LEM

LEM(input_size => hidden_size, [dt];
    return_state=false, init_kernel = glorot_uniform,
    init_recurrent_kernel = glorot_uniform, bias = true)

Long expressive memory network ^[Rusch2022]. See LEMCell for a layer that processes a single sequence.

Arguments

• input_size => hidden_size: input and inner dimension of the layer.
• dt: timestep. Defaul is 1.0.

Keyword arguments

• init_kernel: initializer for the input to hidden weights. Default is glorot_uniform.
• init_recurrent_kernel: initializer for the hidden to hidden weights. Default is glorot_uniform.
• bias: include a bias or not. Default is true.
• return_state: Option to return the last state together with the output. Default is false.

Equations

\[\begin{aligned} \boldsymbol{\Delta t}(t) &= \Delta \hat{t} \, \hat{\sigma} \left( \mathbf{W}^{1}_{hh} \mathbf{h}(t-1) + \mathbf{W}^{1}_{ih} \mathbf{x}(t) + \mathbf{b}^{1} \right), \\ \overline{\boldsymbol{\Delta t}}(t) &= \Delta \hat{t} \, \hat{\sigma} \left( \mathbf{W}^{2}_{hh} \mathbf{h}(t-1) + \mathbf{W}^{2}_{ih} \mathbf{x}(t) + \mathbf{b}^{2} \right), \\ \mathbf{z}(t) &= \left( 1 - \boldsymbol{\Delta t}(t) \right) \odot \mathbf{z}(t-1) + \boldsymbol{\Delta t}(t) \odot \sigma \left( \mathbf{W}^{z}_{hh} \mathbf{h}(t-1) + \mathbf{W}^{z}_{ih} \mathbf{x}(t) + \mathbf{b}^{z} \right), \\ \mathbf{h}(t) &= \left( 1 - \boldsymbol{\Delta t}(t) \right) \odot \mathbf{h}(t-1) + \boldsymbol{\Delta t}(t) \odot \sigma \left( \mathbf{W}^{h}_{zh} \mathbf{z}(t) + \mathbf{W}^{h}_{ih} \mathbf{x}(t) + \mathbf{b}^{h} \right) \end{aligned}\]

Forward

lem(inp, (state, zstate))
lem(inp)

Arguments

• inp: The input to the LEM. It should be a vector of size input_size x len or a matrix of size input_size x len x batch_size.
• (state, cstate): A tuple containing the hidden and cell states of the LEM. They should be vectors of size hidden_size or matrices of size hidden_size x batch_size. If not provided, they are assumed to be vectors of zeros, initialized by Flux.initialstates.

Returns

• New hidden states new_states as an array of size hidden_size x len x batch_size. When return_state = true it returns a tuple of the hidden stats new_states and the last state of the iteration.