# Convolutional layers

A convolution is an integral that expresses the amount of overlap of one function `g`

as it is shifted over another function `f`

. It therefore "blends" one function with another. The neural network package supports convolution, pooling, subsampling and other relevant facilities. These are divided based on the dimensionality of the input and output Tensors:

- Temporal Modules apply to sequences with a one-dimensional relationship
(e.g. sequences of words, phonemes and letters. Strings of some kind).
- TemporalConvolution : a 1D convolution over an input sequence ;
- TemporalSubSampling : a 1D sub-sampling over an input sequence ;
- TemporalMaxPooling : a 1D max-pooling operation over an input sequence ;
- LookupTable : a convolution of width
`1`

, commonly used for word embeddings ; - TemporalRowConvolution : a row-oriented 1D convolution over an input sequence ;

- Spatial Modules apply to inputs with two-dimensional relationships (e.g. images):
- SpatialConvolution : a 2D convolution over an input image ;
- SpatialFullConvolution : a 2D full convolution over an input image ;
- SpatialDilatedConvolution : a 2D dilated convolution over an input image ;
- SpatialConvolutionLocal : a 2D locally-connected layer over an input image ;
- SpatialSubSampling : a 2D sub-sampling over an input image ;
- SpatialMaxPooling : a 2D max-pooling operation over an input image ;
- SpatialDilatedMaxPooling : a 2D dilated max-pooling operation over an input image ;
- SpatialFractionalMaxPooling : a 2D fractional max-pooling operation over an input image ;
- SpatialAveragePooling : a 2D average-pooling operation over an input image ;
- SpatialAdaptiveMaxPooling : a 2D max-pooling operation which adapts its parameters dynamically such that the output is of fixed size ;
- SpatialAdaptiveAveragePooling : a 2D average-pooling operation which adapts its parameters dynamically such that the output is of fixed size ;
- SpatialMaxUnpooling : a 2D max-unpooling operation ;
- SpatialLPPooling : computes the
`p`

norm in a convolutional manner on a set of input images ; - SpatialConvolutionMap : a 2D convolution that uses a generic connection table ;
- SpatialZeroPadding : pads a feature map with specified number of zeros ;
- SpatialReflectionPadding : pads a feature map with the reflection of the input ;
- SpatialReplicationPadding : pads a feature map with the value at the edge of the input borders ;
- SpatialSubtractiveNormalization : a spatial subtraction operation on a series of 2D inputs using
- SpatialCrossMapLRN : a spatial local response normalization between feature maps ;
- SpatialBatchNormalization: mean/std normalization over the mini-batch inputs and pixels, with an optional affine transform that follows a kernel for computing the weighted average in a neighborhood ;
- SpatialUpSamplingNearest: A simple nearest neighbor upsampler applied to every channel of the feature map.
- SpatialUpSamplingBilinear: A simple bilinear upsampler applied to every channel of the feature map.

- Volumetric Modules apply to inputs with three-dimensional relationships (e.g. videos) :
- VolumetricConvolution : a 3D convolution over an input video (a sequence of images) ;
- VolumetricFullConvolution : a 3D full convolution over an input video (a sequence of images) ;
- VolumetricDilatedConvolution : a 3D dilated convolution over an input image ;
- VolumetricMaxPooling : a 3D max-pooling operation over an input video.
- VolumetricDilatedMaxPooling : a 3D dilated max-pooling operation over an input video ;
- VolumetricFractionalMaxPooling : a 3D fractional max-pooling operation over an input image ;
- VolumetricAveragePooling : a 3D average-pooling operation over an input video.
- VolumetricMaxUnpooling : a 3D max-unpooling operation.
- VolumetricReplicationPadding : Pads a volumetric feature map with the value at the edge of the input borders. ;

## Temporal Modules

Excluding an optional first batch dimension, temporal layers expect a 2D Tensor as input. The
first dimension is the number of frames in the sequence (e.g. `nInputFrame`

), the last dimension
is the number of features per frame (e.g. `inputFrameSize`

). The output will normally have the same number
of dimensions, although the size of each dimension may change. These are commonly used for processing acoustic signals or sequences of words, i.e. in Natural Language Processing.

Note: The LookupTable is special in that while it does output a temporal Tensor of size `nOutputFrame x outputFrameSize`

,
its input is a 1D Tensor of indices of size `nIndices`

. Again, this is excluding the option first batch dimension.

## TemporalConvolution

```
module = nn.TemporalConvolution(inputFrameSize, outputFrameSize, kW, [dW])
```

Applies a 1D convolution over an input sequence composed of `nInputFrame`

frames. The `input`

tensor in
`forward(input)`

is expected to be a 2D tensor (`nInputFrame x inputFrameSize`

) or a 3D tensor (`nBatchFrame x nInputFrame x inputFrameSize`

).

The parameters are the following:
* `inputFrameSize`

: The input frame size expected in sequences given into `forward()`

.
* `outputFrameSize`

: The output frame size the convolution layer will produce.
* `kW`

: The kernel width of the convolution
* `dW`

: The step of the convolution. Default is `1`

.

Note that depending of the size of your kernel, several (of the last) frames of the sequence might be lost. It is up to the user to add proper padding frames in the input sequences.

If the input sequence is a 2D tensor of dimension `nInputFrame x inputFrameSize`

, the output sequence will be
`nOutputFrame x outputFrameSize`

where

```
nOutputFrame = (nInputFrame - kW) / dW + 1
```

If the input sequence is a 3D tensor of dimension `nBatchFrame x nInputFrame x inputFrameSize`

, the output sequence will be
`nBatchFrame x nOutputFrame x outputFrameSize`

.

The parameters of the convolution can be found in `self.weight`

(Tensor of
size `outputFrameSize x (kW x inputFrameSize)`

) and `self.bias`

(Tensor of
size `outputFrameSize`

). The corresponding gradients can be found in
`self.gradWeight`

and `self.gradBias`

.

For a 2D input, the output value of the layer can be precisely described as:

```
output[t][i] = bias[i]
+ sum_j sum_{k=1}^kW weight[i][k][j]
* input[dW*(t-1)+k)][j]
```

Here is a simple example:

```
inp=5; -- dimensionality of one sequence element
outp=1; -- number of derived features for one sequence element
kw=1; -- kernel only operates on one sequence element per step
dw=1; -- we step once and go on to the next sequence element
mlp=nn.TemporalConvolution(inp,outp,kw,dw)
x=torch.rand(7,inp) -- a sequence of 7 elements
print(mlp:forward(x))
```

which gives:

```
-0.9109
-0.9872
-0.6808
-0.9403
-0.9680
-0.6901
-0.6387
[torch.Tensor of dimension 7x1]
```

This is equivalent to:

```
weights=torch.reshape(mlp.weight,inp) -- weights applied to all
bias= mlp.bias[1];
for i=1,x:size(1) do -- for each sequence element
element= x[i]; -- features of ith sequence element
print(element:dot(weights) + bias)
end
```

which gives:

```
-0.91094998687717
-0.98721705771773
-0.68075004276185
-0.94030132495887
-0.96798754116609
-0.69008470895581
-0.63871422284166
```

## TemporalMaxPooling

```
module = nn.TemporalMaxPooling(kW, [dW])
```

Applies 1D max-pooling operation in `kW`

regions by step size
`dW`

steps. Input sequence composed of `nInputFrame`

frames. The `input`

tensor in
`forward(input)`

is expected to be a 2D tensor (`nInputFrame x inputFrameSize`

)
or a 3D tensor (`nBatchFrame x nInputFrame x inputFrameSize`

).

If the input sequence is a 2D tensor of dimension `nInputFrame x inputFrameSize`

, the output sequence will be
`nOutputFrame x inputFrameSize`

where

```
nOutputFrame = (nInputFrame - kW) / dW + 1
```

## TemporalSubSampling

```
module = nn.TemporalSubSampling(inputFrameSize, kW, [dW])
```

Applies a 1D sub-sampling over an input sequence composed of `nInputFrame`

frames. The `input`

tensor in
`forward(input)`

is expected to be a 2D tensor (`nInputFrame x inputFrameSize`

). The output frame size
will be the same as the input one (`inputFrameSize`

).

The parameters are the following:
* `inputFrameSize`

: The input frame size expected in sequences given into `forward()`

.
* `kW`

: The kernel width of the sub-sampling
* `dW`

: The step of the sub-sampling. Default is `1`

.

Note that depending of the size of your kernel, several (of the last) frames of the sequence might be lost. It is up to the user to add proper padding frames in the input sequences.

If the input sequence is a 2D tensor `nInputFrame x inputFrameSize`

, the output sequence will be
`inputFrameSize x nOutputFrame`

where

```
nOutputFrame = (nInputFrame - kW) / dW + 1
```

The parameters of the sub-sampling can be found in `self.weight`

(Tensor of
size `inputFrameSize`

) and `self.bias`

(Tensor of
size `inputFrameSize`

). The corresponding gradients can be found in
`self.gradWeight`

and `self.gradBias`

.

The output value of the layer can be precisely described as:

```
output[t][i] = bias[i] + weight[i] * sum_{k=1}^kW input[dW*(t-1)+k][i]
```

## LookupTable

```
module = nn.LookupTable(nIndex, size, [paddingValue], [maxNorm], [normType])
```

This layer is a particular case of a convolution, where the width of the convolution would be `1`

.
When calling `forward(input)`

, it assumes `input`

is a 1D or 2D tensor filled with indices.
If the input is a matrix, then each row is assumed to be an input sample of given batch. Indices start
at `1`

and can go up to `nIndex`

. For each index, it outputs a corresponding `Tensor`

of size
specified by `size`

.

LookupTable can be very slow if a certain input occurs frequently compared to other inputs;
this is often the case for input padding. During the backward step, there is a separate thread
for each input symbol which results in a bottleneck for frequent inputs.
generating a `n x size1 x size2 x ... x sizeN`

tensor, where `n`

is the size of a 1D `input`

tensor.

Again with a 1D input, when only `size1`

is provided, the `forward(input)`

is equivalent to
performing the following matrix-matrix multiplication in an efficient manner:

```
P M
```

where `M`

is a 2D matrix of size `nIndex x size1`

containing the parameters of the lookup-table and
`P`

is a 2D matrix of size `n x nIndex`

, where for each `i`

th row vector, every element is zero except the one at index `input[i]`

where it is `1`

.

1D example:

```
-- a lookup table containing 10 tensors of size 3
module = nn.LookupTable(10, 3)
input = torch.Tensor{1,2,1,10}
print(module:forward(input))
```

Outputs something like:

```
-1.4415 -0.1001 -0.1708
-0.6945 -0.4350 0.7977
-1.4415 -0.1001 -0.1708
-0.0745 1.9275 1.0915
[torch.DoubleTensor of dimension 4x3]
```

Note that the first row vector is the same as the 3rd one!

Given a 2D input tensor of size `m x n`

, the output is a `m x n x size`

tensor, where `m`

is the number of samples in
the batch and `n`

is the number of indices per sample.

2D example:

```
-- a lookup table containing 10 tensors of size 3
module = nn.LookupTable(10, 3)
-- a batch of 2 samples of 4 indices each
input = torch.Tensor({{1,2,4,5},{4,3,2,10}})
print(module:forward(input))
```

Outputs something like:

```
(1,.,.) =
-0.0570 -1.5354 1.8555
-0.9067 1.3392 0.6275
1.9662 0.4645 -0.8111
0.1103 1.7811 1.5969
(2,.,.) =
1.9662 0.4645 -0.8111
0.0026 -1.4547 -0.5154
-0.9067 1.3392 0.6275
-0.0193 -0.8641 0.7396
[torch.DoubleTensor of dimension 2x4x3]
```

LookupTable supports max-norm regularization. One can activate the max-norm constraints by setting non-nil maxNorm in constructor or using setMaxNorm function. In the implementation, the max-norm constraint is enforced in the forward pass. That is the output of the LookupTable always obeys the max-norm constraint, even though the module weights may temporarily exceed the max-norm constraint.

max-norm regularization example:

```
-- a lookup table with max-norm constraint: 2-norm <= 1
module = nn.LookupTable(10, 3, 0, 1, 2)
input = torch.Tensor{1,2,1,10}
print(module.weight)
-- output of the module always obey max-norm constraint
print(module:forward(input))
-- the rows accessed should be re-normalized
print(module.weight)
```

Outputs something like:

```
0.2194 1.4759 -1.1829
0.7069 0.2436 0.9876
-0.2955 0.3267 1.1844
-0.0575 -0.2957 1.5079
-0.2541 0.5331 -0.0083
0.8005 -1.5994 -0.4732
-0.0065 2.3441 -0.6354
0.2910 0.4230 0.0975
1.2662 1.1846 1.0114
-0.4095 -1.0676 -0.9056
[torch.DoubleTensor of size 10x3]
0.1152 0.7751 -0.6212
0.5707 0.1967 0.7973
0.1152 0.7751 -0.6212
-0.2808 -0.7319 -0.6209
[torch.DoubleTensor of size 4x3]
0.1152 0.7751 -0.6212
0.5707 0.1967 0.7973
-0.2955 0.3267 1.1844
-0.0575 -0.2957 1.5079
-0.2541 0.5331 -0.0083
0.8005 -1.5994 -0.4732
-0.0065 2.3441 -0.6354
0.2910 0.4230 0.0975
1.2662 1.1846 1.0114
-0.2808 -0.7319 -0.6209
[torch.DoubleTensor of size 10x3]
```

Note that the 1st, 2nd and 10th rows of the module.weight are updated to obey the max-norm constraint, since their indices appear in the "input".

### TemporalRowConvolution

```
module = nn.TemporalRowConvolution(inputFrameSize, kW, [dW], [featFirst]))
```

Applies a 1D row-oriented convolution over an input sequence composed of `nInputFrame`

frames. The input tensor in `forward(input)`

is expected to be a 2D tensor (`nInputFrame x inputFrameSize`

) or a 3D tensor (`nBatchFrame x nInputFrame x inputFrameSize`

). The layer can be used without a bias by `module:noBias()`

.

The parameters are the following:
* `inputFrameSize`

: The input frame size expected in sequences given into `forward()`

.
* `kW`

: The kernel width of the convolution.
* `dW`

: The step of the convolution Default is `1`

.
* `featFirst`

: Expects input to be in the form `nBatchFrame x inputFrameSize x nInputFrame`

is `true`

. Default is `false`

.

If the input sequence is a 2D tensor of dimension `nInputFrame x inputFrameSize`

, the output sequence will be `nOutputFrame x inputFrameSize`

where

```
lua
nOutputFrame = (nInputFrame - kW) / dW + 1
```

If the input sequence is a 3D tensor of dimension `nBatchFrame x nInputFrame x inputFrameSize`

, the output sequence will be `nBatch x nOutputFrame x outputFrameSize`

.

The parameters are the convolution can be found in `self.weight`

(Tensor of size `inputFrameSize x kW`

) and `self.bias`

(Tensor of size `inputFrameSize`

). The corresponding gradients can be found in `self.gradWeight`

and `self.gradBias`

.

For a 2D input, the output value of the layer can be precisely described as:

```
lua
output[t][i] = bias[i] + sum_{k=1}^kW weight[i][k] * input[dW(t-1)+k][i]
```

Here is a simple example: ```lua inp = 5; kw = 3; dw = 1;

-- row convolution with a kernel width of 3 (future context of 2) module = nn.TemporalRowConvolution(inp, kw, dw)

x = torch.rand(8, inp) print(module:forward(x)) ```

which gives

```
lua
0.1188 0.1945 0.1065 -0.0077 -0.3433
0.0630 0.4354 0.1954 -0.2103 -0.3506
0.0340 0.2222 0.3039 -0.2012 -0.3814
0.0820 0.3489 0.2533 -0.0940 -0.3298
0.1964 0.1533 0.1750 -0.1493 -0.3059
0.2651 0.2474 0.0521 -0.1134 -0.4024
[torch.Tensor of dimension 8x5]
```

More information about the layer can be found here.

## Spatial Modules

Excluding an optional batch dimension, spatial layers expect a 3D Tensor as input. The
first dimension is the number of features (e.g. `frameSize`

), the last two dimensions
are spatial (e.g. `height x width`

). These are commonly used for processing images.

### SpatialConvolution

```
module = nn.SpatialConvolution(nInputPlane, nOutputPlane, kW, kH, [dW], [dH], [padW], [padH])
```

Applies a 2D convolution over an input image composed of several input planes. The `input`

tensor in
`forward(input)`

is expected to be a 3D tensor (`nInputPlane x height x width`

).

The parameters are the following:
* `nInputPlane`

: The number of expected input planes in the image given into `forward()`

.
* `nOutputPlane`

: The number of output planes the convolution layer will produce.
* `kW`

: The kernel width of the convolution
* `kH`

: The kernel height of the convolution
* `dW`

: The step of the convolution in the width dimension. Default is `1`

.
* `dH`

: The step of the convolution in the height dimension. Default is `1`

.
* `padW`

: Additional zeros added to the input plane data on both sides of width axis. Default is `0`

. `(kW-1)/2`

is often used here.
* `padH`

: Additional zeros added to the input plane data on both sides of height axis. Default is `0`

. `(kH-1)/2`

is often used here.

Note that depending of the size of your kernel, several (of the last) columns or rows of the input image might be lost. It is up to the user to add proper padding in images.

If the input image is a 3D tensor `nInputPlane x height x width`

, the output image size
will be `nOutputPlane x oheight x owidth`

where

```
owidth = floor((width + 2*padW - kW) / dW + 1)
oheight = floor((height + 2*padH - kH) / dH + 1)
```

The parameters of the convolution can be found in `self.weight`

(Tensor of
size `nOutputPlane x nInputPlane x kH x kW`

) and `self.bias`

(Tensor of
size `nOutputPlane`

). The corresponding gradients can be found in
`self.gradWeight`

and `self.gradBias`

.

The output value of the layer can be precisely described as:

```
output[i][j][k] = bias[k]
+ sum_l sum_{s=1}^kW sum_{t=1}^kH weight[s][t][l][k]
* input[dW*(i-1)+s)][dH*(j-1)+t][l]
```

### SpatialConvolutionMap

```
module = nn.SpatialConvolutionMap(connectionMatrix, kW, kH, [dW], [dH])
```

This class is a generalization of nn.SpatialConvolution. It uses a generic connection table between input and output features. The nn.SpatialConvolution is equivalent to using a full connection table. One can specify different types of connection tables.

#### Full Connection Table

```
table = nn.tables.full(nin,nout)
```

This is a precomputed table that specifies connections between every input and output node.

#### One to One Connection Table

```
table = nn.tables.oneToOne(n)
```

This is a precomputed table that specifies a single connection to each output node from corresponding input node.

#### Random Connection Table

```
table = nn.tables.random(nin,nout, nto)
```

This table is randomly populated such that each output unit has
`nto`

incoming connections. The algorithm tries to assign uniform
number of outgoing connections to each input node if possible.

### SpatialFullConvolution

```
module = nn.SpatialFullConvolution(nInputPlane, nOutputPlane, kW, kH, [dW], [dH], [padW], [padH], [adjW], [adjH])
```

Applies a 2D full convolution over an input image composed of several input planes. The `input`

tensor in
`forward(input)`

is expected to be a 3D or 4D tensor. Note that instead of setting `adjW`

and `adjH`

, SpatialFullConvolution also accepts a table input with two tensors: `{convInput, sizeTensor}`

where `convInput`

is the standard input on which the full convolution
is applied, and the size of `sizeTensor`

is used to set the size of the output. Using the two-input version of forward
will ignore the `adjW`

and `adjH`

values used to construct the module. The layer can be used without a bias by module:noBias().

Other frameworks call this operation "In-network Upsampling", "Fractionally-strided convolution", "Backwards Convolution," "Deconvolution", or "Upconvolution."

The parameters are the following:
* `nInputPlane`

: The number of expected input planes in the image given into `forward()`

.
* `nOutputPlane`

: The number of output planes the convolution layer will produce.
* `kW`

: The kernel width of the convolution
* `kH`

: The kernel height of the convolution
* `dW`

: The step of the convolution in the width dimension. Default is `1`

.
* `dH`

: The step of the convolution in the height dimension. Default is `1`

.
* `padW`

: Additional zeros added to the input plane data on both sides of width axis. Default is `0`

. `(kW-1)/2`

is often used here.
* `padH`

: Additional zeros added to the input plane data on both sides of height axis. Default is `0`

. `(kH-1)/2`

is often used here.
* `adjW`

: Extra width to add to the output image. Default is `0`

. Cannot be greater than dW-1.
* `adjH`

: Extra height to add to the output image. Default is `0`

. Cannot be greater than dH-1.

If the input image is a 3D tensor `nInputPlane x height x width`

, the output image size
will be `nOutputPlane x oheight x owidth`

where

```
owidth = (width - 1) * dW - 2*padW + kW + adjW
oheight = (height - 1) * dH - 2*padH + kH + adjH
```

Further information about the full convolution can be found in the following paper: Fully Convolutional Networks for Semantic Segmentation.

### SpatialDilatedConvolution

```
module = nn.SpatialDilatedConvolution(nInputPlane, nOutputPlane, kW, kH, [dW], [dH], [padW], [padH], [dilationW], [dilationH])
```

Also sometimes referred to as **atrous convolution**.
Applies a 2D dilated convolution over an input image composed of several input planes. The `input`

tensor in
`forward(input)`

is expected to be a 3D or 4D tensor.

The parameters are the following:
* `nInputPlane`

: The number of expected input planes in the image given into `forward()`

.
* `nOutputPlane`

: The number of output planes the convolution layer will produce.
* `kW`

: The kernel width of the convolution
* `kH`

: The kernel height of the convolution
* `dW`

: The step of the convolution in the width dimension. Default is `1`

.
* `dH`

: The step of the convolution in the height dimension. Default is `1`

.
* `padW`

: Additional zeros added to the input plane data on both sides of width axis. Default is `0`

. `(kW-1)/2`

is often used here.
* `padH`

: Additional zeros added to the input plane data on both sides of height axis. Default is `0`

. `(kH-1)/2`

is often used here.
* `dilationW`

: The number of pixels to skip. Default is `1`

. `1`

makes it a SpatialConvolution
* `dilationH`

: The number of pixels to skip. Default is `1`

. `1`

makes it a SpatialConvolution

If the input image is a 3D tensor `nInputPlane x height x width`

, the output image size
will be `nOutputPlane x oheight x owidth`

where

```
owidth = floor((width + 2 * padW - dilationW * (kW-1) - 1) / dW) + 1
oheight = floor((height + 2 * padH - dilationH * (kH-1) - 1) / dH) + 1
```

Further information about the dilated convolution can be found in the following paper: Multi-Scale Context Aggregation by Dilated Convolutions.

### SpatialConvolutionLocal

```
module = nn.SpatialConvolutionLocal(nInputPlane, nOutputPlane, iW, iH, kW, kH, [dW], [dH], [padW], [padH])
```

Applies a 2D locally-connected layer over an input image composed of several input planes. The `input`

tensor in
`forward(input)`

is expected to be a 3D or 4D tensor.

A locally-connected layer is similar to a convolution layer but without weight-sharing.

The parameters are the following:
* `nInputPlane`

: The number of expected input planes in the image given into `forward()`

.
* `nOutputPlane`

: The number of output planes the locally-connected layer will produce.
* `iW`

: The input width.
* `iH`

: The input height.
* `kW`

: The kernel width.
* `kH`

: The kernel height.
* `dW`

: The step in the width dimension. Default is `1`

.
* `dH`

: The step in the height dimension. Default is `1`

.
* `padW`

: Additional zeros added to the input plane data on both sides of width axis. Default is `0`

.
* `padH`

: Additional zeros added to the input plane data on both sides of height axis. Default is `0`

.

If the input image is a 3D tensor `nInputPlane x iH x iW`

, the output image size
will be `nOutputPlane x oH x oW`

where

```
oW = floor((iW + 2*padW - kW) / dW + 1)
oH = floor((iH + 2*padH - kH) / dH + 1)
```

### SpatialLPPooling

```
module = nn.SpatialLPPooling(nInputPlane, pnorm, kW, kH, [dW], [dH])
```

Computes the `p`

norm in a convolutional manner on a set of 2D input planes.

### SpatialMaxPooling

```
module = nn.SpatialMaxPooling(kW, kH [, dW, dH, padW, padH])
```

Applies 2D max-pooling operation in `kWxkH`

regions by step size
`dWxdH`

steps. The number of output features is equal to the number of
input planes.

`nInputPlane x height x width`

, the output
image size will be `nOutputPlane x oheight x owidth`

where

```
owidth = op((width + 2*padW - kW) / dW + 1)
oheight = op((height + 2*padH - kH) / dH + 1)
```

`op`

is a rounding operator. By default, it is `floor`

. It can be changed
by calling `:ceil()`

or `:floor()`

methods.

### SpatialDilatedMaxPooling

```
module = nn.SpatialDilatedMaxPooling(kW, kH [, dW, dH, padW, padH, dilationW, dilationH])
```

Also sometimes referred to as **atrous pooling**.
Applies 2D dilated max-pooling operation in `kWxkH`

regions by step size
`dWxdH`

steps. The number of output features is equal to the number of
input planes. If `dilationW`

and `dilationH`

are not provided, this is equivalent to performing normal `nn.SpatialMaxPooling`

.

`nInputPlane x height x width`

, the output
image size will be `nOutputPlane x oheight x owidth`

where

```
owidth = op((width - (dilationW * (kW - 1) + 1) + 2*padW) / dW + 1)
oheight = op((height - (dilationH * (kH - 1) + 1) + 2*padH) / dH + 1)
```

`op`

is a rounding operator. By default, it is `floor`

. It can be changed
by calling `:ceil()`

or `:floor()`

methods.

### SpatialFractionalMaxPooling

```
module = nn.SpatialFractionalMaxPooling(kW, kH, outW, outH)
-- the output should be the exact size (outH x outW)
OR
module = nn.SpatialFractionalMaxPooling(kW, kH, ratioW, ratioH)
-- the output should be the size (floor(inH x ratioH) x floor(inW x ratioW))
-- ratios are numbers between (0, 1) exclusive
```

Applies 2D Fractional max-pooling operation as described in the paper "Fractional Max Pooling" by Ben Graham in the "pseudorandom" mode.

The max-pooling operation is applied in `kWxkH`

regions by a stochastic step size determined by the target output size.
The number of output features is equal to the number of input planes.

There are two constructors available.

Constructor 1:

```
module = nn.SpatialFractionalMaxPooling(kW, kH, outW, outH)
```

Constructor 2:

```
module = nn.SpatialFractionalMaxPooling(kW, kH, ratioW, ratioH)
```

If the input image is a 3D tensor `nInputPlane x height x width`

, the output
image size will be `nOutputPlane x oheight x owidth`

where

```
owidth = floor(width * ratioW)
oheight = floor(height * ratioH)
```

ratios are numbers between (0, 1) exclusive

### SpatialAveragePooling

```
module = nn.SpatialAveragePooling(kW, kH [, dW, dH, padW, padH])
```

Applies 2D average-pooling operation in `kWxkH`

regions by step size
`dWxdH`

steps. The number of output features is equal to the number of
input planes.

`nInputPlane x height x width`

, the output
image size will be `nOutputPlane x oheight x owidth`

where

```
owidth = op((width + 2*padW - kW) / dW + 1)
oheight = op((height + 2*padH - kH) / dH + 1)
```

`op`

is a rounding operator. By default, it is `floor`

. It can be changed
by calling `:ceil()`

or `:floor()`

methods.

By default, the output of each pooling region is divided by the number of
elements inside the padded image (which is usually `kW*kH`

, except in some
corner cases in which it can be smaller). You can also divide by the number
of elements inside the original non-padded image. To switch between different
division factors, call `:setCountIncludePad()`

or `:setCountExcludePad()`

. If
`padW=padH=0`

, both options give the same results.

### SpatialAdaptiveMaxPooling

```
module = nn.SpatialAdaptiveMaxPooling(W, H)
```

Applies 2D max-pooling operation in an image such that the output is of
size `WxH`

, for any input size. The number of output features is equal
to the number of input planes.

For an output of dimensions `(owidth,oheight)`

, the indexes of the pooling
region `(j,i)`

in the input image of dimensions `(iwidth,iheight)`

are
given by:

```
x_j_start = floor((j /owidth) * iwidth)
x_j_end = ceil(((j+1)/owidth) * iwidth)
y_i_start = floor((i /oheight) * iheight)
y_i_end = ceil(((i+1)/oheight) * iheight)
```

### SpatialAdaptiveAveragePooling

```
module = nn.SpatialAdaptiveAveragePooling(W, H)
```

Applies 2D average-pooling operation in an image such that the output is of
size `WxH`

, for any input size. The number of output features is equal
to the number of input planes.

The pooling region algorithm is the same as that in SpatialAdaptiveMaxPooling.

### SpatialMaxUnpooling

```
module = nn.SpatialMaxUnpooling(poolingModule)
```

Applies 2D "max-unpooling" operation using the indices previously computed
by the SpatialMaxPooling module `poolingModule`

.

When `B = poolingModule:forward(A)`

is called, the indices of the maximal
values (corresponding to their position within each map) are stored:
`B[{n,k,i,j}] = A[{n,k,indices[{n,k,i}],indices[{n,k,j}]}]`

.
If `C`

is a tensor of same size as `B`

, `module:updateOutput(C)`

outputs a
tensor `D`

of same size as `A`

such that:
`D[{n,k,indices[{n,k,i}],indices[{n,k,j}]}] = C[{n,k,i,j}]`

.

Module inspired by: "Visualizing and understanding convolutional networks" (2014) by Matthew Zeiler, Rob Fergus

### SpatialSubSampling

```
module = nn.SpatialSubSampling(nInputPlane, kW, kH, [dW], [dH])
```

Applies a 2D sub-sampling over an input image composed of several input planes. The `input`

tensor in
`forward(input)`

is expected to be a 3D tensor (`nInputPlane x height x width`

). The number of output
planes will be the same as `nInputPlane`

.

The parameters are the following:
* `nInputPlane`

: The number of expected input planes in the image given into `forward()`

.
* `kW`

: The kernel width of the sub-sampling
* `kH`

: The kernel height of the sub-sampling
* `dW`

: The step of the sub-sampling in the width dimension. Default is `1`

.
* `dH`

: The step of the sub-sampling in the height dimension. Default is `1`

.

Note that depending of the size of your kernel, several (of the last) columns or rows of the input image might be lost. It is up to the user to add proper padding in images.

If the input image is a 3D tensor `nInputPlane x height x width`

, the output image size
will be `nInputPlane x oheight x owidth`

where

```
owidth = (width - kW) / dW + 1
oheight = (height - kH) / dH + 1 .
```

The parameters of the sub-sampling can be found in `self.weight`

(Tensor of
size `nInputPlane`

) and `self.bias`

(Tensor of size `nInputPlane`

). The
corresponding gradients can be found in `self.gradWeight`

and
`self.gradBias`

.

The output value of the layer can be precisely described as:

```
output[i][j][k] = bias[k]
+ weight[k] sum_{s=1}^kW sum_{t=1}^kH input[dW*(i-1)+s)][dH*(j-1)+t][k]
```

### SpatialUpSamplingNearest

```
module = nn.SpatialUpSamplingNearest(scale)
```

Applies a 2D up-sampling over an input image composed of several input planes. The `input`

tensor in
`forward(input)`

is expected to be a 3D or 4D tensor (i.e. for 4D: `nBatchPlane x nInputPlane x height x width`

). The number of output planes will be the same. The v dimension is assumed to be the second last dimension (i.e. for 4D it will be the 3rd dim), and the u dimension is assumed to be the last dimension.

The parameters are the following:
* `scale`

: The upscale ratio. Must be a positive integer

The up-scaling method is simple nearest neighbor, ie:

```
output(u,v) = input(floor((u-1)/scale)+1, floor((v-1)/scale)+1)
```

Where `u`

and `v`

are index from 1 (as per lua convention). There are no learnable parameters.

### SpatialUpSamplingBilinear

```
module = nn.SpatialUpSamplingBilinear(scale)
module = nn.SpatialUpSamplingBilinear({oheight=H, owidth=W})
```

Applies a 2D up-sampling over an input image composed of several input planes. The `input`

tensor in
`forward(input)`

is expected to be a 3D or 4D tensor (i.e. for 4D: `nBatchPlane x nInputPlane x height x width`

). The number of output planes will be the same. The v dimension is assumed to be the second last dimension (i.e. for 4D it will be the 3rd dim), and the u dimension is assumed to be the last dimension.

The parameters are the following:
* `scale`

: The upscale ratio. Must be a positive integer
* Or a table `{oheight=H, owidth=W}`

: The required output height and width, should be positive integers.

The up-scaling method is bilinear.
If `scale`

is specified, given an input of height iH and width iW, output height and width will be:

```
oH = (iH - 1)(scale - 1) + iH
oW = (iW - 1)(scale - 1) + iW
```

There are no learnable parameters.

### SpatialZeroPadding

```
module = nn.SpatialZeroPadding(padLeft, padRight, padTop, padBottom)
```

Each feature map of a given input is padded with specified number of zeros. If padding values are negative, then input is cropped.

### SpatialReflectionPadding

```
module = nn.SpatialReflectionPadding(padLeft, padRight, padTop, padBottom)
```

Each feature map of a given input is padded with the reflection of the input boundary

### SpatialReplicationPadding

```
module = nn.SpatialReplicationPadding(padLeft, padRight, padTop, padBottom)
```

Each feature map of a given input is padded with the replication of the input boundary

### SpatialSubtractiveNormalization

```
module = nn.SpatialSubtractiveNormalization(ninputplane, kernel)
```

Applies a spatial subtraction operation on a series of 2D inputs using
`kernel`

for computing the weighted average in a neighborhood. The
neighborhood is defined for a local spatial region that is the size as
kernel and across all features. For a an input image, since there is
only one feature, the region is only spatial. For an RGB image, the
weighted average is taken over RGB channels and a spatial region.

If the `kernel`

is 1D, then it will be used for constructing and seperable
2D kernel. The operations will be much more efficient in this case.

The kernel is generally chosen as a gaussian when it is believed that the correlation of two pixel locations decrease with increasing distance. On the feature dimension, a uniform average is used since the weighting across features is not known.

For this example we use an external package image

```
require 'image'
require 'nn'
lena = image.rgb2y(image.lena())
ker = torch.ones(11)
m=nn.SpatialSubtractiveNormalization(1,ker)
processed = m:forward(lena)
w1=image.display(lena)
w2=image.display(processed)
```

### SpatialCrossMapLRN

```
module = nn.SpatialCrossMapLRN(size [,alpha] [,beta] [,k])
```

Applies Spatial Local Response Normalization between different feature maps.
By default, `alpha = 0.0001`

, `beta = 0.75`

and `k = 1`

The operation implemented is:

```
x_f
y_f = -------------------------------------------------
(k+(alpha/size)* sum_{l=l1 to l2} (x_l^2))^beta
```

where `x_f`

is the input at spatial locations `h,w`

(not shown for simplicity) and feature map `f`

,
`l1`

corresponds to `max(0,f-ceil(size/2))`

and `l2`

to `min(F, f-ceil(size/2) + size)`

. Here, `F`

is the number of feature maps.
More information can be found here.

## SpatialBatchNormalization

`module`

= `nn.SpatialBatchNormalization(N [,eps] [, momentum] [,affine])`

where N = number of input feature maps
eps is a small value added to the standard-deviation to avoid divide-by-zero. Defaults to 1e-5
`affine`

is a boolean. When set to false, the learnable affine transform is disabled. Defaults to true

Implements Batch Normalization as described in the paper: "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift" by Sergey Ioffe, Christian Szegedy

The operation implemented is:

```
y = ( x - mean(x) )
-------------------- * gamma + beta
standard-deviation(x)
```

where the mean and standard-deviation are calculated per feature-map over the mini-batches and pixels and where gamma and beta are learnable parameter vectors of size N (where N = number of feature maps). The learning of gamma and beta is optional.

In training time, this layer keeps a running estimate of it's computed mean and std.
The running sum is kept with a default momentup of 0.1 (unless over-ridden)
In test time, this running mean/std is used to normalize. (**Note that the running mean/std will not be saved if one only checkpoints a model's parameters. In order to correctly use the calculated running mean/std, one needs to checkpoint the model itself (call clearState() first to save space).**)

The module only accepts 4D inputs.

```
-- with learnable parameters
model = nn.SpatialBatchNormalization(m)
A = torch.randn(b, m, h, w)
C = model:forward(A) -- C will be of size `b x m x h x w`
-- without learnable parameters
model = nn.SpatialBatchNormalization(m, nil, nil, false)
A = torch.randn(b, m, h, w)
C = model:forward(A) -- C will be of size `b x m x h x w`
```

## Volumetric Modules

Excluding an optional batch dimension, volumetric layers expect a 4D Tensor as input. The
first dimension is the number of features (e.g. `frameSize`

), the second is sequential (e.g. `time`

) and the
last two dimensions are spatial (e.g. `height x width`

). These are commonly used for processing videos (sequences of images).

### VolumetricConvolution

```
module = nn.VolumetricConvolution(nInputPlane, nOutputPlane, kT, kW, kH [, dT, dW, dH, padT, padW, padH])
```

Applies a 3D convolution over an input image composed of several input planes. The `input`

tensor in
`forward(input)`

is expected to be a 4D tensor (`nInputPlane x time x height x width`

).

The parameters are the following:
* `nInputPlane`

: The number of expected input planes in the image given into `forward()`

.
* `nOutputPlane`

: The number of output planes the convolution layer will produce.
* `kT`

: The kernel size of the convolution in time
* `kW`

: The kernel width of the convolution
* `kH`

: The kernel height of the convolution
* `dT`

: The step of the convolution in the time dimension. Default is `1`

.
* `dW`

: The step of the convolution in the width dimension. Default is `1`

.
* `dH`

: The step of the convolution in the height dimension. Default is `1`

.
* `padT`

: Additional zeros added to the input plane data on both sides of time axis. Default is `0`

. `(kT-1)/2`

is often used here.
* `padW`

: Additional zeros added to the input plane data on both sides of width axis. Default is `0`

. `(kW-1)/2`

is often used here.
* `padH`

: Additional zeros added to the input plane data on both sides of height axis. Default is `0`

. `(kH-1)/2`

is often used here.

Note that depending of the size of your kernel, several (of the last) columns or rows of the input image might be lost. It is up to the user to add proper padding in images. This layer can be used without a bias by module:noBias().

If the input image is a 4D tensor `nInputPlane x time x height x width`

, the output image size
will be `nOutputPlane x otime x oheight x owidth`

where

```
otime = floor((time + 2*padT - kT) / dT + 1)
owidth = floor((width + 2*padW - kW) / dW + 1)
oheight = floor((height + 2*padH - kH) / dH + 1)
```

The parameters of the convolution can be found in `self.weight`

(Tensor of
size `nOutputPlane x nInputPlane x kT x kH x kW`

) and `self.bias`

(Tensor of
size `nOutputPlane`

). The corresponding gradients can be found in
`self.gradWeight`

and `self.gradBias`

.

### VolumetricFullConvolution

```
module = nn.VolumetricFullConvolution(nInputPlane, nOutputPlane, kT, kW, kH, [dT], [dW], [dH], [padT], [padW], [padH], [adjT], [adjW], [adjH])
```

Applies a 3D full convolution over an input image composed of several input planes. The `input`

tensor in
`forward(input)`

is expected to be a 4D or 5D tensor. Note that instead of setting `adjT`

, `adjW`

and `adjH`

, VolumetricFullConvolution also accepts a table input with two tensors: `{convInput, sizeTensor}`

where `convInput`

is the standard input on which the full convolution is applied, and the size of `sizeTensor`

is used to set the size of the output. Using the two-input version of forward
will ignore the `adjT`

, `adjW`

and `adjH`

values used to construct the module.

This can be used as 3D deconvolution, or 3D upsampling. So that the 3D FCN can be easly implemented. This layer can be used without a bias by module:noBias().

The parameters are the following:
* nInputPlane: The number of expected input planes in the image given into forward().
*

`nOutputPlane`

: The number of output planes the convolution layer will produce.

`kT`

: The kernel depth of the convolution
`kW`

: The kernel width of the convolution

`kH`

: The kernel height of the convolution
`dT`

: The step of the convolution in the depth dimension. Default is `1`

.

`dW`

: The step of the convolution in the width dimension. Default is `1`

.
`dH`

: The step of the convolution in the height dimension. Default is `1`

.

`padT`

: Additional zeros added to the input plane data on both sides of time (depth) axis. Default is `0`

. `(kT-1)/2`

is often used here.
`padW`

: Additional zeros added to the input plane data on both sides of width axis. Default is `0`

. `(kW-1)/2`

is often used here.

`padH`

: Additional zeros added to the input plane data on both sides of height axis. Default is `0`

. `(kH-1)/2`

is often used here.
`adjT`

: Extra depth to add to the output image. Default is `0`

. Cannot be greater than dT-1.

`adjW`

: Extra width to add to the output image. Default is `0`

. Cannot be greater than dW-1.
`adjH`

: Extra height to add to the output image. Default is `0`

. Cannot be greater than dH-1.If the input image is a 3D tensor `nInputPlane x depth x height x width`

, the output image size
will be `nOutputPlane x odepth x oheight x owidth`

where

```
odepth = (depth - 1) * dT - 2*padT + kT + adjT
owidth = (width - 1) * dW - 2*padW + kW + adjW
oheight = (height - 1) * dH - 2*padH + kH + adjH
```

### VolumetricDilatedConvolution

```
module = nn.VolumetricDilatedConvolution(nInputPlane, nOutputPlane, kT, kW, kH, [dT], [dW], [dH], [padT], [padW], [padH], [dilationT], [dilationW], [dilationH])
```

Applies a 3D dilated convolution over an input image composed of several input planes. The `input`

tensor in
`forward(input)`

is expected to be a 4D or 5D tensor.

The parameters are the following:
* `nInputPlane`

: The number of expected input planes in the image given into `forward()`

.
* `nOutputPlane`

: The number of output planes the convolution layer will produce.
* `kT`

: The kernel depth of the convolution
* `kW`

: The kernel width of the convolution
* `kH`

: The kernel height of the convolution
* `dT`

: The step of the convolution in the depth dimension. Default is `1`

.
* `dW`

: The step of the convolution in the width dimension. Default is `1`

.
* `dH`

: The step of the convolution in the height dimension. Default is `1`

.
* `padT`

: Additional zeros added to the input plane data on both sides of time (depth) axis. Default is `0`

. `(kT-1)/2`

is often used here.
* `padW`

: Additional zeros added to the input plane data on both sides of width axis. Default is `0`

. `(kW-1)/2`

is often used here.
* `padH`

: Additional zeros added to the input plane data on both sides of height axis. Default is `0`

. `(kH-1)/2`

is often used here.
* `dilationT`

: The number of pixels to skip. Default is `1`

. `1`

makes it a VolumetricConvolution
* `dilationW`

: The number of pixels to skip. Default is `1`

. `1`

makes it a VolumetricConvolution
* `dilationH`

: The number of pixels to skip. Default is `1`

. `1`

makes it a VolumetricConvolution

If the input image is a 4D tensor `nInputPlane x depth x height x width`

, the output image size
will be `nOutputPlane x odepth x oheight x owidth`

where

```
odepth = floor((depth + 2 * padT - dilationT * (kT-1) + 1) / dT) + 1
owidth = floor((width + 2 * padW - dilationW * (kW-1) + 1) / dW) + 1
oheight = floor((height + 2 * padH - dilationH * (kH-1) + 1) / dH) + 1
```

Further information about the dilated convolution can be found in the following paper: Multi-Scale Context Aggregation by Dilated Convolutions.

### VolumetricMaxPooling

```
module = nn.VolumetricMaxPooling(kT, kW, kH [, dT, dW, dH, padT, padW, padH])
```

Applies 3D max-pooling operation in `kTxkWxkH`

regions by step size
`dTxdWxdH`

steps. The number of output features is equal to the number of
input planes / dT. The input can optionally be padded with zeros. Padding should be smaller than half of kernel size. That is, `padT < kT/2`

, `padW < kW/2`

and `padH < kH/2`

.

### VolumetricDilatedMaxPooling

```
module = nn.VolumetricDilatedMaxPooling(kT, kW, kH [, dT, dW, dH, padT, padW, padH, dilationT, dilationW, dilationH])
```

Also sometimes referred to as **atrous pooling**.
Applies 3D dilated max-pooling operation in `kTxkWxkH`

regions by step size
`dTxdWxdH`

steps. The number of output features is equal to the number of
input planes. If `dilationT`

, `dilationW`

and `dilationH`

are not provided, this is equivalent to performing normal `nn.VolumetricMaxPooling`

.

If the input image is a 4D tensor `nInputPlane x depth x height x width`

, the output
image size will be `nOutputPlane x otime x oheight x owidth`

where

```
otime = op((depth - (dilationT * (kT - 1) + 1) + 2*padT) / dT + 1)
owidth = op((width - (dilationW * (kW - 1) + 1) + 2*padW) / dW + 1)
oheight = op((height - (dilationH * (kH - 1) + 1) + 2*padH) / dH + 1)
```

`op`

is a rounding operator. By default, it is `floor`

. It can be changed
by calling `:ceil()`

or `:floor()`

methods.

### VolumetricFractionalMaxPooling

```
module = nn.VolumetricFractionalMaxPooling(kT, kW, kH, outT, outW, outH)
-- the output should be the exact size (outH x outW x outT)
OR
module = nn.VolumetricFractionalMaxPooling(kT, kW, kH, ratioT, ratioW, ratioH)
-- the output should be the size (floor(inH x ratioH) x floor(inW x ratioW) x floor(inT x ratioT))
-- ratios are numbers between (0, 1) exclusive
```

Applies 3D Fractional max-pooling operation in the "pseudorandom" mode, analogous to SpatialFractionalMaxPooling.

The max-pooling operation is applied in `kTxkWxkH`

regions by a stochastic step size determined by the target output size.
The number of output features is equal to the number of input planes.

There are two constructors available.

Constructor 1:

```
module = nn.VolumetricFractionalMaxPooling(kT, kW, kH, outT, outW, outH)
```

Constructor 2:

```
module = nn.VolumetricFractionalMaxPooling(kT, kW, kH, ratioT, ratioW, ratioH)
```

If the input image is a 4D tensor `nInputPlane x height x width x time`

, the output
image size will be `nOutputPlane x oheight x owidth x otime`

where

```
otime = floor(time * ratioT)
owidth = floor(width * ratioW)
oheight = floor(height * ratioH)
```

ratios are numbers between (0, 1) exclusive

### VolumetricAveragePooling

```
module = nn.VolumetricAveragePooling(kT, kW, kH [, dT, dW, dH])
```

Applies 3D average-pooling operation in `kTxkWxkH`

regions by step size
`dTxdWxdH`

steps. The number of output features is equal to the number of
input planes / dT.

### VolumetricMaxUnpooling

```
module = nn.VolumetricMaxUnpooling(poolingModule)
```

Applies 3D "max-unpooling" operation using the indices previously computed
by the VolumetricMaxPooling module `poolingModule`

.

When `B = poolingModule:forward(A)`

is called, the indices of the maximal
values (corresponding to their position within each map) are stored:
`B[{n,k,t,i,j}] = A[{n,k,indices[{n,k,t}],indices[{n,k,i}],indices[{n,k,j}]}]`

.
If `C`

is a tensor of same size as `B`

, `module:updateOutput(C)`

outputs a
tensor `D`

of same size as `A`

such that:
`D[{n,k,indices[{n,k,t}],indices[{n,k,i}],indices[{n,k,j}]}] = C[{n,k,t,i,j}]`

.

### VolumetricReplicationPadding

```
module = nn.VolumetricReplicationPadding(padLeft, padRight, padTop, padBottom,
padFront, padBack)
```

Each feature map of a given input is padded with the replication of the input boundary.