Algorithm to recursively generate a hierarchical modular connectivity mask using PyTorch

hierarchical_modular_mask.py

"""A function to generate a modular connectivity mask.

You can use it to mask a weight matrix, and then mask the gradients using a hook to fully disable those connections.

    self.hh = Parameter(torch.normal(...))

    mask = hierarchical_modular_mask(200, level=3, density=0.05, scale=2.0)
    self.hh.data *= mask
    self.hh.register_hook(lambda grad: grad * mask)
"""

def hierarchical_modular_mask(
    size: int,
    *,
    level: int = 3,
    density: float = 0.05,
    scale: float = 2.0,
    dale_mask: Optional[Tensor] = None,
    inhibitory_distance: int = 0
):
    """Generate a connectivity matrix mask of hierarchical modules.
    
    Modules are densely connected locally, with increasing sparsity to nodes in more distal modules. If a
    `dale_mask` is given, the inhibitory units (0 values in the mask) will be disabled for all but the
    immediately local module. This gives a "local inhibition" effect, with long-range excitatory 
    connections. If you want more distal inhibition, you can configure it with the `inhibitory_distance`
    argument.
    
    Args:
        size:                   The side-length of the square mask.
        level:                  The number of hierarchical levels to generate.
        density:                The density of the highest level of the hierarchy (i.e.g the off-diagonal regions).
        scale:                  The scaling factor for the density at each subsequent level of the hierarchy.
        dale_mask:              A tensor of shape (size), specifying a 1 for positive units and a 0 for negative units.
        inhibitory_distance:    A number specifying the number of hierarchical steps inhibitory connections may span.
    
    Returns:
        A (size, size) mask matrix of 1's and 0's
    
    ref: https://www.nature.com/articles/srep22057
    """
    assert inhibitory_distance <= level, "inhibitory_distance cannot be larger than the total number of hierarchical levels"
    first_half = math.ceil(size/2)
    second_half = math.floor(size/2)
    
    bg = (torch.rand((size, size)) <= density).float()
    blank_diag = 1 - torch.block_diag(
        torch.ones(first_half, first_half),
        torch.ones(second_half, second_half),
    )

    if level > 0:
        inner_diag = torch.zeros(size, size) + torch.block_diag(
            hierarchical_modular_mask(
                first_half, level-1, density * scale, scale, dale_mask[:first_half] if dale_mask is not None else dale_mask
            ),
            hierarchical_modular_mask(
                second_half, level-1, density * scale, scale, dale_mask[first_half:] if dale_mask is not None else dale_mask
            ),
        )
    else:
        inner_diag = torch.zeros(size, size) + torch.block_diag(
            (torch.rand(first_half, first_half) <= density * scale).float(),
            (torch.rand(second_half, second_half) <= density * scale).float(),
        )
        
   if dale_mask is not None and level > inhibitory_distance:
        # Disable non-local inhibitory connections
        bg *= dale_mask

    mask = (bg * blank_diag) + inner_diag
    return mask

Result:

Now with Dale's Law/local inhibition. Columns with blue are inhibitory, and do not have any synapses outside their inner local module: