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 optional properties using one or more namevalue arguments.layer
= fullyConnectedLayer(outputSize
,Name=Value
)
Input Arguments
outputSize
— Output size
positive integer
Output size for the fully connected layer, specified as a positive integer.
Example:
10
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Namevalue arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value,
and enclose
Name
in quotes.
Example: fullyConnectedLayer(10,Name="fc1")
creates a fully
connected layer with an output size of 10 and the name
'fc1'
.
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 these values:
"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 have the form
bias = func(sz)
, wheresz
is the size of the biases.
The layer initializes the biases only when the
Bias
property is empty.
Data Types: char
 string
 function_handle
Weights
— Layer weights
[]
(default)  matrix
Initial layer weights, specified as a matrix.
The layer weights are learnable parameters. You can specify the
initial value of 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 the trainnet
and trainNetwork
functions
use the Weights
property as the
initial value. If the Weights
property is empty, then the software 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
Initial layer biases, specified as a matrix.
The layer biases are learnable parameters. When you train a neural
network, if Bias
is nonempty,
then the trainnet
and trainNetwork
functions
use the Bias
property as the
initial value. If Bias
is empty,
then software uses the initializer specified by BiasInitializer
.
At training time, Bias
is an
OutputSize
by1
matrix.
Data Types: single
 double
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
Name
— Layer name
""
(default)  character vector  string scalar
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 these values:
"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 have the form
bias = func(sz)
, wheresz
is the size of the biases.
The layer initializes the biases only when the Bias
property is
empty.
The FullyConnectedLayer
object stores this property as a character vector or a
function handle.
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 of 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 the trainnet
function uses the Weights
property as the initial value.
If the Weights
property is empty, then the software 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 the trainnet
function uses the Bias
property as the initial value. If
Bias
is empty, then software 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
NumInputs
— Number of inputs
1
(default)
This property is readonly.
Number of inputs to the layer, returned as 1
. This layer accepts a
single input only.
Data Types: double
InputNames
— Input names
{'in'}
(default)
This property is readonly.
Input names, returned as {'in'}
. This layer accepts a single input
only.
Data Types: cell
NumOutputs
— Number of outputs
1
(default)
This property is readonly.
Number of outputs from the layer, returned as 1
. This layer has a
single output only.
Data Types: double
OutputNames
— Output names
{'out'}
(default)
This property is readonly.
Output names, returned as {'out'}
. 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: [] Use properties method to see a list of 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]
layers = 6x1 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
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: [] Use properties method to see a list of 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: [] Use properties method to see a list of 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] Use properties method to see a list of 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.
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 the software uses the global training parameters defined by the
trainingOptions
function.
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 and 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 consist of one or
more of these characters:
"S"
— Spatial"C"
— Channel"B"
— Batch"T"
— Time"U"
— Unspecified
For example, you can describe 2D image data that is 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, as having the format "SSCB"
(spatial, spatial, channel, batch).
You can interact with these dlarray
objects in automatic differentiation
workflows, such as those for 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 software passes the output of the layer 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 according to the formats
in this table. The formats listed here are only a subset. The layer may support additional
formats such as formats with additional "S"
(spatial) or
"U"
(unspecified) dimensions.
Input Format  Output Format 



 
 
 




 
 


 
 


 
 

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 2015 IEEE International Conference on Computer Vision (ICCV), 1026–34. Santiago, Chile: IEEE, 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.” Preprint, submitted February 19, 2014. https://arxiv.org/abs/1312.6120.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Code generation does not support passing
dlarray
objects with unspecified (U) dimensions to this layer.
GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.
Refer to the usage notes and limitations in the C/C++ Code Generation section. The same limitations apply to GPU code generation.
Version History
Introduced in R2016aR2024a: DAGNetwork
and SeriesNetwork
objects are not recommend
Starting in R2024a, DAGNetwork
and SeriesNetwork
objects are not recommended, use dlnetwork
objects
instead.
There are no plans to remove support for DAGNetwork
and
SeriesNetwork
objects. However, dlnetwork
objects have these advantages and are recommended instead:
dlnetwork
objects are a unified data type that supports network building, prediction, builtin training, visualization, compression, verification, and custom training loops.dlnetwork
objects support a wider range of network architectures that you can create or import from external platforms.The
trainnet
function supportsdlnetwork
objects, which enables you to easily specify loss functions. You can select from builtin loss functions or specify a custom loss function.Training and prediction with
dlnetwork
objects is typically faster thanLayerGraph
andtrainNetwork
workflows.
To convert a trained DAGNetwork
or SeriesNetwork
object to a dlnetwork
object, use the dag2dlnetwork
function.
Fully connected layers behave slightly differently in dlnetwork
objects when compared to DAGNetwork
and
SeriesNetwork
objects. 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.
R2019a: 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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