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.

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

Fully Connected

OutputSize — Output size

positive integer

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

Example: 10

InputSize — Input size

'auto' (default) | positive integer

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

WeightsInitializer — Function to initialize weights

'glorot' (default) | 'he' | 'orthogonal' | 'narrow-normal' | 'zeros' | 'ones' | function handle

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

BiasInitializer — Function to initialize bias

'zeros' (default) | 'narrow-normal' | 'ones' | 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

Weights — Layer weights

[] (default) | matrix

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

Bias — Layer biases

[] (default) | matrix

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

WeightLearnRateFactor — Learning rate factor for weights

1 (default) | nonnegative scalar

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

BiasLearnRateFactor — Learning rate factor for biases

1 (default) | nonnegative scalar

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

WeightL2Factor — L₂ regularization factor for weights

1 (default) | nonnegative scalar

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

BiasL2Factor — L₂ regularization factor for biases

0 (default) | nonnegative scalar

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.

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

Layer

Name — Layer name

'' (default) | character vector | string scalar

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 Name set to ''.

Data Types: char | string

NumInputs — Number of inputs

1 (default)

This property is read-only.

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

Data Types: double

InputNames — Input names

{'in'} (default)

This property is read-only.

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

Data Types: cell

NumOutputs — Number of outputs

1 (default)

This property is read-only.

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

Data Types: double

OutputNames — Output names

{'out'} (default)

This property is read-only.

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

Data Types: cell

Examples

Create Fully Connected Layer

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

Specify Initial Weights and Biases in Fully Connected Layer

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.

Algorithms

Fully Connected Layer

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

The convolutional (and down-sampling) layers are followed by one or more fully connected layers.

As the name suggests, all neurons in a fully connected layer connect to all the neurons in the previous layer. This layer combines all of the features (local information) learned by the previous layers across the image to identify the larger patterns. For classification problems, the last fully connected layer combines the features to classify the images. This is the reason that the outputSize argument of the last fully connected layer of the network is equal to the number of classes of the data set. For regression problems, the output size must be equal to the number of response variables.

You can also adjust the learning rate and the regularization parameters for this layer using the related name-value pair arguments when creating the fully connected layer. If you choose not to adjust them, then trainNetwork uses the global training parameters defined by the trainingOptions function. For details on global and layer training options, see Set Up Parameters and Train Convolutional Neural Network.

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

If the input to the layer is a sequence (for example, in an LSTM network), then the fully connected layer acts independently on each time step. For example, if the layer before the fully connected layer outputs an array X of size D-by-N-by-S, then the fully connected layer outputs an array Z of size outputSize-by-N-by-S. At time step t, the corresponding entry of Z is $$W{X}_t+b$$, where $$X_t$$ denotes time step t of X.

The fully connected layer flattens the output. It reshapes the array such that the spatial data is encoded in the channel dimension.

For sequence input, the layer applies the fully connect operation independently to each time step of the input.

Layer Input and Output Formats

Layers in a layer array or layer graph pass data specified as formatted dlarray objects.

You can interact with these dlarray objects in automatic differentiation workflows such as when developing a custom layer, using a functionLayer object, or using the forward and predict functions with dlnetwork objects.

This table shows the supported input formats of a FullyConnectedLayer object and the corresponding output format. If the output of the layer is passed to a custom layer that does not inherit from the nnet.layer.Formattable class, or a FunctionLayer object with the Formattable option set to false, then the layer receives an unformatted dlarray object with dimensions ordered corresponding to the formats outlined in this table.

Input Format	Output Format
"CB" (channel, batch)	"CB" (channel, batch)
"SCB" (spatial, channel, batch)	"CB" (channel, batch)
"SSCB" (spatial, spatial, channel, batch)	"CB" (channel, batch)
"SSSCB" (spatial, spatial, spatial, channel, batch)	"CB" (channel, batch)
"CBT" (channel, batch, time)	"CBT" (channel, batch, time)

In dlnetwork objects, FullyConnectedLayer objects also support the following input and output format combinations.

Input Format	Output Format
"SCBT" (spatial, channel, batch)	"CBT" (channel, batch, time)
"SSCBT" (spatial, spatial, channel, batch, time)	"CBT" (channel, batch, time)
"SSSCBT" (spatial, spatial, spatial, channel, batch, time)	"CBT" (channel, batch, time)

To use these input formats in trainNetwork workflows, first convert the data to "CBT" (channel, batch, time) format using flattenLayer.

Compatibility Considerations

Default weights initialization is Glorot

Behavior changed in R2019a

Starting in R2019a, the software, by default, initializes the layer weights of this layer using the Glorot initializer. This behavior helps stabilize training and usually reduces the training time of deep networks.

In previous releases, the software, by default, initializes the layer weights by sampling from a normal distribution with zero mean and variance 0.01. To reproduce this behavior, set the 'WeightsInitializer' option of the layer to 'narrow-normal'.

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).

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