, prepend])	Registers a backward hook on the module.
register_full_backward_pre_hook(hook[, prepend])	Registers a backward pre-hook on the module.
register_load_state_dict_post_hook(hook)	Registers a post hook to be run after module's load_state_dict is called.
register_module(name, module)	Alias for add_module().
register_parameter(name, param)	Adds a parameter to the module.
register_state_dict_pre_hook(hook)	These hooks will be called with arguments: self, prefix, and keep_vars before calling state_dict on self.
requires_grad_([requires_grad])	Change if autograd should record operations on parameters in this module.
reset_parameters()	Reset learnable parameters.
set_extra_state(state)	This function is called from load_state_dict() to handle any extra state found within the state_dict.
share_memory()	See torch.Tensor.share_memory_()
state_dict(*args[, destination, prefix, ...])	Returns a dictionary containing references to the whole state of the module.
to(*args, **kwargs)	Moves and/or casts the parameters and buffers.
to_empty(*, device)	Moves the parameters and buffers to the specified device without copying storage.
train([mode])	Sets the module in training mode.
type(dst_type)	Casts all parameters and buffers to dst_type.
xpu([device])	Moves all model parameters and buffers to the XPU.
zero_grad([set_to_none])	Sets gradients of all model parameters to zero.

__call__

Notes

Architecture proposed for r-simplex (r>0) classification on simplicial complices.

forward(x, laplacian_up, laplacian_down, projection_mat)

Forward pass of the SAN Layer.

\[\mathcal N = \{\mathcal N_1, \mathcal N_2,...,\mathcal N_{2p+1}\} = \{A_{\uparrow, r}, A_{\downarrow, r}, A_{\uparrow, r}^2, A_{\downarrow, r}^2,...,A_{\uparrow, r}^p, A_{\downarrow, r}^p, Q_r\},\]

\[\begin{split}\begin{align*} &🟥\quad m_{(y \rightarrow x),k}^{(r)} = \alpha_k(h_x^t,h_y^t) = a_k(h_x^{t}, h_y^{t}) \cdot \psi_k^t(h_x^{t})\quad \forall \mathcal N_k \in \mathcal{N}\\ &🟧\quad m_{x,k}^{(r)} = \bigoplus_{y \in \mathcal{N}_k(x)} m^{(r)}_{(y \rightarrow x),k}\\ &🟩\quad m_{x}^{(r)} = \bigotimes_{\mathcal{N}_k\in\mathcal N}m_{x,k}^{(r)}\\ &🟦\quad h_x^{t+1,(r)} = \phi^{t}(h_x^t, m_{x}^{(r)}) \end{align*}\end{split}\]

Parameters:

xtorch.Tensor, shape = (…, n_cells, in_channels)

Input tensor.

laplacian_uptorch.Tensor

laplacian_downtorch.Tensor

The up- and down-laplacians of the simplicial complex.

projection_mattorch.Tensor

The projection matrix used.

Returns:

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

Output tensor.

reset_parameters() → None

Reset learnable parameters.