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lstmLayer

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

Description

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

The layer performs additive interactions, which can help improve gradient flow over long sequences during training.

Creation

Description

example

layer = lstmLayer(numHiddenUnits) creates an LSTM layer and sets the NumHiddenUnits property.

example

layer = lstmLayer(numHiddenUnits,Name,Value) sets additional OutputMode, Activations, State, Parameters and Initialization, Learning Rate and Regularization, and Name properties using one or more name-value pair arguments. You can specify multiple name-value pair arguments. Enclose each property name in quotes.

Properties

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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. To split your sequences into smaller sequences for when you use the trainnet and trainNetwork functions, use the SequenceLength training option.

The layer outputs data with NumHiddenUnits channels.

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

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 LSTMLayer object stores this property as a character vector.

To set this property, use the corresponding name-value argument when you create the LSTMLayer object. After you create a LSTMLayer 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 and CellState properties must be empty.

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 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". If InputSize is "auto", then the software automatically assigns the input size at training time.

If InputSize is "auto", then the LSTMLayer 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|.

The layer 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 LSTMLayer 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+ex)1.

  • "hard-sigmoid" — Use the hard sigmoid function,

    σ(x)={00.2x+0.51if x<2.5if2.5x2.5if x>2.5.

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

The LSTMLayer object stores this property as a character vector.

To set this property, use the corresponding name-value argument when you create the LSTMLayer object. After you create a LSTMLayer 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 a NumHiddenUnits-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 the following:

  • 'glorot' – Initialize the input weights with the Glorot initializer [2] (also known as Xavier initializer). The Glorot initializer independently samples from a uniform distribution with zero mean and variance 2/(InputSize + numOut), where numOut = 4*NumHiddenUnits.

  • 'he' – Initialize the input weights with the He initializer [3]. The He initializer samples from a normal distribution with zero mean and variance 2/InputSize.

  • 'orthogonal' – Initialize the input weights with Q, the orthogonal matrix given by the QR decomposition of Z = QR for a random matrix Z sampled from a unit normal distribution. [4]

  • 'narrow-normal' – Initialize the input weights by independently sampling from a normal distribution with zero mean and standard deviation 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 form weights = func(sz), where sz is the size of the input weights.

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

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 given by the QR decomposition of Z = QR for a random matrix Z sampled from a unit normal distribution. [4]

  • 'glorot' – Initialize the recurrent weights with the Glorot initializer [2] (also known as Xavier initializer). The Glorot initializer independently samples from a uniform distribution with zero mean and variance 2/(numIn + numOut), where numIn = NumHiddenUnits and numOut = 4*NumHiddenUnits.

  • 'he' – Initialize the recurrent weights with the He initializer [3]. The He initializer samples from a normal distribution with zero mean and variance 2/NumHiddenUnits.

  • 'narrow-normal' – Initialize the recurrent weights by independently sampling from a normal distribution with zero mean and standard deviation 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 form weights = func(sz), where sz is the size of the recurrent weights.

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

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 form bias = func(sz), where sz is the size of the bias.

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

The LSTMLayer 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 LSTM layer. The four matrices are concatenated vertically in the following 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 or trainNetwork function, if InputWeights is nonempty, then the software uses the InputWeights property as the initial value. If InputWeights is empty, then the software uses the initializer specified by InputWeightsInitializer.

At training time, InputWeights is a 4*NumHiddenUnits-by-InputSize 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 LSTM layer. The four matrices are vertically concatenated in the following 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 or trainNetwork function, if RecurrentWeights is nonempty, then the software uses the RecurrentWeights property as the initial value. If RecurrentWeights is empty, then the software uses the initializer specified by RecurrentWeightsInitializer.

At training time RecurrentWeights is a 4*NumHiddenUnits-by-NumHiddenUnits matrix.

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 and trainNetwork functions use the Bias property as the initial value. If Bias is empty, then software uses the initializer specified by BiasInitializer.

At training time, Bias is a 4*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, if InputWeightsLearnRateFactor 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 the trainingOptions function.

To control the value of the learning rate factor for the four individual matrices in InputWeights, 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, if RecurrentWeightsLearnRateFactor 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 the trainingOptions function.

To control the value of the learning rate factor for the four individual matrices in RecurrentWeights, 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 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 in Bias, specify a 1-by-4 vector. The entries of BiasLearnRateFactor 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 in InputWeights, 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 is 2, 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 in RecurrentWeights, 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 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

Layer name, specified as a character vector or a string scalar. For Layer array input, the trainnet, trainNetwork, assembleNetwork, layerGraph, and dlnetwork functions automatically assign names to layers with the name "".

The LSTMLayer object stores this property as a character vector.

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 and CellState 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 and CellState properties must be empty.

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

Examples

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Create an LSTM layer with the name 'lstm1' and 100 hidden units.

layer = lstmLayer(100,'Name','lstm1')
layer = 
  LSTMLayer with properties:

                       Name: 'lstm1'
                 InputNames: {'in'}
                OutputNames: {'out'}
                  NumInputs: 1
                 NumOutputs: 1
             HasStateInputs: 0
            HasStateOutputs: 0

   Hyperparameters
                  InputSize: 'auto'
             NumHiddenUnits: 100
                 OutputMode: 'sequence'
    StateActivationFunction: 'tanh'
     GateActivationFunction: 'sigmoid'

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

   State Parameters
                HiddenState: []
                  CellState: []

Use properties method to see a list of all properties.

Include an LSTM layer in a Layer array.

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

layers = [ ...
    sequenceInputLayer(inputSize)
    lstmLayer(numHiddenUnits)
    fullyConnectedLayer(numClasses)
    softmaxLayer
    classificationLayer]
layers = 
  5x1 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

Train a deep learning LSTM network for sequence-to-label classification.

Load the example data from WaveformData.mat. The data is a numObservations-by-1 cell array of sequences, where numObservations is the number of sequences. Each sequence is a numChannels-by-numTimeSteps numeric array, where numChannels is the number of channels of the sequence and numTimeSteps is the number of time steps of the sequence.

load WaveformData

Visualize some of the sequences in a plot.

numChannels = size(data{1},1);

idx = [3 4 5 12];
figure
tiledlayout(2,2)
for i = 1:4
    nexttile
    stackedplot(data{idx(i)}',DisplayLabels="Channel "+string(1:numChannels))
    
    xlabel("Time Step")
    title("Class: " + string(labels(idx(i))))
end

Set aside data for testing. Partition the data into a training set containing 90% of the data and a test set containing the remaining 10% of the data. To partition the data, use the trainingPartitions function, attached to this example as a supporting file. To access this file, open the example as a live script.

numObservations = numel(data);
[idxTrain,idxTest] = trainingPartitions(numObservations, [0.9 0.1]);
XTrain = data(idxTrain);
TTrain = labels(idxTrain);

XTest = data(idxTest);
TTest = labels(idxTest);

Define the LSTM network architecture. Specify the input size as the number of channels of the input data. Specify an LSTM layer to have 120 hidden units and to output the last element of the sequence. Finally, include a fully connected with an output size that matches the number of classes, followed by a softmax layer and a classification layer.

numHiddenUnits = 120;
numClasses = numel(categories(TTrain));

layers = [ ...
    sequenceInputLayer(numChannels)
    lstmLayer(numHiddenUnits,OutputMode="last")
    fullyConnectedLayer(numClasses)
    softmaxLayer
    classificationLayer]
layers = 
  5×1 Layer array with layers:

     1   ''   Sequence Input          Sequence input with 3 dimensions
     2   ''   LSTM                    LSTM with 120 hidden units
     3   ''   Fully Connected         4 fully connected layer
     4   ''   Softmax                 softmax
     5   ''   Classification Output   crossentropyex

Specify the training options. Train using the Adam solver with a learn rate of 0.01 and a gradient threshold of 1. Set the maximum number of epochs to 150 and shuffle every epoch. The software, by default, trains on a GPU if one is available. Using a GPU requires Parallel Computing Toolbox and a supported GPU device. For information on supported devices, see GPU Computing Requirements (Parallel Computing Toolbox).

options = trainingOptions("adam", ...
    MaxEpochs=150, ...
    InitialLearnRate=0.01,...
    Shuffle="every-epoch", ...
    GradientThreshold=1, ...
    Verbose=false, ...
    Plots="training-progress");

Train the LSTM network with the specified training options.

net = trainNetwork(XTrain,TTrain,layers,options);

Classify the test data. Specify the same mini-batch size used for training.

YTest = classify(net,XTest);

Calculate the classification accuracy of the predictions.

acc = mean(YTest == TTest)
acc = 0.8400

Display the classification results in a confusion chart.

figure
confusionchart(TTest,YTest)

To create an LSTM network for sequence-to-label classification, create a layer array containing a sequence input layer, an LSTM layer, a fully connected layer, a softmax layer, and a classification output layer.

Set the size of the sequence input layer to the number of features of the input data. Set the size of the fully connected layer to the number of classes. You do not need to specify the sequence length.

For the LSTM layer, specify the number of hidden units and the output mode 'last'.

numFeatures = 12;
numHiddenUnits = 100;
numClasses = 9;
layers = [ ...
    sequenceInputLayer(numFeatures)
    lstmLayer(numHiddenUnits,'OutputMode','last')
    fullyConnectedLayer(numClasses)
    softmaxLayer
    classificationLayer];

For an example showing how to train an LSTM network for sequence-to-label classification and classify new data, see Sequence Classification Using Deep Learning.

To create an LSTM network for sequence-to-sequence classification, use the same architecture as for sequence-to-label classification, but set the output mode of the LSTM layer to 'sequence'.

numFeatures = 12;
numHiddenUnits = 100;
numClasses = 9;
layers = [ ...
    sequenceInputLayer(numFeatures)
    lstmLayer(numHiddenUnits,'OutputMode','sequence')
    fullyConnectedLayer(numClasses)
    softmaxLayer
    classificationLayer];

To create an LSTM network for sequence-to-one regression, create a layer array containing a sequence input layer, an LSTM layer, a fully connected layer, and a regression output layer.

Set the size of the sequence input layer to the number of features of the input data. Set the size of the fully connected layer to the number of responses. You do not need to specify the sequence length.

For the LSTM layer, specify the number of hidden units and the output mode 'last'.

numFeatures = 12;
numHiddenUnits = 125;
numResponses = 1;

layers = [ ...
    sequenceInputLayer(numFeatures)
    lstmLayer(numHiddenUnits,'OutputMode','last')
    fullyConnectedLayer(numResponses)
    regressionLayer];

To create an LSTM network for sequence-to-sequence regression, use the same architecture as for sequence-to-one regression, but set the output mode of the LSTM layer to 'sequence'.

numFeatures = 12;
numHiddenUnits = 125;
numResponses = 1;

layers = [ ...
    sequenceInputLayer(numFeatures)
    lstmLayer(numHiddenUnits,'OutputMode','sequence')
    fullyConnectedLayer(numResponses)
    regressionLayer];

For an example showing how to train an LSTM network for sequence-to-sequence regression and predict on new data, see Sequence-to-Sequence Regression Using Deep Learning.

You can make LSTM networks deeper by inserting extra LSTM layers with the output mode 'sequence' before the LSTM layer. To prevent overfitting, you can insert dropout layers after the LSTM layers.

For sequence-to-label classification networks, the output mode of the last LSTM layer must be 'last'.

numFeatures = 12;
numHiddenUnits1 = 125;
numHiddenUnits2 = 100;
numClasses = 9;
layers = [ ...
    sequenceInputLayer(numFeatures)
    lstmLayer(numHiddenUnits1,'OutputMode','sequence')
    dropoutLayer(0.2)
    lstmLayer(numHiddenUnits2,'OutputMode','last')
    dropoutLayer(0.2)
    fullyConnectedLayer(numClasses)
    softmaxLayer
    classificationLayer];

For sequence-to-sequence classification networks, the output mode of the last LSTM layer must be 'sequence'.

numFeatures = 12;
numHiddenUnits1 = 125;
numHiddenUnits2 = 100;
numClasses = 9;
layers = [ ...
    sequenceInputLayer(numFeatures)
    lstmLayer(numHiddenUnits1,'OutputMode','sequence')
    dropoutLayer(0.2)
    lstmLayer(numHiddenUnits2,'OutputMode','sequence')
    dropoutLayer(0.2)
    fullyConnectedLayer(numClasses)
    softmaxLayer
    classificationLayer];

Algorithms

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References

[1] Hochreiter, S, and J. Schmidhuber, 1997. Long short-term memory. Neural computation, 9(8), pp.1735–1780.

[2] 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

[3] 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

[4] 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

Version History

Introduced in R2017b

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