# fullyConnectedLayer

Fully connected layer

## Description

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

## Creation

### 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, Learning 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 a mean of zero and a standard deviation of 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 you train 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 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 an OutputSize-by-1 matrix.

Data Types: single | double

### Learning 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 you specify using the trainingOptions function.

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

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 you specify using the trainingOptions function.

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

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.

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

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. The software determines the global L2 regularization factor based on the settings you specify using the trainingOptions function.

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

### 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 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   ''   2-D Convolution         20 5x5 convolutions with stride [1  1] and padding [0  0  0  0]
3   ''   ReLU                    ReLU
4   ''   2-D 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;

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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## 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 R2016a

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