GRUProjectedLayer - Gated recurrent unit (GRU) projected layer for recurrent neural network

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Gated recurrent unit (GRU) projected layer for recurrent neural network (RNN)

Since R2023b

Description

A GRU projected layer is an RNN layer that learns 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 a GRU layer rather than reducing the number of hidden units of the GRU 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 GRU

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 GRUProjectedLayer object. After you create aGRUProjectedLayer 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 GRU layer operation uses three 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 GRU projected layer operation instead uses multiplications of the from RQoQo⊤ht−1, where Qo is a NumHiddenUnits-by-OutputProjectorSize matrix known as the output projector. The layer uses the same projector Qo for each of the three multiplications.

To perform the three multiplications of the form Rht−1, a GRU layer stores three recurrent weights matrices R, which necessitates storing 3*NumHiddenUnits^2 learnable parameters. By instead storing the 3*NumHiddenUnits-by-OutputProjectorSize matrix R′=RQo and Qo, a GRU projected layer can perform the multiplication RQoQo⊤ht−1 and store only 4*NumHiddenUnits*OutputProjectorSize learnable parameters.

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

Tip

To ensure that RQoQo⊤ht−1 requires fewer learnable parameters, set theOutputProjectorSize property to a value less than3*NumHiddenUnits/4.

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

Input projector size, specified as a positive integer.

The GRU layer operation uses three matrix multiplications of the form Wxt, where W denotes the input weights and xt denotes the layer input at time step t.

The GRU projected layer operation instead uses multiplications of the from WQiQi⊤xt, where Qi is an InputSize-by-InputProjectorSize matrix known as the input projector. The layer uses the same projector Qi for each of the three multiplications.

To perform the three multiplications of the form Wxt, a GRU layer stores three weight matrices W, which necessitates storing 3*NumHiddenUnits*InputSize learnable parameters. By instead storing the 3*NumHiddenUnits-by-InputProjectorSize matrix W′=WQi and Qi, a GRU projected layer can perform the multiplication WQiQi⊤xt and store only (3*NumHiddenUnits+InputSize)*InputProjectorSize learnable parameters.

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

Tip

To ensure that WQiQi⊤xt requires fewer learnable parameters, set theInputProjectorSize property to a value less than3*NumHiddenUnits*inputSize/(3*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 GRUProjectedLayer object stores this property as a character vector.

To set this property, use the corresponding name-value argument when you create the GRUProjectedLayer object. After you create a GRUProjectedLayer object, 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 property for the layer operation.

If the HasStateInputs property is 1 (true), then the layer has two inputs with the names"in" and "hidden", which correspond to the input data and hidden state, respectively. In this case, the layer uses the values that the network passes to these inputs for the layer operation. If HasStateInputs is 1 (true), then theHiddenState property must be empty.

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

Flag for state outputs from the layer, specified as 0 (false) or 1 (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 two outputs with the names"out" and "hidden", which correspond to the output data and hidden state, respectively. In this case, the layer also outputs the state values computed during the layer operation.

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

Reset gate mode, specified as one of these values:

• "after-multiplication" — Apply the reset gate after matrix multiplication. This option is cuDNN compatible.
• "before-multiplication" — Apply the reset gate before matrix multiplication.
• "recurrent-bias-after-multiplication" — Apply the reset gate after matrix multiplication and use an additional set of bias terms for the recurrent weights.

For more information about the reset gate calculations, see Gated Recurrent Unit Layer.

The GRUProjectedLayer object stores this property as a character vector.

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

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 theGRUProjectedLayer object stores this property as a character vector.

Data Types: double | char | string

Activations

Activation function to update the 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 R2024b) — Use the rectified linear unit (ReLU) function ReLU(x)={x,x>00,x≤0.

The software uses this option as the function σs in the calculations to update the hidden state.

The GRUProjectedLayer object stores this property as a character vector.

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

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 GRUProjectedLayer object stores this property as a character vector.

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

State

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 = 3*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 have the formweights = func(sz), where sz is the size of the input weights.

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

The GRUProjectedLayer 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 these values:

• "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 = 3*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 have the formweights = func(sz), where sz is the size of the recurrent weights.

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

The GRUProjectedLayer 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 GRUProjectedLayer 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 GRUProjectedLayer 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:

• "zeros" — Initialize the bias with zeros.
• "narrow-normal" — Initialize the bias by independently sampling from a normal distribution with a mean of zero and standard deviation 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 have the formbias = func(sz), where sz is the size of the bias.

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

The GRUProjectedLayer 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 three input weight matrices for the components in the layer operation. The layer concatenates the matrices vertically in this order:

1. Reset gate
2. Update gate
3. Candidate state

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 a3*NumHiddenUnits-by-InputProjectorSize matrix.

Recurrent weights, specified as a matrix.

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

1. Reset gate
2. Update gate
3. Candidate state

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 a3*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.

If ResetGateMode is"after-multiplication" or"before-multiplication", then the bias vector is a concatenation of three bias vectors for the components in the layer operation. The layer concatenates the vectors vertically in this order:

1. Reset gate
2. Update gate
3. Candidate state

In this case, at training time, Bias is a 3*NumHiddenUnits-by-1 numeric vector.

If ResetGateMode is"recurrent-bias-after-multiplication", then the bias vector is a concatenation of six bias vectors for the components in the GRU layer. The layer concatenates the vectors vertically in this order:

1. Reset gate
2. Update gate
3. Candidate state
4. Reset gate (recurrent bias)
5. Update gate (recurrent bias)
6. Candidate state (recurrent bias)

In this case, at training time, Bias is a 6*NumHiddenUnits-by-1 numeric vector.

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.

For more information about the reset gate calculations, see Gated Recurrent Unit Layer.

Learning Rate and Regularization

Learning rate factor for the input weights, specified as a numeric scalar or a 1-by-3 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 three individual matrices inInputWeights, specify a 1-by-3 vector. The entries ofInputWeightsLearnRateFactor correspond to the learning rate factor of these values:

1. Reset gate
2. Update gate
3. Candidate state

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

Example: 2

Example: [1 2 1]

Learning rate factor for the recurrent weights, specified as a numeric scalar or a 1-by-3 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 three individual matrices inRecurrentWeights, specify a 1-by-3 vector. The entries ofRecurrentWeightsLearnRateFactor correspond to the learning rate factor of these values:

1. Reset gate
2. Update gate
3. Candidate state

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

Example: 2

Example: [1 2 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-3 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 three individual vectors inBias, specify a 1-by-3 vector. The entries ofBiasLearnRateFactor correspond to the learning rate factor of these values:

1. Reset gate
2. Update gate
3. Candidate state

If ResetGateMode is"recurrent-bias-after-multiplication", then the software uses the same vector for the recurrent bias vectors.

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

Example: 2

Example: [1 2 1]

L2 regularization factor for the input weights, specified as a numeric scalar or a 1-by-3 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 three individual matrices inInputWeights, specify a 1-by-3 vector. The entries ofInputWeightsL2Factor correspond to the_L2_ regularization factor of these values:

1. Reset gate
2. Update gate
3. Candidate state

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

Example: 2

Example: [1 2 1]

L2 regularization factor for the recurrent weights, specified as a numeric scalar or a 1-by-3 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 three individual matrices inRecurrentWeights, specify a 1-by-3 vector. The entries ofRecurrentWeightsL2Factor correspond to the_L2_ regularization factor of these values:

1. Reset gate
2. Update gate
3. Candidate state

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

Example: 2

Example: [1 2 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-3 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 individual vectors in Bias, specify a 1-by-3 vector. The entries of BiasL2Factor correspond to the_L2_ regularization factor of these values:

1. Reset gate
2. Update gate
3. Candidate state

If ResetGateMode is"recurrent-bias-after-multiplication", then the software uses the same vector for the recurrent bias vectors.

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

Example: 2

Example: [1 2 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 property for the layer operation.

If the HasStateInputs property is 1 (true), then the layer has two inputs with the names"in" and "hidden", which correspond to the input data and hidden state, respectively. In this case, the layer uses the values that the network passes to these inputs for the layer operation. If HasStateInputs is 1 (true), then theHiddenState property must be empty.

Data Types: double

This property is read-only.

Layer input names.

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 property for the layer operation.

If the HasStateInputs property is 1 (true), then the layer has two inputs with the names"in" and "hidden", which correspond to the input data and hidden state, respectively. In this case, the layer uses the values that the network passes to these inputs for the layer operation. If HasStateInputs is 1 (true), then theHiddenState property must be empty.

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

This property is read-only.

Number of outputs from 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 two outputs with the names"out" and "hidden", which correspond to the output data and hidden state, respectively. In this case, the layer also outputs the state values computed during the layer operation.

Data Types: double

This property is read-only.

Layer output names.

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 two outputs with the names"out" and "hidden", which correspond to the output data and hidden state, respectively. In this case, the layer also outputs the state values computed during the layer operation.

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

Examples

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

layer = gruProjectedLayer(100,30,16,Name="grup")

layer = GRUProjectedLayer with properties:

                   Name: 'grup'
             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' ResetGateMode: 'after-multiplication'

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

State Parameters HiddenState: []

Show all properties

Include a GRU 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) gruProjectedLayer(numHiddenUnits,outputProjectorSize,inputProjectorSize) fullyConnectedLayer(10) softmaxLayer];

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

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

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

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

net = dlnetwork(layers);

Analyze the network using the analyzeNetwork function. The network has approximately 34,800 learnable parameters.

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

For the GRU projected layer:

• Specify the same number of hidden units as the GRU 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) gruProjectedLayer(numHiddenUnits,outputProjectorSize,inputProjectorSize,OutputMode="last") fullyConnectedLayer(numClasses) softmaxLayer];

netProjected = dlnetwork(layersProjected);

Analyze the network using the analyzeNetwork function. The network has approximately 14,000 learnable parameters, which is a reduction of about 60%. The sizes of the learnable parameters of the layers following the projected layer have the same sizes as the network without the GRU projected layer. Reducing the number of learnable parameters by projecting a GRU layer rather than reducing the number of hidden units of the GRU 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(netProjected)

Algorithms

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A GRU layer is an RNN layer that learns dependencies between time steps in time-series and sequence data.

The hidden state of the layer at time step t contains the output of the GRU layer for this time step. At each time step, the layer adds information to or removes information from the state. The layer controls these updates using gates.

These components control the hidden state of the layer.

The learnable weights of a GRU layer are the input weights W (InputWeights), the recurrent weights R (RecurrentWeights), and the bias b (Bias). If the ResetGateMode property is "recurrent-bias-after-multiplication", then the gate and state calculations require two sets of bias values. The matrices W and_R_ are concatenations of the input weights and the recurrent weights of each component, respectively. The layer concatenates the matrices in this order:

where r, z, and h˜ denote the reset gate, update gate, and candidate state, respectively.

The bias vector depends on the ResetGateMode property. IfResetGateMode is"after-multiplication" or "before-multiplication", then the bias vector is a concatenation of three vectors:

where the subscript W indicates that this bias corresponds to the input weights multiplication.

If ResetGateMode is"recurrent-bias-after-multiplication", then the bias vector is a concatenation of six vectors:

where the subscript R indicates that this is the bias corresponding to the recurrent weights multiplication.

The hidden state at time step t is given by this equation:

These formulas describe the components at time step t.

In these calculations, σg and σs denote the gate and state activation functions, respectively. ThegruLayer function, by default, uses the sigmoid function given by σ(x)=(1+e−x)−1 to compute the gate activation function and the hyperbolic tangent function (tanh) to compute the state activation function. To specify the state and gate activation functions, use the StateActivationFunction and GateActivationFunction properties, respectively.

A GRU projected layer is an RNN layer that learns 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 a GRU layer rather than reducing the number of hidden units of the GRU 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 GRU layer operation uses three 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 GRU projected layer operation instead uses multiplications of the from RQoQo⊤ht−1, where Qo is a NumHiddenUnits-by-OutputProjectorSize matrix known as the output projector. The layer uses the same projector Qo for each of the three multiplications.

To perform the three multiplications of the form Rht−1, a GRU layer stores three recurrent weights matrices R, which necessitates storing 3*NumHiddenUnits^2 learnable parameters. By instead storing the 3*NumHiddenUnits-by-OutputProjectorSize matrix R′=RQo and Qo, a GRU projected layer can perform the multiplication RQoQo⊤ht−1 and store only 4*NumHiddenUnits*OutputProjectorSize learnable parameters.

Input Projection

The GRU layer operation uses three matrix multiplications of the form Wxt, where W denotes the input weights and xt denotes the layer input at time step t.

The GRU projected layer operation instead uses multiplications of the from WQiQi⊤xt, where Qi is an InputSize-by-InputProjectorSize matrix known as the input projector. The layer uses the same projector Qi for each of the three multiplications.

To perform the three multiplications of the form Wxt, a GRU layer stores three weight matrices W, which necessitates storing 3*NumHiddenUnits*InputSize learnable parameters. By instead storing the 3*NumHiddenUnits-by-InputProjectorSize matrix W′=WQi and Qi, a GRU projected layer can perform the multiplication WQiQi⊤xt and store only (3*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 GRUProjectedLayer 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, GRUProjectedLayer objects also support these input and output format combinations.

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

If the HasStateOutputs property is 1 (true), then the layer has two outputs with the names"out" and "hidden", which correspond to the output data and hidden state, respectively. This additional output has output with the 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:

• GRU 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:

• GRU 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 R2023b