# lstmProjectedLayer

Long short-term memory (LSTM) projected layer

## Description

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

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

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

## Creation

### Syntax

``layer = lstmProjectedLayer(numHiddenUnits,outputProjectorSize,inputProjectorSize)``
``layer = lstmProjectedLayer(___,Name=Value)``

### Description

example

````layer = lstmProjectedLayer(numHiddenUnits,outputProjectorSize,inputProjectorSize)` creates an LSTM projected layer and sets the `NumHiddenUnits`, `OutputProjectorSize`, and `InputProjectorSize` properties.```

example

````layer = lstmProjectedLayer(___,Name=Value)` sets the `OutputMode`, `HasStateInputs`, `HasStateOutputs`, Activations, State, Parameters and Initialization, Learning Rate and Regularization, and `Name` properties using one or more name-value arguments.```

## Properties

expand all

### Projected LSTM

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

The number of hidden units corresponds to the amount of information that the layer remembers between time steps (the hidden state). The hidden state can contain information from all the previous time steps, regardless of the sequence length. If the number of hidden units is too large, then the layer might 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 `trainNetwork` function, use the `SequenceLength` training option.

The layer outputs data with `NumHiddenUnits` channels.

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

Output projector size, specified as a positive integer.

The LSTM layer operation uses four matrix multiplications of the form $R{h}_{t-1}$, where R denotes the recurrent weights and ht denotes the hidden state (or, equivalently, the layer output) at time step t.

The LSTM projected layer operation instead uses multiplications of the from $R{Q}_{o}{Q}_{o}^{\top }{h}_{t-1}$, where Qo is a `NumHiddenUnits`-by-`OutputProjectorSize` matrix known as the output projector. The layer uses the same projector Qo for each of the four multiplications.

To perform the four multiplications of the form $R{h}_{t-1}$, an LSTM layer stores four recurrent weights R, which necessitates storing `4*NumHiddenUnits^2` learnable parameters. By instead storing the `4*NumHiddenUnits`-by-`OutputProjectorSize` matrix $R\prime =R{Q}_{o}$ and Qo, an LSTM projected layer can perform the multiplication $R{Q}_{o}{Q}_{o}^{\top }{h}_{t-1}$ and store only `5*NumHiddenUnits*OutputProjectorSize` learnable parameters.

Tip

To ensure that $R{Q}_{o}{Q}_{o}^{\top }{h}_{t-1}$ requires fewer learnable parameters, set the `OutputProjectorSize` property to a value less than `(4/5)*NumHiddenUnits`.

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

Input projector size, specified as a positive integer.

The LSTM layer operation uses four matrix multiplications of the form $W{x}_{t}$, where W denotes the recurrent weights and xt denotes the layer input at time step t.

The LSTM projected layer operation instead uses multiplications of the from $W{Q}_{i}{Q}_{i}^{\top }{x}_{t}$, where Qi is an `InputSize`-by-`InputProjectorSize` matrix known as the input projector. The layer uses the same projector Qi for each of the four multiplications.

To perform the four multiplications of the form $W{x}_{t}$, an LSTM layer stores four weight matrices W, which necessitates storing `4*NumHiddenUnits*InputSize` learnable parameters. By instead storing the `4*NumHiddenUnits`-by-`InputProjectorSize` matrix $W\prime =W{Q}_{i}$ and Qi, an LSTM projected layer can perform the multiplication $W{Q}_{i}{Q}_{i}^{\top }{x}_{t}$ and store only `(4*NumHiddenUnits+InputSize)*InputProjectorSize` learnable parameters.

Tip

To ensure that $W{Q}_{i}{Q}_{i}^{\top }{x}_{t}$ requires fewer learnable parameters, set the `InputProjectorSize` property to a value less than `(4*numHiddenUnits*inputSize)/(4*numHiddenUnits+inputSize)`.

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

Output mode, specified as one of these values:

• `'sequence'` — Output the complete sequence.

• `'last'` — Output the last time step of the sequence.

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.

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.

Input size, specified as a positive integer or `'auto'`. If `InputSize` is `'auto'`, then the software automatically assigns the input size at training time.

Data Types: `double` | `char`

### Activations

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

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 an LSTM layer uses activation functions, see Long Short-Term Memory Layer.

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

• `'sigmoid'` — Use the sigmoid function $\sigma \left(x\right)={\left(1+{e}^{-x}\right)}^{-1}$.

• `'hard-sigmoid'` — Use the hard sigmoid function

The layer uses this option as the function ${\sigma }_{g}$ in the calculations for the layer gates.

### 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 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 zero mean and a variance of ```2/(InputProjectorSize + numOut)```, where `numOut = 4*NumHiddenUnits`.

• `'he'` — Initialize the input weights with the He initializer [2]. The He initializer samples from a normal distribution with zero mean and a variance of `2/InputProjectorSize`.

• `'orthogonal'` — Initialize the input weights with Q, the orthogonal matrix in the QR decomposition of Z = QR 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 zero mean and a standard deviation of 0.01.

• `'zeros'` — Initialize the input weights with zeros.

• `'ones'` — Initialize the input weights with ones.

• Function handle — Initialize the input weights with a custom function. If you specify a function handle, then the function must be of the 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 in the QR decomposition of Z = QR 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 zero mean and a variance of ```2/(numIn + numOut)```, where `numIn = OutputProjectorSize` and `numOut = 4*NumHiddenUnits`.

• `'he'` — Initialize the recurrent weights with the He initializer [2]. The He initializer samples from a normal distribution with zero mean and a variance of `2/OutputProjectorSize`.

• `'narrow-normal'` — Initialize the recurrent weights by independently sampling from a normal distribution with zero mean and a standard deviation of 0.01.

• `'zeros'` — Initialize the recurrent weights with zeros.

• `'ones'` — Initialize the recurrent weights with ones.

• Function handle — Initialize the recurrent weights with a custom function. If you specify a function handle, then the function must be of the 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 input projector, specified as one of the following:

• `'orthogonal'` — Initialize the input projector with Q, the orthogonal matrix in the QR decomposition of Z = QR 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 zero mean 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 zero mean and a variance of `2/InputSize`.

• `'narrow-normal'` — Initialize the input projector by independently sampling from a normal distribution with zero mean 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 projector 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 projector.

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

Data Types: `char` | `string` | `function_handle`

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

• `'orthogonal'` — Initialize the output projector with Q, the orthogonal matrix in the QR decomposition of Z = QR 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 zero mean and a variance of ```2/(NumHiddenUnits + OutputProjectorSize)```.

• `'he'` — Initialize the output projector with the He initializer [2]. The He initializer samples from a normal distribution with zero mean and a variance of `2/NumHiddenUnits`.

• `'narrow-normal'` — Initialize the output projector by independently sampling from a normal distribution with zero mean 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 be of the form `weights = func(sz)`, where `sz` is the size of the output projector.

The layer only initializes the output projector when the `OutputProjector` 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.

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 layer vertically concatenates the four matrices in this order:

1. Input gate

2. Forget gate

3. Cell candidate

4. Output gate

The input weights are learnable parameters. When you train a network using the `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-`InputProjectorSize` matrix.

Recurrent weights, specified as a matrix.

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

1. Input gate

2. Forget gate

3. Cell candidate

4. Output gate

The recurrent weights are learnable parameters. When you train a network using the `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-`OutputProjectorSize` matrix.

Input projector, specified as a matrix.

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

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

Data Types: `single` | `double`

Layer biases, specified as a numeric vector.

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

1. Input gate

2. Forget gate

3. Cell candidate

4. Output gate

The layer biases are learnable parameters. When you train a network, if `Bias` is nonempty, then `trainNetwork` uses the `Bias` property as the initial value. If `Bias` is empty, then `trainNetwork` 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 input projector, specified as a nonnegative scalar.

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

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

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

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

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

To control the value of the learning rate factor for the four individual vectors 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 input projector, specified as a nonnegative scalar.

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

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

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

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

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

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

1. Input gate

2. Forget gate

3. Cell candidate

4. Output gate

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

Example: `2`

Example: `[1 2 1 1]`

### Layer

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

Data Types: `char` | `string`

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`

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.

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`

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.

## Examples

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

`layer = lstmProjectedLayer(100,30,16,Name="lstmp")`
```layer = LSTMProjectedLayer with properties: Name: 'lstmp' InputNames: {'in'} OutputNames: {'out'} NumInputs: 1 NumOutputs: 1 HasStateInputs: 0 HasStateOutputs: 0 Hyperparameters InputSize: 'auto' NumHiddenUnits: 100 InputProjectorSize: 16 OutputProjectorSize: 30 OutputMode: 'sequence' StateActivationFunction: 'tanh' GateActivationFunction: 'sigmoid' Learnable Parameters InputWeights: [] RecurrentWeights: [] Bias: [] InputProjector: [] OutputProjector: [] State Parameters HiddenState: [] CellState: [] Show all properties ```

Include an LSTM projected layer in a layer array.

```inputSize = 12; numHiddenUnits = 100; outputProjectorSize = max(1,floor(0.75*numHiddenUnits)); inputProjectorSize = max(1,floor(0.25*inputSize)); layers = [ sequenceInputLayer(inputSize) lstmProjectedLayer(numHiddenUnits,outputProjectorSize,inputProjectorSize) fullyConnectedLayer(10) softmaxLayer classificationLayer];```

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

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

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

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

`analyzeNetwork(layers)`

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

For the LSTM projected layer:

• Specify the same number of hidden units as the LSTM layer

• Specify an output projector size of 25% of the number of hidden units.

• Specify an input projector size of 75% of the input size.

• Ensure that the output and input projector sizes are positive by taking the maximum of the sizes and 1.

```outputProjectorSize = max(1,floor(0.25*numHiddenUnits)); inputProjectorSize = max(1,floor(0.75*inputSize)); layersProjected = [ ... sequenceInputLayer(inputSize) lstmProjectedLayer(numHiddenUnits,outputProjectorSize,inputProjectorSize,OutputMode="last") fullyConnectedLayer(numClasses) softmaxLayer classificationLayer];```

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

`analyzeNetwork(layers)`

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## 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.

[2] He, Kaiming, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. "Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification." In Proceedings of the 2015 IEEE International Conference on Computer Vision, 1026–1034. Washington, DC: IEEE Computer Vision Society, 2015.

[3] Saxe, Andrew M., James L. McClelland, and Surya Ganguli. "Exact solutions to the nonlinear dynamics of learning in deep linear neural networks." arXiv preprint arXiv:1312.6120 (2013).

## Version History

Introduced in R2022b