fullyConnectedLayer
Fully connected layer
Description
A fully connected layer multiplies the input by a weight matrix and then adds a bias vector.
Creation
Description
returns a fully connected layer and specifies the layer
= fullyConnectedLayer(outputSize
)OutputSize
property.
sets the optional Parameters and Initialization, Learning Rate and Regularization, and
layer
= fullyConnectedLayer(outputSize
,Name,Value
)Name
properties using namevalue 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 namevalue 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'
 'narrownormal'
 '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 variance2/(InputSize + OutputSize)
.'he'
– Initialize the weights with the He initializer [2]. The He initializer samples from a normal distribution with zero mean and variance2/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]'narrownormal'
– 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)
, wheresz
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 biases
'zeros'
(default)  'narrownormal'
 'ones'
 function handle
Function to initialize the biases, specified as one of the following:
'zeros'
— Initialize the biases with zeros.'ones'
— Initialize the biases with ones.'narrownormal'
— Initialize the biases by independently sampling from a normal distribution with a mean of zero and a standard deviation of 0.01.Function handle — Initialize the biases with a custom function. If you specify a function handle, then the function must be of the form
bias = func(sz)
, wheresz
is the size of the biases.
The layer only initializes the biases 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
byInputSize
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
neural 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
by1
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
— L_{2} regularization factor for weights
1 (default)  nonnegative scalar
L_{2} regularization factor for the weights, specified as a nonnegative scalar.
The software multiplies this factor by the global L_{2} regularization factor to determine the L_{2} regularization for the weights in this layer. For example, if WeightL2Factor
is 2
, then the L_{2} regularization for the weights in this layer is twice the global L_{2} regularization factor. You can specify the global L_{2} regularization factor using the trainingOptions
function.
Data Types: single
 double
 int8
 int16
 int32
 int64
 uint8
 uint16
 uint32
 uint64
BiasL2Factor
— L_{2} regularization factor for biases
0
(default)  nonnegative scalar
L_{2} regularization factor for the biases, specified as a nonnegative scalar.
The software multiplies this factor by the global
L_{2} regularization factor to
determine the L_{2} regularization for the biases in
this layer. For example, if BiasL2Factor
is 2
, then
the L_{2} regularization for the biases in this layer
is twice the global L_{2} regularization factor. The
software determines the global L_{2} 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
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 the name ''
.
Data Types: char
 string
NumInputs
— Number of inputs
1
(default)
This property is readonly.
Number of inputs of the layer. This layer accepts a single input only.
Data Types: double
InputNames
— Input names
{"in"}
(default)
This property is readonly.
Input names of the layer. This layer accepts a single input only.
Data Types: cell
NumOutputs
— Number of outputs
1
(default)
This property is readonly.
Number of outputs of the layer. This layer has a single output only.
Data Types: double
OutputNames
— Output names
{'out'}
(default)
This property is readonly.
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 '' 2D Convolution 20 5x5 convolutions with stride [1 1] and padding [0 0 0 0] 3 '' ReLU ReLU 4 '' 2D 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 nonempty, 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 downsampling) 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 namevalue 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 DbyNbyS, then the fully connected layer outputs an array Z of size outputSize
byNbyS. At time step t, the corresponding entry of Z is $$W{X}_{t}+b$$, where $${X}_{t}$$ denotes time step t of X.
Fully connected layers flatten the output. They encode the spatial data in the
channel dimension by reshaping the output data. Fully connected layers in
SeriesNetwork
and DAGNetwork
objects output
data with the same number of the spatial dimensions as the input by outputting data
with spatial dimensions of size one. Fully connected layers in
dlnetwork
objects remove the spatial dimensions of the
output.
Layer Input and Output Formats
Layers in a layer array or layer graph pass data to subsequent layers as formatted dlarray
objects. The format of a dlarray
object is a string of characters, in which each character describes the corresponding dimension of the data. The formats consists of one or more of these characters:
"S"
— Spatial"C"
— Channel"B"
— Batch"T"
— Time"U"
— Unspecified
For example, 2D image data represented as a 4D array, where the first two dimensions
correspond to the spatial dimensions of the images, the third dimension corresponds to the
channels of the images, and the fourth dimension corresponds to the batch dimension, can be
described as having the format "SSCB"
(spatial, spatial, channel,
batch).
You can interact with these dlarray
objects in automatic differentiation workflows such as 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 FullyConnectedLayer
objects 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
property set to 0
(false), then the layer receives an unformatted dlarray
object with dimensions ordered corresponding to the formats in this table.
Input Format  Output Format  

SeriesNetwork and
DAGNetwork Objects  dletwork Objects  




 

 

 






 

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



 
 


 
 

To use these input formats in trainNetwork
workflows, first convert the data to "CBT"
(channel, batch, time)
format using flattenLayer
.
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. https://proceedings.mlr.press/v9/glorot10a/glorot10a.pdf
[2] He, Kaiming, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. "Delving Deep into Rectifiers: Surpassing HumanLevel Performance on ImageNet Classification." In Proceedings of the 2015 IEEE International Conference on Computer Vision, 1026–1034. Washington, DC: IEEE Computer Vision Society, 2015. https://doi.org/10.1109/ICCV.2015.123
[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).
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.
Version History
Introduced in R2016aR2019a: Default weights initialization is Glorot
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
'narrownormal'
.
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