MultiplicativeLSTMCell

MultiplicativeLSTMCell(in_dims => out_dims;
    use_bias=true, train_state=false, train_memory=false,
    init_bias=nothing, init_recurrent_bias=nothing,
    init_multiplicative_bias=nothing, init_weight=nothing,
    init_recurrent_weight=nothing, init_multiplicative_weight=nothing,
    init_state=zeros32, init_memory=zeros32)

Multiplicative long short term memory cell.

Equations

\[\begin{aligned} \mathbf{m}(t) &= \left( \mathbf{W}_{ih}^{m} \mathbf{x}(t) + \mathbf{b}_{ih}^{m} \right) \circ \left( \mathbf{W}_{hh}^{m} \mathbf{h}(t-1) + \mathbf{b}_{hh}^{m} \right) \\ \hat{\mathbf{h}}(t) &= \mathbf{W}_{ih}^{h} \mathbf{x}(t) + \mathbf{b}_{ih}^{h} + \mathbf{W}_{mh}^{h} \mathbf{m}(t) + \mathbf{b}_{mh}^{h} \\ \mathbf{i}(t) &= \sigma\left( \mathbf{W}_{ih}^{i} \mathbf{x}(t) + \mathbf{b}_{ih}^{i} + \mathbf{W}_{mh}^{i} \mathbf{m}(t) + \mathbf{b}_{mh}^{i} \right) \\ \mathbf{o}(t) &= \sigma\left( \mathbf{W}_{ih}^{o} \mathbf{x}(t) + \mathbf{b}_{ih}^{o} + \mathbf{W}_{mh}^{o} \mathbf{m}(t) + \mathbf{b}_{mh}^{o} \right) \\ \mathbf{f}(t) &= \sigma\left( \mathbf{W}_{ih}^{f} \mathbf{x}(t) + \mathbf{b}_{ih}^{f} + \mathbf{W}_{mh}^{f} \mathbf{m}(t) + \mathbf{b}_{mh}^{f} \right) \\ \mathbf{c}(t) &= \mathbf{f}(t) \circ \mathbf{c}(t-1) + \mathbf{i}(t) \circ \tanh\left( \hat{\mathbf{h}}(t) \right) \\ \mathbf{h}(t) &= \tanh\left( \mathbf{c}(t) \right) \circ \mathbf{o}(t) \end{aligned}\]

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 $\mathbf{b}_{ih}^{m}, \mathbf{b}_{ih}^{h}, \mathbf{b}_{ih}^{i}, \mathbf{b}_{ih}^{o}, \mathbf{b}_{ih}^{f}$. Must be a tuple containing 5 functions. If a single value is passed, it is copied into a 5-element tuple. If set to nothing, weights are initialized from a uniform distribution within [-bound, bound] where bound = inv(sqrt(out_dims)). The functions are applied in order: the first initializes $\mathbf{b}_{ih}^{m}$, the second $\mathbf{b}_{ih}^{h}$, the third $\mathbf{b}_{ih}^{i}$, the fourth $\mathbf{b}_{ih}^{o}$, and the fifth $\mathbf{b}_{ih}^{f}$.
• init_recurrent_bias: Initializer for hidden-to-hidden biases $\mathbf{b}_{hh}^{m}$. Must be a tuple containing 1 function. If a single value is passed, it is used directly. If set to nothing, weights are initialized from a uniform distribution within [-bound, bound] where bound = inv(sqrt(out_dims)).
• init_multiplicative_bias: Initializer for multiplicative-to-hidden biases $\mathbf{b}_{mh}^{h}, \mathbf{b}_{mh}^{i}, \mathbf{b}_{mh}^{o}, \mathbf{b}_{mh}^{f}$. Must be a tuple containing 4 functions. If a single value is passed, it is copied into a 4-element tuple. If set to nothing, weights are initialized from a uniform distribution within [-bound, bound] where bound = inv(sqrt(out_dims)). The functions are applied in order: the first initializes $\mathbf{b}_{mh}^{h}$, the second $\mathbf{b}_{mh}^{i}$, the third $\mathbf{b}_{mh}^{o}$, and the fourth $\mathbf{b}_{mh}^{f}$.
• init_weight: Initializer for input-to-hidden weights $\mathbf{W}_{ih}^{m}, \mathbf{W}_{ih}^{h}, \mathbf{W}_{ih}^{i}, \mathbf{W}_{ih}^{o}, \mathbf{W}_{ih}^{f}$. Must be a tuple containing 5 functions. If a single value is passed, it is copied into a 5-element tuple. If set to nothing, weights are initialized from a uniform distribution within [-bound, bound] where bound = inv(sqrt(out_dims)). The functions are applied in order: the first initializes $\mathbf{W}_{ih}^{m}$, the second $\mathbf{W}_{ih}^{h}$, the third $\mathbf{W}_{ih}^{i}$, the fourth $\mathbf{W}_{ih}^{o}$, and the fifth $\mathbf{W}_{ih}^{f}$.
• init_recurrent_weight: Initializer for hidden-to-hidden weights $\mathbf{W}_{hh}^{m}$. Must be a tuple containing 1 function. If a single value is passed, it is used directly. If set to nothing, weights are initialized from a uniform distribution within [-bound, bound] where bound = inv(sqrt(out_dims)).
• init_multiplicative_weight: Initializer for multiplicative-to-hidden weights $\mathbf{W}_{mh}^{h}, \mathbf{W}_{mh}^{i}, \mathbf{W}_{mh}^{o}, \mathbf{W}_{mh}^{f}$. Must be a tuple containing 4 functions. If a single value is passed, it is copied into a 4-element tuple. If set to nothing, weights are initialized from a uniform distribution within [-bound, bound] where bound = inv(sqrt(out_dims)). The functions are applied in order: the first initializes $\mathbf{W}_{mh}^{h}$, the second $\mathbf{W}_{mh}^{i}$, the third $\mathbf{W}_{mh}^{o}$, and the fourth $\mathbf{W}_{mh}^{f}$.
• init_state: Initializer for hidden state. Default set to zeros32.
• init_memory: Initializer for memory. Default set to zeros32.

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: Input-to-hidden weights $\{ \mathbf{W}_{ih}^{m}, \mathbf{W}_{ih}^{h}, \mathbf{W}_{ih}^{i}, \mathbf{W}_{ih}^{o}, \mathbf{W}_{ih}^{f} \}$
• weight_hh: Hidden-to-hidden weights $\{ \mathbf{W}_{hh}^{m} \}$
• weight_mh: Multiplicative-to-hidden weights $\{ \mathbf{W}_{mh}^{h}, \mathbf{W}_{mh}^{i}, \mathbf{W}_{mh}^{o}, \mathbf{W}_{mh}^{f} \}$
• bias_ih: Input-to-hidden biases (if use_bias=true) $\{ \mathbf{b}_{ih}^{m}, \mathbf{b}_{ih}^{h}, \mathbf{b}_{ih}^{i}, \mathbf{b}_{ih}^{o}, \mathbf{b}_{ih}^{f} \}$
• bias_hh: Hidden-to-hidden biases (if use_bias=true) $\{ \mathbf{b}_{hh}^{m} \}$
• bias_mh: Multiplicative-to-hidden biases (if use_bias=true) $\{ \mathbf{b}_{mh}^{h}, \mathbf{b}_{mh}^{i}, \mathbf{b}_{mh}^{o}, \mathbf{b}_{mh}^{f} \}$
• 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