complexConvolution2dLayer

R2026b

Complex 2-D convolutional layer

Since R2026b

Description

A complex 2-D convolutional layer applies sliding complex convolutional filters to 2-D input. The layer convolves the input by first moving the filters along the input vertically and horizontally and computing the dot product of the weights and the input, and then adding a bias term.

The dimensions that the layer convolves over depends on the layer input:

• For 2-D image input (data with four dimensions corresponding to pixels in two spatial dimensions, the channels, and the observations), the layer convolves over the spatial dimensions.

• For 2-D image sequence input (data with five dimensions corresponding to the pixels in two spatial dimensions, the channels, the observations, and the time steps), the layer convolves over the two spatial dimensions.

• For 1-D image sequence input (data with four dimensions corresponding to the pixels in one spatial dimension, the channels, the observations, and the time steps), the layer convolves over the spatial and time dimensions.

Creation

Syntax

layer = complexConvolution2dLayer(filterSize,numFilters)

layer = complexConvolution2dLayer(filterSize,numFilters,Name=Value)

Description

layer = complexConvolution2dLayer(filterSize,numFilters) creates a complex 2-D convolutional layer and sets the FilterSize and NumFilters properties.

layer = complexConvolution2dLayer(filterSize,numFilters,Name=Value) specifies additional options using one or more name-value arguments. For example, complexConvolution2dLayer(filterSize,numFilters,Name="complex-conv2d") specifies the name "complex-conv2d".

Input Arguments

filterSize — Height and width of filters

vector of two positive integers | positive integer

Height and width of the filters, specified as a vector [h w] of two positive integers, where h is the height and w is the width. filterSize defines the size of the local regions to which the neurons connect in the input.

When you create the layer, you can specify filterSize as a scalar to use the same value for the height and width.

Example: [5 5] specifies filters with a height of 5 and a width of 5.

numFilters — Number of filters

positive integer

Number of filters, specified as a positive integer. This number corresponds to the number of neurons in the layer that connect to the same region in the input. This parameter determines the number of channels (feature maps) in the layer output.

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

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Example: complexConvolution2dLayer(3,16,Padding="same") creates a complex 2-D convolutional layer with 16 filters of size [3 3] and 'same' padding. At training time, the software calculates and sets the size of the padding so that the layer output has the same size as the input.

Stride — Step size for traversing input

[1 1] (default) | vector of two positive integers | positive integer

Step size for traversing the input vertically and horizontally, specified as a vector [a b] of two positive integers, where a is the vertical step size and b is the horizontal step size. When creating the layer, you can specify Stride as a scalar to use the same value for both step sizes.

Example: [2 3] specifies a vertical step size of 2 and a horizontal step size of 3.

DilationFactor — Factor for dilated convolution

[1 1] (default) | vector of two positive integers | positive integer

Factor for dilated convolution (also known as atrous convolution), specified as a vector [h w] of two positive integers, where h is the vertical dilation and w is the horizontal dilation. When creating the layer, you can specify DilationFactor as a scalar to use the same value for both horizontal and vertical dilations.

Use dilated convolutions to increase the receptive field (the area of the input which the layer can see) of the layer without increasing the number of parameters or computation.

The layer expands the filters by inserting zeros between each filter element. The dilation factor determines the step size for sampling the input or equivalently the upsampling factor of the filter. It corresponds to an effective filter size of (Filter Size – 1) .* Dilation Factor + 1. For example, a 3-by-3 filter with the dilation factor [2 2] is equivalent to a 5-by-5 filter with zeros between the elements.

Example: [2 3]

Padding — Input edge padding

[0 0 0 0] (default) | vector of nonnegative integers | "same"

Input edge padding, specified as one of these values:

• "same" — Add padding of size calculated by the software at training or prediction time so that the output has the same size as the input when the stride equals 1. If the stride is larger than 1, then the output size is ceil(inputSize/Stride), where inputSize is the height or width of the input and stride is the stride in the corresponding dimension. The software adds the same amount of padding to the top and bottom, and to the left and right, if possible. If the padding that must be added vertically has an odd value, then the software adds extra padding to the bottom. If the padding that must be added horizontally has an odd value, then the software adds extra padding to the right.

• Nonnegative integer p — Add padding of size p to all the edges of the input.

• Vector [a b] of nonnegative integers — Add padding of size a to the top and bottom of the input and padding of size b to the left and right.

• Vector [t b l r] of nonnegative integers — Add padding of size t to the top, b to the bottom, l to the left, and r to the right of the input.

Example: Padding=1 adds one row of padding to the top and bottom, and one column of padding to the left and right of the input.

Example: Padding="same" adds padding so that the output has the same size as the input (if the stride equals 1).

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

PaddingValue — Value with which to pad input data

0 (default) | scalar | "symmetric-include-edge" | "symmetric-exclude-edge" | "replicate"

Value with which to pad the input data, specified as one of the following:

PaddingValue	Description	Example
Scalar	Pad with the specified scalar value.	$$\left[\begin{array}{ccc}3& 1& 4\\ 1& 5& 9\\ 2& 6& 5\end{array}\right]\to \left[\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 3& 1& 4& 0& 0\\ 0& 0& 1& 5& 9& 0& 0\\ 0& 0& 2& 6& 5& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$
"symmetric-include-edge"	Pad using mirrored values of the input, including the edge values.	$$\left[\begin{array}{ccc}3& 1& 4\\ 1& 5& 9\\ 2& 6& 5\end{array}\right]\to \left[\begin{array}{ccccccc}5& 1& 1& 5& 9& 9& 5\\ 1& 3& 3& 1& 4& 4& 1\\ 1& 3& 3& 1& 4& 4& 1\\ 5& 1& 1& 5& 9& 9& 5\\ 6& 2& 2& 6& 5& 5& 6\\ 6& 2& 2& 6& 5& 5& 6\\ 5& 1& 1& 5& 9& 9& 5\end{array}\right]$$
"symmetric-exclude-edge"	Pad using mirrored values of the input, excluding the edge values.	$$\left[\begin{array}{ccc}3& 1& 4\\ 1& 5& 9\\ 2& 6& 5\end{array}\right]\to \left[\begin{array}{ccccccc}5& 6& 2& 6& 5& 6& 2\\ 9& 5& 1& 5& 9& 5& 1\\ 4& 1& 3& 1& 4& 1& 3\\ 9& 5& 1& 5& 9& 5& 1\\ 5& 6& 2& 6& 5& 6& 2\\ 9& 5& 1& 5& 9& 5& 1\\ 4& 1& 3& 1& 4& 1& 3\end{array}\right]$$
"replicate"	Pad using repeated border elements of the input	$$\left[\begin{array}{ccc}3& 1& 4\\ 1& 5& 9\\ 2& 6& 5\end{array}\right]\to \left[\begin{array}{ccccccc}3& 3& 3& 1& 4& 4& 4\\ 3& 3& 3& 1& 4& 4& 4\\ 3& 3& 3& 1& 4& 4& 4\\ 1& 1& 1& 5& 9& 9& 9\\ 2& 2& 2& 6& 5& 5& 5\\ 2& 2& 2& 6& 5& 5& 5\\ 2& 2& 2& 6& 5& 5& 5\end{array}\right]$$

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | char | string
Complex Number Support: Yes

NumChannels — Number of input channels

"auto" (default) | positive integer

Number of input channels, specified as one of the following:

• "auto" — Automatically determine the number of input channels at training time.

• Positive integer — Configure the layer for the specified number of input channels. NumChannels and the number of channels in the layer input data must match. For example, if the input is an RGB image, then NumChannels must be 3. If the input is the output of a convolutional layer with 16 filters, then NumChannels must be 16.

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

WeightsInitializer — Function to initialize weights

"complex-glorot-normal" (default) | "complex-glorot-uniform-square" | "complex-he-normal" | "complex-he-uniform-square" | "complex-narrow-normal" | "zeros" | "ones" | function handle

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

• "complex-glorot-normal" – Initialize the weights with the complex normal Glorot initializer [1]. The complex normal Glorot initializer independently samples real and imaginary parts of the weights from a normal distribution with zero mean and variance 1/(numIn + numOut).

• "complex-glorot-uniform-square" – Initialize the weights with the complex uniform square Glorot initializer [1]. The complex uniform square Glorot initializer independently samples real and imaginary parts of the weights from a uniform distribution in the interval (-sqrt(3/(numIn + numOut)), sqrt(3/(numIn + numOut))).

• "complex-he-normal" – Initialize the weights with the complex normal He initializer [1]. The complex normal He initializer samples real and imaginary parts of the weights from a normal distribution with zero mean and variance 1/numIn.

• "complex-he-uniform-square" – Initialize the weights with the complex uniform square He initializer [1]. The complex uniform square He initializer samples real and imaginary parts of the weights from a uniform distribution in the interval (-sqrt(3/numIn), sqrt(3/numIn)).

• "complex-narrow-normal" — Initialize the real and imaginary parts of the weights by independently sampling from a normal distribution with a mean of zero and a variance of 1e-4/2, such that the total variance is 1e-4.

• "zeros" – Initialize the weights with a real array of zeros.

• "ones" – Initialize the weights with a real array of ones.

• Function handle — Initialize the weights with a custom function. If you specify a function handle, then the function syntax 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.

Here, numIn = FilterSize(1)*FilterSize(2)*NumChannels and numOut = FilterSize(1)*FilterSize(2)*NumFilters.

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) | "complex-narrow-normal" | "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.

• "complex-narrow-normal" — Initialize the real and imaginary parts of the weights by independently sampling from a normal distribution with a mean of zero and a variance of 1e-4/2, such that the total variance is 1e-4.

• 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), where sz 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) | numeric array

Layer weights for the convolutional layer, specified as a numeric array.

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 a FilterSize(1)-by-FilterSize(2)-by-NumChannels-by-NumFilters array.

Data Types: single | double
Complex Number Support: Yes

Bias — Layer biases

[] (default) | numeric array

Layer biases for the convolutional layer, specified as a numeric array.

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 a 1-by-1-by-NumFilters array.

Data Types: single | double
Complex Number Support: Yes

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. 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

Name — Layer name

"" (default) | character vector | string scalar

Layer name, specified as a character vector or a string scalar. For Layer array input, the trainnet and dlnetwork functions automatically assign names to unnamed layers.

This argument sets the Name property.

Data Types: char | string

Properties

2-D Convolution

FilterSize — Height and width of filters

vector of two positive integers

Height and width of the filters, specified as a vector [h w] of two positive integers, where h is the height and w is the width. FilterSize defines the size of the local regions to which the neurons connect in the input.

When you create the layer, you can specify FilterSize as a scalar to use the same value for the height and width.

Example: [5 5] specifies filters with a height of 5 and a width of 5.

NumFilters — Number of filters

positive integer

This property is read-only after object creation. To set this property, use the corresponding positional input argument when you create the ComplexConvolution2DLayer object.

Number of filters, specified as a positive integer. This number determines the number of channels (feature maps) in the layer output.

Data Types: double

Stride — Step size for traversing input

[1 1] (default) | vector of two positive integers

Step size for traversing the input vertically and horizontally, specified as a vector [a b] of two positive integers, where a is the vertical step size and b is the horizontal step size.

When you set this property, you can also specify a scalar value to use the same value for both dimensions.

If the stride sizes are less than the corresponding pooling window sizes, then the pooling regions overlap.

Example: [2 3] specifies a vertical step size of 2 and a horizontal step size of 3.

Data Types: double

DilationFactor — Factor for dilated convolution

[1 1] (default) | vector of two positive integers

Factor for dilated convolution (also known as atrous convolution), specified as a vector [h w] of two positive integers, where h is the vertical dilation and w is the horizontal dilation. When creating the layer, you can specify DilationFactor as a scalar to use the same value for both horizontal and vertical dilations.

Use dilated convolutions to increase the receptive field (the area of the input which the layer can see) of the layer without increasing the number of parameters or computation.

The layer expands the filters by inserting zeros between each filter element. The dilation factor determines the step size for sampling the input or equivalently the upsampling factor of the filter. It corresponds to an effective filter size of (Filter Size – 1) .* Dilation Factor + 1. For example, a 3-by-3 filter with the dilation factor [2 2] is equivalent to a 5-by-5 filter with zeros between the elements.

Example: [2 3]

PaddingSize — Size of padding

[0 0 0 0] (default) | vector of four nonnegative integers

Size of padding to apply to input borders, specified as a vector [t b l r] of four nonnegative integers, where t is the padding applied to the top, b is the padding applied to the bottom, l is the padding applied to the left, and r is the padding applied to the right.

When you create a layer, use the Padding name-value argument to specify the padding size.

Example: [1 1 2 2] adds one row of padding to the top and bottom, and two columns of padding to the left and right of the input.

Data Types: double

PaddingMode — Method to determine padding size

Read-only: 'manual' (default) | 'same'

This property is read-only.

Method to determine padding size, represented as one of these:

• 'manual' – Pad using the integer or vector specified by the Padding name-value argument.

• 'same' – Apply padding such that the output has the same size as the input for a stride of one. If the stride is larger than 1, then the output size is ceil(inputSize/stride), where inputSize is the height or width of the input and stride is the stride in the corresponding dimension. The software adds the same amount of padding to the top and bottom, and to the left and right, if possible. If the padding that must be added vertically has an odd value, then the software adds extra padding to the bottom. If the padding that must be added horizontally has an odd value, then the software adds extra padding to the right.

When you create a layer, use the Padding name-value argument to specify the method to determine padding size.

PaddingValue — Value to pad data

0 (default) | scalar | "symmetric-include-edge" | "symmetric-exclude-edge" | "replicate"

Value to pad data, specified as one of these values:

PaddingValue	Description	Example
Scalar	Pad with the specified scalar value.	$$\left[\begin{array}{ccc}3& 1& 4\\ 1& 5& 9\\ 2& 6& 5\end{array}\right]\to \left[\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 3& 1& 4& 0& 0\\ 0& 0& 1& 5& 9& 0& 0\\ 0& 0& 2& 6& 5& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$
"symmetric-include-edge"	Pad using mirrored values of the input, including the edge values.	$$\left[\begin{array}{ccc}3& 1& 4\\ 1& 5& 9\\ 2& 6& 5\end{array}\right]\to \left[\begin{array}{ccccccc}5& 1& 1& 5& 9& 9& 5\\ 1& 3& 3& 1& 4& 4& 1\\ 1& 3& 3& 1& 4& 4& 1\\ 5& 1& 1& 5& 9& 9& 5\\ 6& 2& 2& 6& 5& 5& 6\\ 6& 2& 2& 6& 5& 5& 6\\ 5& 1& 1& 5& 9& 9& 5\end{array}\right]$$
"symmetric-exclude-edge"	Pad using mirrored values of the input, excluding the edge values.	$$\left[\begin{array}{ccc}3& 1& 4\\ 1& 5& 9\\ 2& 6& 5\end{array}\right]\to \left[\begin{array}{ccccccc}5& 6& 2& 6& 5& 6& 2\\ 9& 5& 1& 5& 9& 5& 1\\ 4& 1& 3& 1& 4& 1& 3\\ 9& 5& 1& 5& 9& 5& 1\\ 5& 6& 2& 6& 5& 6& 2\\ 9& 5& 1& 5& 9& 5& 1\\ 4& 1& 3& 1& 4& 1& 3\end{array}\right]$$
"replicate"	Pad using repeated border elements of the input	$$\left[\begin{array}{ccc}3& 1& 4\\ 1& 5& 9\\ 2& 6& 5\end{array}\right]\to \left[\begin{array}{ccccccc}3& 3& 3& 1& 4& 4& 4\\ 3& 3& 3& 1& 4& 4& 4\\ 3& 3& 3& 1& 4& 4& 4\\ 1& 1& 1& 5& 9& 9& 9\\ 2& 2& 2& 6& 5& 5& 5\\ 2& 2& 2& 6& 5& 5& 5\\ 2& 2& 2& 6& 5& 5& 5\end{array}\right]$$

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | char | string
Complex Number Support: Yes

NumChannels — Number of input channels

"auto" (default) | positive integer

This property is read-only after object creation. To set this property, use the corresponding name-value argument when you create the ComplexConvolution2DLayer object.

Number of input channels, specified as one of these values:

• "auto" — Automatically determine the number of input channels at training time.

• Positive integer — Configure the layer for the specified number of input channels. NumChannels and the number of channels in the layer input data must match. For example, if the input is an RGB image, then NumChannels must be 3. If the input is the output of a convolutional layer with 16 filters, then NumChannels must be 16.

The ComplexConvolution2DLayer object stores this property as a character vector or double type.

Data Types: double | char

Parameters and Initialization

WeightsInitializer — Function to initialize weights

"complex-glorot-normal" (default) | "complex-glorot-uniform-square" | "complex-he-normal" | "complex-he-uniform-square" | "complex-narrow-normal" | "zeros" | "ones" | function handle

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

• "complex-glorot-normal" – Initialize the weights with the complex normal Glorot initializer [1]. The complex normal Glorot initializer independently samples real and imaginary parts of the weights from a normal distribution with zero mean and variance 1/(numIn + numOut).

• "complex-glorot-uniform-square" – Initialize the weights with the complex uniform square Glorot initializer [1]. The complex uniform square Glorot initializer independently samples real and imaginary parts of the weights from a uniform distribution in the interval (-sqrt(3/(numIn + numOut)), sqrt(3/(numIn + numOut))).

• "complex-he-normal" – Initialize the weights with the complex normal He initializer [1]. The complex normal He initializer samples real and imaginary parts of the weights from a normal distribution with zero mean and variance 1/numIn.

• "complex-he-uniform-square" – Initialize the weights with the complex uniform square He initializer [1]. The complex uniform square He initializer samples real and imaginary parts of the weights from a uniform distribution in the interval (-sqrt(3/numIn), sqrt(3/numIn)).

• "complex-narrow-normal" — Initialize the real and imaginary parts of the weights by independently sampling from a normal distribution with a mean of zero and a variance of 1e-4/2, such that the total variance is 1e-4.

• "zeros" – Initialize the weights with a real array of zeros.

• "ones" – Initialize the weights with a real array of ones.

• Function handle — Initialize the weights with a custom function. If you specify a function handle, then the function syntax 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.

Here, numIn = FilterSize(1)*FilterSize(2)*NumChannels and numOut = FilterSize(1)*FilterSize(2)*NumFilters.

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) | "complex-narrow-normal" | "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.

• "complex-narrow-normal" — Initialize the real and imaginary parts of the weights by independently sampling from a normal distribution with a mean of zero and a variance of 1e-4/2, such that the total variance is 1e-4.

• 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), where sz 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) | numeric array

Layer weights for the convolutional layer, specified as a numeric array.

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 a FilterSize(1)-by-FilterSize(2)-by-NumChannels-by-NumFilters array.

Data Types: single | double
Complex Number Support: Yes

Bias — Layer biases

[] (default) | numeric array

Layer biases for the convolutional layer, specified as a numeric array.

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 the BiasInitializer property.

At training time, Bias is a 1-by-1-by-NumFilters array.

Data Types: single | double
Complex Number Support: Yes

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: double

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.

The ComplexConvolution2DLayer object stores this property as double type.

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: double

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

The ComplexConvolution2DLayer object stores this property as double type.

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

Layer

Name — Layer name

'' (default) | character vector

Layer name, specified as a character vector. For Layer array input, the trainnet and dlnetwork functions automatically assign names to unnamed layers.

Data Types: char

NumInputs — Number of inputs

Read-only: 1 (default)

This property is read-only.

Number of inputs to the layer, represented as 1. This layer has a single input only.

Data Types: double

InputNames — Input name

Read-only: {'in'} (default)

This property is read-only.

Input name, represented as {'in'}. This layer has a single input only.

NumOutputs — Number of outputs

Read-only: 1 (default)

This property is read-only.

Number of outputs from the layer, represented as 1. This layer has a single output only.

Data Types: double

OutputNames — Output name

Read-only: {'out'} (default)

This property is read-only.

Output name, represented as {'out'}. This layer has a single output only.

Examples

Create Complex 2-D Convolutional Layer

Create a complex 2-D convolutional layer with 96 filters, each with a height and width of 11. Use a stride (step size) of 4 in the horizontal and vertical directions.

layer = complexConvolution2dLayer(11,96,Stride=4)

layer = 
  ComplexConvolution2DLayer with properties:

              Name: ''

   Hyperparameters
        FilterSize: [11 11]
       NumChannels: 'auto'
        NumFilters: 96
            Stride: [4 4]
    DilationFactor: [1 1]
       PaddingMode: 'manual'
       PaddingSize: [0 0 0 0]
      PaddingValue: 0

   Learnable Parameters
           Weights: []
              Bias: []

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Algorithms

Complex Numbers

Complex 2-D convolutional layers support real-valued and complex-valued inputs. The output is always complex, even if the inputs, weights, and biases are real.

Complex 2-D Convolutional Layer

A complex 2-D convolutional layer applies sliding complex convolutional filters to 2-D input. The layer convolves the input by moving the filters along the input vertically and horizontally and computing the dot product of the weights and the input, and then adding a bias term.

The dimensions that the layer convolves over depends on the layer input:

• For 2-D input (data with four dimensions corresponding to two spatial dimensions, the channels, and the observations), the layer convolves over the spatial dimensions.

• For 2-D sequence input (data with five dimensions corresponding to two spatial dimensions, the channels, the observations, and the time steps), the layer convolves over the two spatial dimensions.

• For 1-D sequence input (data with four dimensions corresponding to one spatial dimension, the channels, the observations, and the time steps), the layer convolves over the spatial and time dimensions.

The complex convolutional layer consists of various components.¹

Filters and Stride

A complex convolutional layer consists of neurons that connect to subregions of the input images or the outputs of the previous layer. The layer learns the features localized by these regions while scanning through an image. When creating a layer using the complexConvolution2dLayer function, you can specify the size of these regions using the filterSize input argument.

For each region, the layer computes a dot product of the weights and the input, and then adds a bias term. A set of weights that is applied to a region in the image is called a filter. The filter moves along the input image vertically and horizontally, repeating the same computation for each region. In other words, the filter convolves the input.

This image shows a 3-by-3 filter scanning through the input. The lower map represents the input and the upper map represents the output.

The step size with which the filter moves is called a stride. You can specify the step size with the Stride name-value argument. The local regions that the neurons connect to can overlap depending on the filterSize and 'Stride' values.

This image shows a 3-by-3 filter scanning through the input with a stride of 2. The lower map represents the input and the upper map represents the output.

The number of weights in a filter is h * w * c, where h is the height, and w is the width of the filter, respectively, and c is the number of channels in the input. For example, if the input is a color image, the number of color channels is 3. The number of filters determines the number of channels in the output of a complex convolutional layer. Specify the number of filters using the numFilters argument with the complexConvolution2dLayer function.

Dilated Convolutions

A dilated convolution is a convolution in which the filters are expanded by spaces inserted between the elements of the filter. Specify the dilation factor using the 'DilationFactor' property.

Use dilated convolutions to increase the receptive field (the area of the input which the layer can see) of the layer without increasing the number of parameters or computation.

The layer expands the filters by inserting zeros between each filter element. The dilation factor determines the step size for sampling the input or equivalently the upsampling factor of the filter. It corresponds to an effective filter size of (Filter Size – 1) .* Dilation Factor + 1. For example, a 3-by-3 filter with the dilation factor [2 2] is equivalent to a 5-by-5 filter with zeros between the elements.

This image shows a 3-by-3 filter dilated by a factor of two scanning through the input. The lower map represents the input and the upper map represents the output.

Feature Maps

As a filter moves along the input, it uses the same set of weights and the same bias for the convolution, forming a feature map. Each feature map is the result of a convolution using a different set of weights and a different bias. Hence, the number of feature maps is equal to the number of filters. The total number of parameters in a convolutional layer is ((h*w*c + 1)*Number of Filters), where 1 is the bias.

Padding

You can also apply padding to input image borders vertically and horizontally using the 'Padding' name-value argument. Padding is values appended to the borders of the input to increase its size. By adjusting the padding, you can control the output size of the layer.

This image shows a 3-by-3 filter scanning through the input with padding of size 1. The lower map represents the input and the upper map represents the output.

Output Size

The output height and width of a convolutional layer is (Input Size – ((Filter Size – 1)*Dilation Factor + 1) + 2*Padding)/Stride + 1. This value must be an integer for the whole image to be fully covered. If the combination of these options does not lead the input data to be fully covered, the software by default ignores the remaining part of the data along the right and bottom edges in the convolution.

Number of Neurons

The product of the output height and width gives the total number of neurons in a feature map, say Map Size. The total number of neurons (output size) in a convolutional layer is Map Size*Number of Filters.

Usually, the results from these neurons pass through some form of nonlinearity, such as rectified linear units (ReLU).

Learnable Parameters

You can adjust the learning rates and regularization options for the layer using name-value arguments while defining the complex convolutional layer. If you choose not to specify these options, then the trainnet function uses the global training options defined with the trainingOptions function.

Number of Layers

A complex convolutional neural network can consist of one or multiple complex convolutional layers. The number of complex convolutional layers depends on the amount and complexity of the data.

Layer Input and Output Formats

Most 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 format consists of one or more of these characters:

• "S" — Spatial

• "C" — Channel

• "B" — Batch

• "T" — Time

• "U" — Unspecified

For example, you can describe 2-D image data that is represented as a 4-D 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

• using the forward and predict functions with dlnetwork objects

This table shows the supported input formats of ComplexConvolution2DLayer 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 to 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 of the formats that the layer supports. The layer might support additional formats, such as formats with additional "S" (spatial) or "U" (unspecified) dimensions.

Input Format	Output Format
"SSCB" (spatial, spatial, channel, batch)	"SSCB" (spatial, spatial, channel, batch)
"SCBT" (spatial, channel, batch, time)	"SCBT" (spatial, channel, batch, time)
"SSCBT" (spatial, spatial, channel, batch, time)	"SSCBT" (spatial, spatial, channel, batch, time)
"SSC" (spatial, spatial, channel)	"SSCB" (spatial, spatial, channel, batch)
"SCT" (spatial, channel, time)	"SCBT" (spatial, channel, batch, time)
"SSCT" (spatial, spatial, channel, time)	"SSCT" (spatial, spatial, channel, time)

References

[1] Barrachina, Jose Agustin, Chengfang Ren, Gilles Vieillard, Christelle Morisseau, and Jean-Philippe Ovarlez. "Theory and Implementation of Complex-Valued Neural Networks". Preprint, submitted February 16, 2023. https://arxiv.org/abs/2302.08286

[2] Trabelsi, Chiheb, Olexa Bilaniuk, Ying Zhang, Dmitriy Serdyuk, Sandeep Subramanian, João Felipe Santos, Soroush Mehri, Negar Rostamzadeh, Yoshua Bengio, and Christopher J Pal. "Deep Complex Networks". Preprint, submitted February 25, 2018. https://arxiv.org/abs/1705.09792.

Version History

Introduced in R2026b