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)	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)	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.
update(x_message_on_target)	Update embeddings on each cell (step 4).
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]

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)	Add a child module to the current module.
apply(fn)	Apply 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])	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, laplacian_up, laplacian_down, ...)	Forward pass of the SAN Layer.
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.
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__

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.