LEMCell

LEMCell(in_dims => out_dims;
    use_bias=true, train_state=false, train_memory=false,
    init_bias=nothing, init_recurrent_bias=nothing,
    init_weight=nothing, init_recurrent_weight=nothing,
    init_state=zeros32, init_memory=zeros32, dt=1.0)

Long expressive memory unit.

Equations

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

Arguments

• in_dims: Input Dimension
• out_dims: Output (Hidden State & Memory) Dimension

Keyword Arguments

• use_bias: Flag to use bias in the computation. Default set to true.
• train_state: Flag to set the initial hidden state as trainable. Default set to false.
• train_memory: Flag to set the initial memory state as trainable. Default set to false.
• init_bias: Initializer for input-to-hidden biases $\mathbf{b}_{ih}^{1}, \mathbf{b}_{ih}^{2}, \mathbf{b}_{ih}^{c}, \mathbf{b}_{ih}^{h}$. Must be a tuple of 4 functions, e.g., (glorot_uniform, kaiming_uniform, lecun_normal, zeros). If a single function is passed, it is expanded to a 4-element tuple. If set to nothing, biases are initialized from a uniform distribution within [-bound, bound] where bound = inv(sqrt(out_dims)). Default is nothing.
• init_recurrent_bias: Initializer for hidden-to-hidden biases $\mathbf{b}_{hh}^{1}, \mathbf{b}_{hh}^{2}, \mathbf{b}_{hh}^{c}, \mathbf{b}_{hh}^{h}$. Must be a tuple of 3 functions. If a single function is passed, it is expanded to a 3-element tuple. If set to nothing, biases are initialized from a uniform distribution within [-bound, bound] where bound = inv(sqrt(out_dims)). Default is nothing.
• init_cell_bias: Initializer for bias $\mathbf{b}_{ch}$. If set to nothing, weights are initialized from a uniform distribution within [-bound, bound] where bound = inv(sqrt(out_dims)). Default is nothing.
• init_weight: Initializer for input-to-hidden weights $\mathbf{W}_{ih}^{1}, \mathbf{W}_{ih}^{2}, \mathbf{W}_{ih}^{c}, \mathbf{W}_{ih}^{h}$. Must be a tuple of 4 functions. If a single function is passed, it is expanded to a 4-element tuple. If set to nothing, weights are initialized from a uniform distribution within [-bound, bound] where bound = inv(sqrt(out_dims)). Default is nothing.
• init_recurrent_weight: Initializer for hidden-to-hidden weights $\mathbf{W}_{hh}^{1}, \mathbf{W}_{hh}^{2}, \mathbf{W}_{hh}^{c}, \mathbf{W}_{hh}^{h}$. Must be a tuple of 3 functions. If a single function is passed, it is expanded to a 3-element tuple. If set to nothing, weights are initialized from a uniform distribution within [-bound, bound] where bound = inv(sqrt(out_dims)). Default is nothing.
  • init_cell_weight: Initializer for input to hidden weight $\mathbf{W}_{ch}$. If set to
  nothing, weights are initialized from a uniform distribution within [-bound, bound] where bound = inv(sqrt(out_dims)). Default is nothing.
• init_state: Initializer for hidden state. Default set to zeros32.
• init_memory: Initializer for memory. Default set to zeros32.
• dt: timestep. Defaul is 1.0.

Inputs

• Case 1a: Only a single input x of shape (in_dims, batch_size), train_state is set to false, train_memory is set to false - Creates a hidden state using init_state, hidden memory using init_memory and proceeds to Case 2.
• Case 1b: Only a single input x of shape (in_dims, batch_size), train_state is set to true, train_memory is set to false - Repeats hidden_state vector from the parameters to match the shape of x, creates hidden memory using init_memory and proceeds to Case 2.
• Case 1c: Only a single input x of shape (in_dims, batch_size), train_state is set to false, train_memory is set to true - Creates a hidden state using init_state, repeats the memory vector from parameters to match the shape of x and proceeds to Case 2.
• Case 1d: Only a single input x of shape (in_dims, batch_size), train_state is set to true, train_memory is set to true - Repeats the hidden state and memory vectors from the parameters to match the shape of x and proceeds to Case 2.
• Case 2: Tuple (x, (h, c)) is provided, then the output and a tuple containing the updated hidden state and memory is returned.

Returns

• Tuple Containing

  • Output $h_{new}$ of shape (out_dims, batch_size)
  • Tuple containing new hidden state $h_{new}$ and new memory $c_{new}$

• Updated model state

Parameters

• weight_ih: Concatenated weights mapping from input to internal units $\{ \mathbf{W}_{ih}^{1}, \mathbf{W}_{ih}^{2}, \mathbf{W}_{ih}^{c}, \mathbf{W}_{ih}^{h} \}$ The functions provided in init_weight are applied in order: the first initializes $\mathbf{W}_{ih}^{1}$, the second $\mathbf{W}_{ih}^{2}$, the third $\mathbf{W}_{ih}^{c}$, and the fourth $\mathbf{W}_{ih}^{h}$.
• weight_hh: Concatenated weights mapping from hidden state to internal units $\{ \mathbf{W}_{hh}^{1}, \mathbf{W}_{hh}^{2}, \mathbf{W}_{hh}^{c} \}$ The functions provided in init_recurrent_weight are applied in order: the first initializes $\mathbf{W}_{hh}^{1}$, the second $\mathbf{W}_{hh}^{2}$, and the third $\mathbf{W}_{hh}^{c}$.
• weight_ch: Weights to map from cell space $\mathbf{W}_{ch}$.
• bias_ih: Concatenated input-to-hidden bias vectors (not present if use_bias=false) $\{ \mathbf{b}_{ih}^{1}, \mathbf{b}_{ih}^{2}, \mathbf{b}_{ih}^{c}, \mathbf{b}_{ih}^{h} \}$ The functions provided in init_bias are applied in order: the first initializes $\mathbf{b}_{ih}^{1}$, the second $\mathbf{b}_{ih}^{2}$, the third $\mathbf{b}_{ih}^{c}$, and the fourth $\mathbf{b}_{ih}^{h}$.
• bias_hh: Concatenated hidden-to-hidden bias vectors (not present if use_bias=false) $\{ \mathbf{b}_{hh}^{1}, \mathbf{b}_{hh}^{2}, \mathbf{b}_{hh}^{c} \}$ The functions provided in init_recurrent_bias are applied in order: the first initializes $\mathbf{b}_{hh}^{1}$, the second $\mathbf{b}_{hh}^{2}$, and the third $\mathbf{b}_{hh}^{c}$.
• bias_ch: Bias vector for the cell-hidden connection $\mathbf{b}_{ch}$ (not present if use_bias=false)
• hidden_state: Initial hidden state vector (not present if train_state=false)
• memory: Initial memory vector (not present if train_memory=false)

States

• rng: Controls the randomness (if any) in the initial state generation