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Long short-term memory (LSTM) layer

An LSTM layer 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.

`layer = lstmLayer(numHiddenUnits)`

`layer = lstmLayer(numHiddenUnits,Name,Value)`

creates an LSTM layer and sets the `layer`

= lstmLayer(`numHiddenUnits`

)`NumHiddenUnits`

property.

sets additional `layer`

= lstmLayer(`numHiddenUnits`

,`Name,Value`

)`Name`

,
`OutputMode`

, Activations, and Learn Rate and L2 Factors properties
using one or more name-value pair arguments. You can specify multiple name-value
pair arguments. Enclose each property name in quotes.

`Name`

— Layer name`''`

(default) | character vectorLayer name, specified as a character vector. If `Name`

is set to `''`

, then the software automatically assigns a name at training time.

**Data Types: **`char`

`InputSize`

— Input size`'auto'`

(default) | positive integer Input size, specified as a positive integer or `'auto'`

. If `InputSize`

is `'auto'`

, then the software automatically assigns the input size at training time.

**Example: ** 100

`NumHiddenUnits`

— Number of hidden unitspositive integer

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

**Example: ** 200

`OutputMode`

— Format of output`'sequence'`

(default) | `'last'`

Format of output, specified as one of the following:

`'sequence'`

– Output the complete sequence.`'last'`

– Output the last time step of the sequence.

`StateActivationFunction`

— Activation function to update the cell and hidden state`'tanh'`

(default) | `'softsign'`

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

`'tanh'`

– Use the hyperbolic tangent function (tanh).`'softsign'`

– Use the softsign function $$\text{softsign}(x)=\frac{x}{1+\left|x\right|}$$.

The layer uses this option as the function $${\sigma}_{c}$$ in the calculations to update the cell and hidden state. For more information on how activation functions are used in an LSTM layer, see Long Short-Term Memory Layer.

`GateActivationFunction`

— Activation function to apply to the gates`'sigmoid'`

(default) | `'hard-sigmoid'`

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

`'sigmoid'`

– Use the sigmoid function $$\sigma (x)={(1+{e}^{-x})}^{-1}$$.`'hard-sigmoid'`

– Use the hard sigmoid function$$\sigma (x)=\{\begin{array}{cc}\begin{array}{l}0\hfill \\ 0.2x+0.5\hfill \\ 1\hfill \end{array}& \begin{array}{l}\text{if}x-2.5\hfill \\ \text{if}-2.5\le x\le 2.5\hfill \\ \text{if}x2.5\hfill \end{array}\end{array}.$$

The layer uses this option as the function $${\sigma}_{g}$$ in the calculations for the input, output, and forget gate. For more information on how activation functions are used in an LSTM layer, see Long Short-Term Memory Layer.

`BiasLearnRateFactor`

— Learning rate factor for biases1 (default) | nonnegative scalar | 1-by-4 numeric vector

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 specified with the `trainingOptions`

function.

To control the value of the learning rate factor for the four
individual matrices in `Bias`

, specify a 1-by-4
vector. The entries of `BiasLearnRateFactor`

correspond to the learning rate factor of the following:

Input gate

Forget gate

Cell candidate

Output gate

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

**Example: **
`2`

**Example: **
`[1 2 1 1]`

`BiasL2Factor`

— L2 regularization factor for biases0 (default) | nonnegative scalar | 1-by-4 numeric vector

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 learning rate 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. You can specify the global L2 regularization
factor using the `trainingOptions`

function.

To control the value of the L2 regularization factor for the four
individual matrices in `Bias`

, specify a 1-by-4
vector. The entries of `BiasL2Factor`

correspond to
the L2 regularization factor of the following:

Input gate

Forget gate

Cell candidate

Output gate

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

**Example: **
`2`

**Example: **
`[1 2 1 1]`

`InputWeightsLearnRateFactor`

— Learning rate factor for input weights1 (default) | numeric scalar | 1-by-4 numeric vector

Learning rate factor for the input weights, specified as a numeric 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 specified 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 the following:

Input gate

Forget gate

Cell candidate

Output gate

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

**Example: **`2`

**Example: **
`[1 2 1 1]`

`InputWeightsL2Factor`

— L2 regularization factor for input weights1 (default) | numeric scalar | 1-by-4 numeric vector

L2 regularization factor for the input weights, specified as a numeric 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 specified with 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 the following:

Input gate

Forget gate

Cell candidate

Output gate

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

**Example: **`2`

**Example: **
`[1 2 1 1]`

`RecurrentWeightsLearnRateFactor`

— Learning rate factor for recurrent weights1 (default) | numeric scalar | 1-by-4 numeric vector

Learning rate factor for the recurrent weights, specified as a numeric 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 specified with 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 the following:

Input gate

Forget gate

Cell candidate

Output gate

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

**Example: **`2`

**Example: **
`[1 2 1 1]`

`RecurrentWeightsL2Factor`

— L2 regularization factor for recurrent weights1 (default) | numeric scalar | 1-by-4 numeric vector

L2 regularization factor for the recurrent weights, specified as a numeric 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 specified with 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 the following:

Input gate

Forget gate

Cell candidate

Output gate

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

**Example: **`2`

**Example: **
`[1 2 1 1]`

`CellState`

— Initial value of cell statenumeric vector

Initial value of the cell state, specified as a
`NumHiddenUnits`

-by-1 numeric vector. This value
corresponds to the cell state at time step 0. The value of this property
can change when using `predictAndUpdateState`

and
`classifyAndUpdateState`

.

After setting this property, calls to the
`resetState`

function set the cell state to this
value.

`HiddenState`

— Initial value of the hidden statenumeric vector

Initial value of the hidden state, specified as a
`NumHiddenUnits`

-by-1 numeric vector. This value
corresponds to the hidden state at time step 0. The value of this
property can change when using
`predictAndUpdateState`

and
`classifyAndUpdateState`

.

After setting this property, calls to the
`resetState`

function set the hidden state to
this value.

`Bias`

— Layer biasesnumeric vector

Layer biases for the LSTM layer, specified as a
`4*NumHiddenUnits`

-by-1 numeric vector.

The bias vector is a concatenation of the four bias vectors for the components (gates) in the LSTM layer. The four vectors are concatenated vertically in the following order:

Input gate

Forget gate

Cell candidate

Output gate

`InputWeights`

— Input weightsmatrix

Input weights, specified as a
`4*NumHiddenUnits`

-by-`InputSize`

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:

Input gate

Forget gate

Cell candidate

Output gate

`RecurrentWeights`

— Recurrent weightsmatrix

Recurrent weights, specified as a
`4*NumHiddenUnits`

-by-`NumHiddenUnits`

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:

Input gate

Forget gate

Cell candidate

Output gate

Create an LSTM layer with the name `'lstm1'`

and 100 hidden units.

layer = lstmLayer(100,'Name','lstm1')

layer = LSTMLayer with properties: Name: 'lstm1' Hyperparameters InputSize: 'auto' NumHiddenUnits: 100 OutputMode: 'sequence' StateActivationFunction: 'tanh' GateActivationFunction: 'sigmoid' Learnable Parameters InputWeights: [] RecurrentWeights: [] Bias: [] State Parameters HiddenState: [] CellState: [] Show 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 Japanese Vowels data set as described in [1] and [2]. `XTrain`

is a cell array containing 270 sequences of varying length with a feature dimension of 12. `Y`

is a categorical vector of labels 1,2,...,9. The entries in `XTrain`

are matrices with 12 rows (one row for each feature) and a varying number of columns (one column for each time step).

[XTrain,YTrain] = japaneseVowelsTrainData;

Visualize the first time series in a plot. Each line corresponds to a feature.

figure plot(XTrain{1}') title("Training Observation 1") numFeatures = size(XTrain{1},1); legend("Feature " + string(1:numFeatures),'Location','northeastoutside')

Define the LSTM network architecture. Specify the input size as 12 (the number of features of the input data). Specify an LSTM layer to have 100 hidden units and to output 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 = 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

Specify the training options. Specify the solver as `'adam'`

and `'GradientThreshold'`

as 1. Set the mini-batch size to 27 and set the maximum number of epochs to 100.

Because the mini-batches are small with short sequences, the CPU is better suited for training. Set `'ExecutionEnvironment'`

to `'cpu'`

. To train on a GPU, if available, set `'ExecutionEnvironment'`

to `'auto'`

(the default value).

maxEpochs = 100; miniBatchSize = 27; options = trainingOptions('adam', ... 'ExecutionEnvironment','cpu', ... 'MaxEpochs',maxEpochs, ... 'MiniBatchSize',miniBatchSize, ... 'GradientThreshold',1, ... 'Verbose',false, ... 'Plots','training-progress');

Train the LSTM network with the specified training options.

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

Load the test set and classify the sequences into speakers.

[XTest,YTest] = japaneseVowelsTestData;

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

`YPred = classify(net,XTest,'MiniBatchSize',miniBatchSize);`

Calculate the classification accuracy of the predictions.

acc = sum(YPred == YTest)./numel(YTest)

acc = 0.9270

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.

Specify the size of the sequence input layer to be the number of features of the input data. Specify the size of the fully connected layer to be 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 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.

Specify the size of the sequence input layer to be the number of features of the input data. Specify the size of the fully connected layer to be 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 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.

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') lstmLayer(numHiddenUnits2,'OutputMode','last') 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') lstmLayer(numHiddenUnits2,'OutputMode','sequence') fullyConnectedLayer(numClasses) softmaxLayer classificationLayer];

An LSTM layer 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, where the
layer controls these updates using *gates*.

This table summarizes the components that control the cell state and hidden state of the layer.

Component | Purpose |
---|---|

Input gate (i) | Control level of cell state update |

Forget gate (f) | Control level of cell state reset (forget) |

Cell candidate (g) | Add information to cell state |

Output gate (o) | Control level of cell state added to hidden state |

This diagram illustrates the flow of data at time step *t*. The diagram
highlights 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. These matrices are concatenated as follows:

$$W=\left[\begin{array}{c}{W}_{i}\\ {W}_{f}\\ {W}_{g}\\ {W}_{o}\end{array}\right],R=\left[\begin{array}{c}{R}_{i}\\ {R}_{f}\\ {R}_{g}\\ {R}_{o}\end{array}\right],b=\left[\begin{array}{c}{b}_{i}\\ {b}_{f}\\ {b}_{g}\\ {b}_{o}\end{array}\right],$$

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

$${c}_{t}={f}_{t}\odot {c}_{t-1}+{i}_{t}\odot {g}_{t},$$

where $$\odot $$ denotes the Hadamard product (element-wise multiplication of vectors).

The hidden state at time step *t* is given by

$${h}_{t}={o}_{t}\odot {\sigma}_{c}({c}_{t}),$$

where $${\sigma}_{c}$$ denotes the state activation function. The `lstmLayer`

function, by default, uses the hyperbolic tangent function (tanh) for the state activation
function.

This table shows the formula for each component at time step
*t*.

Component | Formula |
---|---|

Input gate | $${i}_{t}={\sigma}_{g}({W}_{i}{x}_{t}+\text{}{\text{R}}_{i}{h}_{t-1}+{b}_{i})$$ |

Forget gate | $${f}_{t}={\sigma}_{g}({W}_{f}{x}_{t}+\text{}{\text{R}}_{f}{h}_{t-1}+{b}_{f})$$ |

Cell candidate | $${g}_{t}={\sigma}_{c}({W}_{g}{x}_{t}+\text{}{\text{R}}_{g}{h}_{t-1}+{b}_{g})$$ |

Output gate | $${o}_{t}={\sigma}_{g}({W}_{o}{x}_{t}+\text{}{\text{R}}_{o}{h}_{t-1}+{b}_{o})$$ |

In these calculations, $${\sigma}_{g}$$ denotes the gate activation function. The `lstmLayer`

function, by default, uses the sigmoid function given by $$\sigma (x)={(1+{e}^{-x})}^{-1}$$ for the gate activation function.

[1] M. Kudo, J. Toyama, and M. Shimbo.
"Multidimensional Curve Classification Using Passing-Through Regions."
*Pattern Recognition Letters*. Vol. 20, No. 11–13, pages
1103–1111.

[2] *UCI Machine Learning Repository:
Japanese Vowels Dataset*.
https://archive.ics.uci.edu/ml/datasets/Japanese+Vowels

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

`bilstmLayer`

| `classifyAndUpdateState`

| `predictAndUpdateState`

| `resetState`

| `sequenceInputLayer`

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