Documentation

# fullyConnectedLayer

Fully connected layer

## Description

A fully connected layer multiplies the input by a weight matrix and then adds a bias vector.

## Creation

### Syntax

``layer = fullyConnectedLayer(outputSize)``
``layer = fullyConnectedLayer(outputSize,Name,Value)``

### Description

````layer = fullyConnectedLayer(outputSize)` returns a fully connected layer and specifies the `OutputSize` property.```

example

````layer = fullyConnectedLayer(outputSize,Name,Value)` sets the optional Parameters and Initialization, Learn Rate and Regularization, and `Name` properties using name-value pairs. For example, `fullyConnectedLayer(10,'Name','fc1')` creates a fully connected layer with an output size of 10 and the name `'fc1'`. You can specify multiple name-value pairs. Enclose each property name in single quotes.```

## Properties

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### Fully Connected

Output size for the fully connected layer, specified as a positive integer.

Example: `10`

Input size for the fully connected layer, specified as a positive integer or `'auto'`. If `InputSize` is `'auto'`, then the software automatically determines the input size during training.

### Parameters and Initialization

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

• `'glorot'` – Initialize the weights with the Glorot initializer [1] (also known as Xavier initializer). The Glorot initializer independently samples from a uniform distribution with zero mean and variance `2/(InputSize + OutputSize)`.

• `'he'` – Initialize the weights with the He initializer [2]. 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. [3]

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

• `'zeros'` – Initialize the weights with zeros.

• `'ones'` – Initialize the weights with ones.

• Function handle – Initialize the 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 weights. For an example, see Specify Custom Weight Initialization Function.

The layer only initializes the weights when the `Weights` property is empty.

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

Function to initialize the bias, specified as one of the following:

• `'zeros'` – Initialize the bias with zeros.

• `'ones'` – Initialize the bias with ones.

• `'narrow-normal'` – Initialize the bias by independently sampling from a normal distribution with zero mean and standard deviation 0.01.

• 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`

Layer weights, specified as a matrix.

The layer weights are learnable parameters. You can specify the initial value for the weights directly using the `Weights` property of the layer. When training a network, if the `Weights` property of the layer is nonempty, then `trainNetwork` uses the `Weights` property as the initial value. If the `Weights` property is empty, then `trainNetwork` uses the initializer specified by the `WeightsInitializer` property of the layer.

At training time, `Weights` is an `OutputSize`-by-`InputSize` matrix.

Data Types: `single` | `double`

Layer biases, specified as a matrix.

The layer biases are learnable parameters. When training 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 an `OutputSize`-by-`1` matrix.

Data Types: `single` | `double`

### Learn Rate and Regularization

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

The software multiplies this factor by the global learning rate to determine the learning rate for the weights in this layer. For example, if `WeightLearnRateFactor` is 2, then the learning rate for the weights in this layer is twice the current global learning rate. The software determines the global learning rate based on the settings specified with the `trainingOptions` function.

Example: `2`

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

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.

Example: `2`

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

The software multiplies this factor by the global L2 regularization factor to determine the L2 regularization for the weights in this layer. For example, if `WeightL2Factor` is 2, then the L2 regularization for the weights in this layer is twice the global L2 regularization factor. You can specify the global L2 regularization factor using the `trainingOptions` function.

Example: `2`

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

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

Example: `2`

### Layer

Layer name, specified as a character vector or a string scalar. To include a layer in a layer graph, you must specify a nonempty unique layer name. If you train a series network with the layer and `Name` is set to `''`, then the software automatically assigns a name to the layer at training time.

Data Types: `char` | `string`

Number of inputs of the layer. This layer accepts a single input only.

Data Types: `double`

Input names of the layer. This layer accepts a single input only.

Data Types: `cell`

Number of outputs of the layer. This layer has a single output only.

Data Types: `double`

Output names of the layer. This layer has a single output only.

Data Types: `cell`

## Examples

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Create a fully connected layer with an output size of 10 and the name `'fc1'`.

`layer = fullyConnectedLayer(10,'Name','fc1')`
```layer = FullyConnectedLayer with properties: Name: 'fc1' Hyperparameters InputSize: 'auto' OutputSize: 10 Learnable Parameters Weights: [] Bias: [] Show all properties ```

Include a fully connected layer in a `Layer` array.

```layers = [ ... imageInputLayer([28 28 1]) convolution2dLayer(5,20) reluLayer maxPooling2dLayer(2,'Stride',2) fullyConnectedLayer(10) softmaxLayer classificationLayer]```
```layers = 7x1 Layer array with layers: 1 '' Image Input 28x28x1 images with 'zerocenter' normalization 2 '' Convolution 20 5x5 convolutions with stride [1 1] and padding [0 0 0 0] 3 '' ReLU ReLU 4 '' Max Pooling 2x2 max pooling with stride [2 2] and padding [0 0 0 0] 5 '' Fully Connected 10 fully connected layer 6 '' Softmax softmax 7 '' Classification Output crossentropyex ```

To specify the weights and bias initializer functions, use the `WeightsInitializer` and `BiasInitializer` properties respectively. To specify the weights and biases directly, use the `Weights` and `Bias` properties respectively.

Specify Initialization Function

Create a fully connected layer with an output size of 10 and specify the weights initializer to be the He initializer.

```outputSize = 10; layer = fullyConnectedLayer(outputSize,'WeightsInitializer','he')```
```layer = FullyConnectedLayer with properties: Name: '' Hyperparameters InputSize: 'auto' OutputSize: 10 Learnable Parameters Weights: [] Bias: [] Show all properties ```

Note that the `Weights` and `Bias` properties are empty. At training time, the software initializes these properties using the specified initialization functions.

Specify Custom Initialization Function

To specify your own initialization function for the weights and biases, set the `WeightsInitializer` and `BiasInitializer` properties to a function handle. For these properties, specify function handles that take the size of the weights and biases as input and output the initialized value.

Create a fully connected layer with output size 10 and specify initializers that sample the weights and biases from a Gaussian distribution with a standard deviation of 0.0001.

```outputSize = 10; weightsInitializationFcn = @(sz) rand(sz) * 0.0001; biasInitializationFcn = @(sz) rand(sz) * 0.0001; layer = fullyConnectedLayer(outputSize, ... 'WeightsInitializer',@(sz) rand(sz) * 0.0001, ... 'BiasInitializer',@(sz) rand(sz) * 0.0001)```
```layer = FullyConnectedLayer with properties: Name: '' Hyperparameters InputSize: 'auto' OutputSize: 10 Learnable Parameters Weights: [] Bias: [] Show all properties ```

Again, the `Weights` and `Bias` properties are empty. At training time, the software initializes these properties using the specified initialization functions.

Specify Weights and Bias Directly

Create a fully connected layer with an output size of 10 and set the weights and bias to `W` and `b` in the MAT file `FCWeights.mat` respectively.

```outputSize = 10; load FCWeights layer = fullyConnectedLayer(outputSize, ... 'Weights',W, ... 'Bias',b)```
```layer = FullyConnectedLayer with properties: Name: '' Hyperparameters InputSize: 720 OutputSize: 10 Learnable Parameters Weights: [10x720 double] Bias: [10x1 double] Show all properties ```

Here, the `Weights` and `Bias` properties contain the specified values. At training time, if these properties are non-empty, then the software uses the specified values as the initial weights and biases. In this case, the software does not use the initializer functions.

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## Compatibility Considerations

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Behavior changed in R2019a

## 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, pp. 249-256. 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 IEEE international conference on computer vision, pp. 1026-1034. 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).