LSTMProjectedLayer - Long short-term memory (LSTM) projected layer for recurrent neural network

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Long short-term memory (LSTM) projected layer for recurrent neural network (RNN)

Since R2022b

Description

An LSTM projected layer is an RNN layer that learns long-term dependencies between time steps in time-series and sequence data using projected learnable weights.

To compress a deep learning network, you can use projected layers. A projected layer is a type of deep learning layer that enables compression by reducing the number of stored learnable parameters. The layer introduces learnable projector matrices_Q_, replaces multiplications of the form Wx, where W is a learnable matrix, with the multiplication WQQ⊤x, and stores Q and W′=WQ instead of storing W. Projecting x into a lower dimensional space using Q typically requires less memory to store the learnable parameters and can have similarly strong prediction accuracy.

Reducing the number of learnable parameters by projecting an LSTM layer rather than reducing the number of hidden units of the LSTM layer maintains the output size of the layer and, in turn, the sizes of the downstream layers, which can result in better prediction accuracy.

Creation

Syntax

Description

Properties

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Projected LSTM

Number of hidden units (also known as the hidden size), specified as a positive integer.

The number of hidden units corresponds to the amount of information that the layer remembers between time steps (the hidden state). The hidden state can contain information from all the previous time steps, regardless of the sequence length. If the number of hidden units is too large, then the layer can overfit to the training data. The hidden state does not limit the number of time steps that the layer processes in an iteration.

The layer outputs data with NumHiddenUnits channels.

To set this property, use the numHiddenUnits argument when you create the LSTMProjectedLayer object. After you create aLSTMProjectedLayer object, this property is read-only.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

Output projector size, specified as a positive integer.

The LSTM layer operation uses four matrix multiplications of the form Rht−1, where R denotes the recurrent weights and_ht_ denotes the hidden state (or, equivalently, the layer output) at time step t.

The LSTM projected layer operation instead uses multiplications of the from RQoQo⊤ht−1, where Qo is aNumHiddenUnits-by-OutputProjectorSize matrix known as the output projector. The layer uses the same projector_Qo_ for each of the four multiplications.

To perform the four multiplications of the form Rht−1, an LSTM layer stores four recurrent weights matrices_R_, which necessitates storing 4*NumHiddenUnits^2 learnable parameters. By instead storing the4*NumHiddenUnits-by-OutputProjectorSize matrix R′=RQo and Qo, an LSTM projected layer can perform the multiplication RQoQo⊤ht−1 and store only 5*NumHiddenUnits*OutputProjectorSize learnable parameters.

To set this property, use the outputProjectorSize argument when you create the LSTMProjectedLayer object. After you create aLSTMProjectedLayer object, this property is read-only.

Tip

To ensure that RQoQo⊤ht−1 requires fewer learnable parameters, set theOutputProjectorSize property to a value less than4*NumHiddenUnits/5.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

Input projector size, specified as a positive integer.

The LSTM layer operation uses four matrix multiplications of the form Wxt, where W denotes the input weights and_xt_ denotes the layer input at time step_t_.

The LSTM projected layer operation instead uses multiplications of the from WQiQi⊤xt, where Qi is anInputSize-by-InputProjectorSize matrix known as the input projector. The layer uses the same projector_Qi_ for each of the four multiplications.

To perform the four multiplications of the form Wxt, an LSTM layer stores four weight matrices W, which necessitates storing 4*NumHiddenUnits*InputSize learnable parameters. By instead storing the4*NumHiddenUnits-by-InputProjectorSize matrix W′=WQi and Qi, an LSTM projected layer can perform the multiplication WQiQi⊤xt and store only(4*NumHiddenUnits+InputSize)*InputProjectorSize learnable parameters.

To set this property, use the inputProjectorSize argument when you create the LSTMProjectedLayer object. After you create aLSTMProjectedLayer object, this property is read-only.

Tip

To ensure that WQiQi⊤xt requires fewer learnable parameters, set theInputProjectorSize property to a value less than4*NumHiddenUnits*inputSize/(4*NumHiddenUnits+inputSize).

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

Output mode, specified as one of these values:

• "sequence" — Output the complete sequence.
• "last" — Output the last time step of the sequence.

The LSTMProjectedLayer object stores this property as a character vector.

To set this property, use the corresponding name-value argument when you create the LSTMProjectedLayer object. After you create a LSTMProjectedLayer object, this property is read-only.

This property is read-only.

Flag for state inputs to the layer, specified as 0 (false) or 1 (true).

If the HasStateInputs property is 0 (false), then the layer has one input with the name"in", which corresponds to the input data. In this case, the layer uses the HiddenState and CellState properties for the layer operation.

If the HasStateInputs property is 1 (true), then the layer has three inputs with the names"in", "hidden", and "cell", which correspond to the input data, hidden state, and cell state, respectively. In this case, the layer uses the values passed to these inputs for the layer operation. If HasStateInputs is 1 (true), then the HiddenState andCellState properties must be empty.

This property is read-only.

Flag for state outputs from the layer, specified as0 (false) or1 (true).

If the HasStateOutputs property is 0 (false), then the layer has one output with the name"out", which corresponds to the output data.

If the HasStateOutputs property is 1 (true), then the layer has three outputs with the names"out", "hidden", and"cell", which correspond to the output data, hidden state, and cell state, respectively. In this case, the layer also outputs the state values that it computes.

This property is read-only.

Input size, specified as a positive integer or "auto". IfInputSize is "auto", then the software automatically assigns the input size at training time.

If InputSize is "auto", then theLSTMProjectedLayer object stores this property as a character vector.

Data Types: double | char | string

Activations

This property is read-only.

Activation function to update the cell and hidden state, specified as one of these values:

• "tanh" — Use the hyperbolic tangent function (tanh).
• "softsign" — Use the softsign function softsign(x)=x1+|x|.
• "relu" (since R2024a) — Use the rectified linear unit (ReLU) function ReLU(x)={x,x>00,x≤0.

The software uses this option as the function σc in the calculations to update the cell and hidden state.

For more information on how an LSTM layer uses activation functions, see Long Short-Term Memory Layer.

The LSTMProjectedLayer object stores this property as a character vector.

Activation function to apply to the gates, specified as one of these values:

• "sigmoid" — Use the sigmoid function, σ(x)=(1+e−x)−1.
• "hard-sigmoid" — Use the hard sigmoid function,

The software uses this option as the function σg in the calculations for the layer gates.

The LSTMProjectedLayer object stores this property as a character vector.

To set this property, use the corresponding name-value argument when you create the LSTMProjectedLayer object. After you create a LSTMProjectedLayer object, this property is read-only.

State

Cell state to use in the layer operation, specified as a NumHiddenUnits-by-1 numeric vector. This value corresponds to the initial cell state when data is passed to the layer.

After you set this property manually, calls to the resetState function set the cell state to this value.

If HasStateInputs is 1 (true), then the CellState property must be empty.

Data Types: single | double

Hidden state to use in the layer operation, specified as aNumHiddenUnits-by-1 numeric vector. This value corresponds to the initial hidden state when data is passed to the layer.

After you set this property manually, calls to the resetState function set the hidden state to this value.

If HasStateInputs is 1 (true), then the HiddenState property must be empty.

Data Types: single | double

Parameters and Initialization

Function to initialize the input weights, specified as one of these values:

• "glorot" — Initialize the input weights with the Glorot initializer [1] (also known as the Xavier initializer). The Glorot initializer independently samples from a uniform distribution with a mean of zero and a variance of2/(InputProjectorSize + numOut), where numOut = 4*NumHiddenUnits.
• "he" — Initialize the input weights with the He initializer [2]. The He initializer samples from a normal distribution with a mean of zero and a variance of2/InputProjectorSize.
• "orthogonal" — Initialize the input weights with_Q_, the orthogonal matrix in the QR decomposition of_Z_ = Q R for a random matrix Z sampled from a unit normal distribution [3].
• "narrow-normal" — Initialize the input weights by independently sampling from a normal distribution with a mean of zero and a standard deviation of 0.01.
• "zeros" — Initialize the input weights with zeros.
• "ones" — Initialize the input weights with ones.
• Function handle — Initialize the input weights with a custom function. If you specify a function handle, then the function must be of the formweights = func(sz), where sz is the size of the input weights.

The layer only initializes the input weights when the InputWeights property is empty.

The LSTMProjectedLayer object stores this property as a character vector or a function handle.

Data Types: char | string | function_handle

Function to initialize the recurrent weights, specified as one of the following:

• "orthogonal" — Initialize the recurrent weights with_Q_, the orthogonal matrix in the QR decomposition of_Z_ = Q R for a random matrix Z sampled from a unit normal distribution [3].
• "glorot" — Initialize the recurrent weights with the Glorot initializer [1] (also known as the Xavier initializer). The Glorot initializer independently samples from a uniform distribution with a mean of zero and a variance of 2/(numIn + numOut), where numIn = OutputProjectorSize andnumOut = 4*NumHiddenUnits.
• "he" — Initialize the recurrent weights with the He initializer [2]. The He initializer samples from a normal distribution with a mean of zero and a variance of2/OutputProjectorSize.
• "narrow-normal" — Initialize the recurrent weights by independently sampling from a normal distribution with a mean of zero and a standard deviation of 0.01.
• "zeros" — Initialize the recurrent weights with zeros.
• "ones" — Initialize the recurrent weights with ones.
• Function handle — Initialize the recurrent weights with a custom function. If you specify a function handle, then the function must be of the formweights = func(sz), where sz is the size of the recurrent weights.

The layer only initializes the recurrent weights when the RecurrentWeights property is empty.

The LSTMProjectedLayer object stores this property as a character vector or a function handle.

Data Types: char | string | function_handle

Function to initialize the input projector, specified as one of these values:

• "orthogonal" — Initialize the input projector with Q, the orthogonal matrix in the QR decomposition of_Z_ =Q R for a random matrix Z sampled from a unit normal distribution [3].
• "glorot" — Initialize the input projector with the Glorot initializer [1] (also known as the Xavier initializer). The Glorot initializer independently samples from a uniform distribution with a mean of zero and a variance of 2/(InputSize + inputProjectorSize).
• "he" — Initialize the input projector with the He initializer [2]. The He initializer samples from a normal distribution with a mean of zero and a variance of2/InputSize.
• "narrow-normal" — Initialize the input projector by independently sampling from a normal distribution with a mean of zero and a standard deviation of 0.01.
• "zeros" — Initialize the input projector with zeros.
• "ones" — Initialize the input projector with ones.
• Function handle — Initialize the input projector with a custom function. If you specify a function handle, then the function must have the formweights = func(sz), wheresz is the size of the input projector.

The layer initializes the input projector only when the InputProjector property is empty.

The LSTMProjectedLayer object stores this property as a character vector or a function handle.

Data Types: char | string | function_handle

Function to initialize the output projector, specified as one of these values:

• "orthogonal" — Initialize the output projector with_Q_, the orthogonal matrix in the QR decomposition of_Z_ = Q R for a random matrix Z sampled from a unit normal distribution[3].
• "glorot" — Initialize the output projector with the Glorot initializer [1] (also known as the Xavier initializer). The Glorot initializer independently samples from a uniform distribution with a mean of zero and a variance of2/(NumHiddenUnits + OutputProjectorSize).
• "he" — Initialize the output projector with the He initializer [2]. The He initializer samples from a normal distribution with a mean of zero and a variance of 2/NumHiddenUnits.
• "narrow-normal" — Initialize the output projector by independently sampling from a normal distribution with a mean of zero and a standard deviation of 0.01.
• "zeros" — Initialize the output projector with zeros.
• "ones" — Initialize the output projector with ones.
• Function handle — Initialize the output projector with a custom function. If you specify a function handle, then the function must have the formweights = func(sz), where sz is the size of the output projector.

The layer initializes the output projector only when the OutputProjector property is empty.

The LSTMProjectedLayer object stores this property as a character vector or a function handle.

Data Types: char | string | function_handle

Function to initialize the bias, specified as one of these values:

• "unit-forget-gate" — Initialize the forget gate bias with ones and the remaining biases with zeros.
• "narrow-normal" — Initialize the bias by independently sampling from a normal distribution with zero mean and a standard deviation of 0.01.
• "ones" — Initialize the bias with ones.
• Function handle — Initialize the bias with a custom function. If you specify a function handle, then the function must be of the formbias = func(sz), where sz is the size of the bias.

The layer only initializes the bias when the Bias property is empty.

The LSTMProjectedLayer object stores this property as a character vector or a function handle.

Data Types: char | string | function_handle

Input weights, specified as a matrix.

The input weight matrix is a concatenation of the four input weight matrices for the components (gates) in the layer operation. The layer vertically concatenates the four matrices in this order:

1. Input gate
2. Forget gate
3. Cell candidate
4. Output gate

The input weights are learnable parameters. When you train a neural network using the trainnet function, if InputWeights is nonempty, then the software uses theInputWeights property as the initial value. If InputWeights is empty, then the software uses the initializer specified by InputWeightsInitializer.

At training time, InputWeights is a4*NumHiddenUnits-by-InputProjectorSize matrix.

Recurrent weights, specified as a matrix.

The recurrent weight matrix is a concatenation of the four recurrent weight matrices for the components (gates) in the layer operation. The layer vertically concatenates the four matrices in this order:

1. Input gate
2. Forget gate
3. Cell candidate
4. Output gate

The recurrent weights are learnable parameters. When you train an RNN using the trainnet function, if RecurrentWeights is nonempty, then the software uses theRecurrentWeights property as the initial value. IfRecurrentWeights is empty, then the software uses the initializer specified by RecurrentWeightsInitializer.

At training time, RecurrentWeights is a4*NumHiddenUnits-by-OutputProjectorSize matrix.

Input projector, specified as a matrix.

The input projector weights are learnable parameters. When you train a network using thetrainnet function, if InputProjector is nonempty, then the software uses the InputProjector property as the initial value. If InputProjector is empty, then the software uses the initializer specified by InputProjectorInitializer.

At training time, InputProjector is a InputSize-by-InputProjectorSize matrix.

Data Types: single | double

Output projector, specified as a matrix.

The output projector weights are learnable parameters. When you train a network using thetrainnet function, if OutputProjector is nonempty, then the software uses the OutputProjector property as the initial value. If OutputProjector is empty, then the software uses the initializer specified by OutputProjectorInitializer.

At training time, OutputProjector is a NumHiddenUnits-by-OutputProjectorSize matrix.

Data Types: single | double

Layer biases, specified as a numeric vector.

The bias vector is a concatenation of the four bias vectors for the components (gates) in the layer. The layer vertically concatenates the four vectors in this order:

1. Input gate
2. Forget gate
3. Cell candidate
4. Output gate

The layer biases are learnable parameters. When you train a neural network, if Bias is nonempty, then the trainnet function uses the Bias property as the initial value. IfBias is empty, then software uses the initializer specified by BiasInitializer.

At training time, Bias is a4*NumHiddenUnits-by-1 numeric vector.

Learning Rate and Regularization

Learning rate factor for the input weights, specified as a nonnegative scalar or a 1-by-4 numeric vector.

The software multiplies this factor by the global learning rate to determine the learning rate factor for the input weights of the layer. For example, ifInputWeightsLearnRateFactor is 2, then the learning rate factor for the input weights of the layer is twice the current global learning rate. The software determines the global learning rate based on the settings you specify with thetrainingOptions function.

To control the value of the learning rate factor for the four individual matrices inInputWeights, specify a 1-by-4 vector. The entries of InputWeightsLearnRateFactor correspond to the learning rate factor of these components:

1. Input gate
2. Forget gate
3. Cell candidate
4. Output gate

To specify the same value for all the matrices, specify a nonnegative scalar.

Example: 2

Example: [1 2 1 1]

Learning rate factor for the recurrent weights, specified as a nonnegative scalar or a 1-by-4 numeric vector.

The software multiplies this factor by the global learning rate to determine the learning rate for the recurrent weights of the layer. For example, ifRecurrentWeightsLearnRateFactor is 2, then the learning rate for the recurrent weights of the layer is twice the current global learning rate. The software determines the global learning rate based on the settings you specify using thetrainingOptions function.

To control the value of the learning rate factor for the four individual matrices inRecurrentWeights, specify a 1-by-4 vector. The entries of RecurrentWeightsLearnRateFactor correspond to the learning rate factor of these components:

1. Input gate
2. Forget gate
3. Cell candidate
4. Output gate

To specify the same value for all the matrices, specify a nonnegative scalar.

Example: 2

Example: [1 2 1 1]

Learning rate factor for the input projector, specified as a nonnegative scalar.

The software multiplies this factor by the global learning rate to determine the learning rate factor for the input projector of the layer. For example, if InputProjectorLearnRateFactor is 2, then the learning rate factor for the input projector of the layer is twice the current global learning rate. The software determines the global learning rate based on the settings you specify using the trainingOptions function.

Learning rate factor for the output projector, specified as a nonnegative scalar.

The software multiplies this factor by the global learning rate to determine the learning rate factor for the output projector of the layer. For example, if OutputProjectorLearnRateFactor is 2, then the learning rate factor for the output projector of the layer is twice the current global learning rate. The software determines the global learning rate based on the settings you specify using the trainingOptions function.

Learning rate factor for the biases, specified as a nonnegative scalar or a 1-by-4 numeric vector.

The software multiplies this factor by the global learning rate to determine the learning rate for the biases in this layer. For example, if BiasLearnRateFactor is 2, then the learning rate for the biases in the layer is twice the current global learning rate. The software determines the global learning rate based on the settings you specify using the trainingOptions function.

To control the value of the learning rate factor for the four individual vectors inBias, specify a 1-by-4 vector. The entries ofBiasLearnRateFactor correspond to the learning rate factor of these components:

1. Input gate
2. Forget gate
3. Cell candidate
4. Output gate

To specify the same value for all the vectors, specify a nonnegative scalar.

Example: 2

Example: [1 2 1 1]

L2 regularization factor for the input weights, specified as a nonnegative scalar or a 1-by-4 numeric vector.

The software multiplies this factor by the global_L2_ regularization factor to determine the_L2_ regularization factor for the input weights of the layer. For example, if InputWeightsL2Factor is 2, then the L2 regularization factor for the input weights of the layer is twice the current global L2 regularization factor. The software determines the L2 regularization factor based on the settings you specify using the trainingOptions function.

To control the value of the L2 regularization factor for the four individual matrices inInputWeights, specify a 1-by-4 vector. The entries of InputWeightsL2Factor correspond to the_L2_ regularization factor of these components:

1. Input gate
2. Forget gate
3. Cell candidate
4. Output gate

To specify the same value for all the matrices, specify a nonnegative scalar.

Example: 2

Example: [1 2 1 1]

L2 regularization factor for the recurrent weights, specified as a nonnegative scalar or a 1-by-4 numeric vector.

The software multiplies this factor by the global_L2_ regularization factor to determine the_L2_ regularization factor for the recurrent weights of the layer. For example, if RecurrentWeightsL2Factor is2, then the L2 regularization factor for the recurrent weights of the layer is twice the current global_L2_ regularization factor. The software determines the L2 regularization factor based on the settings you specify using the trainingOptions function.

To control the value of the L2 regularization factor for the four individual matrices inRecurrentWeights, specify a 1-by-4 vector. The entries of RecurrentWeightsL2Factor correspond to the_L2_ regularization factor of these components:

1. Input gate
2. Forget gate
3. Cell candidate
4. Output gate

To specify the same value for all the matrices, specify a nonnegative scalar.

Example: 2

Example: [1 2 1 1]

L2 regularization factor for the input projector, specified as a nonnegative scalar.

The software multiplies this factor by the global L2 regularization factor to determine the L2 regularization factor for the input projector of the layer. For example, if InputProjectorL2Factor is 2, then the L2 regularization factor for the input projector of the layer is twice the current global L2 regularization factor. The software determines the global L2 regularization factor based on the settings you specify using the trainingOptions function.

L2 regularization factor for the output projector, specified as a nonnegative scalar.

The software multiplies this factor by the global L2 regularization factor to determine the L2 regularization factor for the output projector of the layer. For example, if OutputProjectorL2Factor is 2, then the L2 regularization factor for the output projector of the layer is twice the current global L2 regularization factor. The software determines the global L2 regularization factor based on the settings you specify using the trainingOptions function.

L2 regularization factor for the biases, specified as a nonnegative scalar or a 1-by-4 numeric vector.

The software multiplies this factor by the global L2 regularization factor to determine the L2 regularization for the biases in this layer. For example, if BiasL2Factor is 2, then the L2 regularization for the biases in this layer is twice the global L2 regularization factor. The software determines the global L2 regularization factor based on the settings you specify using the trainingOptions function.

To control the value of the L2 regularization factor for the four individual vectors in Bias, specify a 1-by-4 vector. The entries of BiasL2Factor correspond to the L2 regularization factor of these components:

1. Input gate
2. Forget gate
3. Cell candidate
4. Output gate

To specify the same value for all the vectors, specify a nonnegative scalar.

Example: 2

Example: [1 2 1 1]

Layer

Data Types: char | string

This property is read-only.

Number of inputs to the layer.

If the HasStateInputs property is 0 (false), then the layer has one input with the name"in", which corresponds to the input data. In this case, the layer uses the HiddenState and CellState properties for the layer operation.

If the HasStateInputs property is 1 (true), then the layer has three inputs with the names"in", "hidden", and "cell", which correspond to the input data, hidden state, and cell state, respectively. In this case, the layer uses the values passed to these inputs for the layer operation. If HasStateInputs is 1 (true), then the HiddenState andCellState properties must be empty.

Data Types: double

This property is read-only.

Input names of the layer.

If the HasStateInputs property is 0 (false), then the layer has one input with the name"in", which corresponds to the input data. In this case, the layer uses the HiddenState and CellState properties for the layer operation.

If the HasStateInputs property is 1 (true), then the layer has three inputs with the names"in", "hidden", and "cell", which correspond to the input data, hidden state, and cell state, respectively. In this case, the layer uses the values passed to these inputs for the layer operation. If HasStateInputs is 1 (true), then the HiddenState andCellState properties must be empty.

The LSTMProjectedLayer object stores this property as a cell array of character vectors.

This property is read-only.

Number of outputs to the layer.

If the HasStateOutputs property is 0 (false), then the layer has one output with the name"out", which corresponds to the output data.

If the HasStateOutputs property is 1 (true), then the layer has three outputs with the names"out", "hidden", and"cell", which correspond to the output data, hidden state, and cell state, respectively. In this case, the layer also outputs the state values that it computes.

Data Types: double

This property is read-only.

Output names of the layer.

If the HasStateOutputs property is 0 (false), then the layer has one output with the name"out", which corresponds to the output data.

If the HasStateOutputs property is 1 (true), then the layer has three outputs with the names"out", "hidden", and"cell", which correspond to the output data, hidden state, and cell state, respectively. In this case, the layer also outputs the state values that it computes.

The LSTMProjectedLayer object stores this property as a cell array of character vectors.

Examples

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Create an LSTM projected layer with 100 hidden units, an output projector size of 30, an input projector size of 16, and the name "lstmp".

layer = lstmProjectedLayer(100,30,16,Name="lstmp")

layer = LSTMProjectedLayer with properties:

                   Name: 'lstmp'
             InputNames: {'in'}
            OutputNames: {'out'}
              NumInputs: 1
             NumOutputs: 1
         HasStateInputs: 0
        HasStateOutputs: 0

Hyperparameters InputSize: 'auto' NumHiddenUnits: 100 InputProjectorSize: 16 OutputProjectorSize: 30 OutputMode: 'sequence' StateActivationFunction: 'tanh' GateActivationFunction: 'sigmoid'

Learnable Parameters InputWeights: [] RecurrentWeights: [] Bias: [] InputProjector: [] OutputProjector: []

State Parameters HiddenState: [] CellState: []

Show all properties

Include an LSTM projected layer in a layer array.

inputSize = 12; numHiddenUnits = 100; outputProjectorSize = max(1,floor(0.75numHiddenUnits)); inputProjectorSize = max(1,floor(0.25inputSize));

layers = [ sequenceInputLayer(inputSize) lstmProjectedLayer(numHiddenUnits,outputProjectorSize,inputProjectorSize) fullyConnectedLayer(10) softmaxLayer];

Compare the sizes of networks that do and do not contain projected layers.

Define an LSTM network architecture. Specify the input size as 12, which corresponds to the number of features of the input data. Configure an LSTM layer with 100 hidden units that outputs the last element of the sequence. Finally, specify nine classes by including a fully connected layer of size 9, followed by a softmax layer and a classification layer.

inputSize = 12; numHiddenUnits = 100; numClasses = 9;

layers = [ ... sequenceInputLayer(inputSize) lstmLayer(numHiddenUnits,OutputMode="last") fullyConnectedLayer(numClasses) softmaxLayer classificationLayer]

layers = 5×1 Layer array with layers:

 1   ''   Sequence Input          Sequence input with 12 dimensions
 2   ''   LSTM                    LSTM with 100 hidden units
 3   ''   Fully Connected         9 fully connected layer
 4   ''   Softmax                 softmax
 5   ''   Classification Output   crossentropyex

Analyze the network using the analyzeNetwork function. The network has approximately 46,100 learnable parameters.

Create an identical network with an LSTM projected layer in place of the LSTM layer.

For the LSTM projected layer:

• Specify the same number of hidden units as the LSTM layer
• Specify an output projector size of 25% of the number of hidden units.
• Specify an input projector size of 75% of the input size.
• Ensure that the output and input projector sizes are positive by taking the maximum of the sizes and 1.

outputProjectorSize = max(1,floor(0.25numHiddenUnits)); inputProjectorSize = max(1,floor(0.75inputSize));

layersProjected = [ ... sequenceInputLayer(inputSize) lstmProjectedLayer(numHiddenUnits,outputProjectorSize,inputProjectorSize,OutputMode="last") fullyConnectedLayer(numClasses) softmaxLayer classificationLayer];

Analyze the network using the analyzeNetwork function. The network has approximately 17,500 learnable parameters, which is a reduction of more than half. The sizes of the learnable parameters of the layers following the projected layer have the same sizes as the network without the LSTM projected layer. Reducing the number of learnable parameters by projecting an LSTM layer rather than reducing the number of hidden units of the LSTM layer maintains the output size of the layer and, in turn, the sizes of the downstream layers, which can result in better prediction accuracy.

analyzeNetwork(layersProjected)

Algorithms

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An LSTM layer is an RNN layer that learns long-term dependencies between time steps in time-series and sequence data.

The state of the layer consists of the hidden state (also known as the_output state_) and the cell state. The hidden state at time step t contains the output of the LSTM layer for this time step. The cell state contains information learned from the previous time steps. At each time step, the layer adds information to or removes information from the cell state. The layer controls these updates using gates.

These components control the cell state and hidden state of the layer.

This diagram illustrates the flow of data at time step t. This diagram shows how the gates forget, update, and output the cell and hidden states.

The learnable weights of an LSTM layer are the input weights W (InputWeights), the recurrent weights R (RecurrentWeights), and the bias b (Bias). The matrices W, R, and b are concatenations of the input weights, the recurrent weights, and the bias of each component, respectively. The layer concatenates the matrices according to these equations:

where i, f, g, and_o_ denote the input gate, forget gate, cell candidate, and output gate, respectively.

The cell state at time step t is given by

where ⊙ denotes the Hadamard product (element-wise multiplication of vectors).

The hidden state at time step t is given by

where σc denotes the state activation function. By default, thelstmLayer function uses the hyperbolic tangent function (tanh) to compute the state activation function.

These formulas describe the components at time step t.

In these calculations, σg denotes the gate activation function. By default, thelstmLayer function, uses the sigmoid function, given by σ(x)=(1+e−x)−1, to compute the gate activation function.

An LSTM projected layer is an RNN layer that learns long-term dependencies between time steps in time-series and sequence data using projected learnable weights.

To compress a deep learning network, you can use projected layers. A projected layer is a type of deep learning layer that enables compression by reducing the number of stored learnable parameters. The layer introduces learnable projector matrices_Q_, replaces multiplications of the form Wx, where W is a learnable matrix, with the multiplication WQQ⊤x, and stores Q and W′=WQ instead of storing W. Projecting x into a lower dimensional space using Q typically requires less memory to store the learnable parameters and can have similarly strong prediction accuracy.

Reducing the number of learnable parameters by projecting an LSTM layer rather than reducing the number of hidden units of the LSTM layer maintains the output size of the layer and, in turn, the sizes of the downstream layers, which can result in better prediction accuracy.

Output Projection

The LSTM layer operation uses four matrix multiplications of the form Rht−1, where R denotes the recurrent weights and_ht_ denotes the hidden state (or, equivalently, the layer output) at time step t.

The LSTM projected layer operation instead uses multiplications of the from RQoQo⊤ht−1, where Qo is aNumHiddenUnits-by-OutputProjectorSize matrix known as the output projector. The layer uses the same projector_Qo_ for each of the four multiplications.

To perform the four multiplications of the form Rht−1, an LSTM layer stores four recurrent weights matrices_R_, which necessitates storing 4*NumHiddenUnits^2 learnable parameters. By instead storing the4*NumHiddenUnits-by-OutputProjectorSize matrix R′=RQo and Qo, an LSTM projected layer can perform the multiplication RQoQo⊤ht−1 and store only 5*NumHiddenUnits*OutputProjectorSize learnable parameters.

Input Projection

The LSTM layer operation uses four matrix multiplications of the form Wxt, where W denotes the input weights and_xt_ denotes the layer input at time step_t_.

The LSTM projected layer operation instead uses multiplications of the from WQiQi⊤xt, where Qi is anInputSize-by-InputProjectorSize matrix known as the input projector. The layer uses the same projector_Qi_ for each of the four multiplications.

To perform the four multiplications of the form Wxt, an LSTM layer stores four weight matrices W, which necessitates storing 4*NumHiddenUnits*InputSize learnable parameters. By instead storing the4*NumHiddenUnits-by-InputProjectorSize matrix W′=WQi and Qi, an LSTM projected layer can perform the multiplication WQiQi⊤xt and store only(4*NumHiddenUnits+InputSize)*InputProjectorSize learnable parameters.

Layers in a layer array or layer graph pass data to subsequent layers as formatted dlarray objects. The format of a dlarray object is a string of characters in which each character describes the corresponding dimension of the data. The format consists of one or more of these characters:

• "S" — Spatial
• "C" — Channel
• "B" — Batch
• "T" — Time
• "U" — Unspecified

For example, you can describe 2-D image data that is represented as a 4-D array, where the first two dimensions correspond to the spatial dimensions of the images, the third dimension corresponds to the channels of the images, and the fourth dimension corresponds to the batch dimension, as having the format "SSCB" (spatial, spatial, channel, batch).

You can interact with these dlarray objects in automatic differentiation workflows, such as those for developing a custom layer, using a functionLayer object, or using the forward and predict functions withdlnetwork objects.

This table shows the supported input formats of LSTMProjectedLayer objects and the corresponding output format. If the software passes the output of the layer to a custom layer that does not inherit from the nnet.layer.Formattable class, or aFunctionLayer object with the Formattable property set to 0 (false), then the layer receives an unformatted dlarray object with dimensions ordered according to the formats in this table. The formats listed here are only a subset. The layer may support additional formats such as formats with additional "S" (spatial) or"U" (unspecified) dimensions.

In dlnetwork objects, LSTMProjectedLayer objects also support these input and output format combinations.

If the HasStateInputs property is 1 (true), then the layer has two additional inputs with the names"hidden" and "cell", which correspond to the hidden state and cell state, respectively. These additional inputs expect input format"CB" (channel, batch).

If the HasStateOutputs property is 1 (true), then the layer has two additional outputs with names"hidden" and "cell", which correspond to the hidden state and cell state, respectively. These additional outputs have output format"CB" (channel, batch).

References

[1] Glorot, Xavier, and Yoshua Bengio. "Understanding the Difficulty of Training Deep Feedforward Neural Networks." In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, 249–356. Sardinia, Italy: AISTATS, 2010. https://proceedings.mlr.press/v9/glorot10a/glorot10a.pdf

[2] He, Kaiming, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. "Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification." In 2015 IEEE International Conference on Computer Vision (ICCV), 1026–34. Santiago, Chile: IEEE, 2015. https://doi.org/10.1109/ICCV.2015.123

[3] Saxe, Andrew M., James L. McClelland, and Surya Ganguli. "Exact Solutions to the Nonlinear Dynamics of Learning in Deep Linear Neural Networks.” Preprint, submitted February 19, 2014. https://arxiv.org/abs/1312.6120.

Extended Capabilities

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Usage notes and limitations:

• LSTM projected layer objects support generic C and C++ code generation only.
• Code generation does not support passingdlarray objects with unspecified (U) dimensions to this layer.
• For code generation, you must pass a dlarray object with a channel (C) dimension as the input to this layer. For example, code generation supports data format such as "SSC" or "SSCBT".

Usage notes and limitations:

• LSTM projected layer objects only support generating CUDA code generation that does not depend on third-party libraries.
• Code generation does not support passingdlarray objects with unspecified (U) dimensions to this layer.
• For code generation, you must pass a dlarray object with a channel (C) dimension as the input to this layer. For example, code generation supports data format such as "SSC" or "SSCBT".

Version History

Introduced in R2022b