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# 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 base on the dimensionality of the input and output Tensors:

## 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 dimenstion 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 (inputFrameSize x kW) ) 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][j][k]
                                * 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[i][t] = bias[i] + weight[i] * sum_{k=1}^kW input[i][dW*(t-1)+k)]
 ## LookupTable ## 
module = nn.LookupTable(nIndex, sizes)

or

module = nn.LookupTable(nIndex, size1, [size2], [size3], ...)

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 sizes (a LongStorage) or size1 x size2 x....

Given a 1D input, the output tensors are concatenated, 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:

M P

where M is a 2D matrix size1 x nIndex containing the parameters of the lookup-table and P is a 2D matrix, where each column vector i is a zero vector except 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 size1 x size2 x ... x sizeN 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]
 ## Spatial Modules ## Excluding and 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 dimenstions 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: The additional zeros added per width to the input planes. Default is 0, a good number is (kW-1)/2.
  • padH: The additional zeros added per height to the input planes. Default is padW, a good number is (kH-1)/2.

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 algorihtm tries to assign uniform number of outgoing connections to each input node if possible.

 ### 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])

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.

 ### SpatialAveragePooling ### 
module = nn.SpatialAveragePooling(kW, kH [, dW, dH])

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.

 ### 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)
 ### 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.

 ### 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.

 ### 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 anerage 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)

## 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.

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 and 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 dimenstions 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])

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.

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 4D tensor nInputPlane x time x height x width, the output image size will be nOutputPlane x otime x owidth x oheight where

otime   = (time  - kT)  / dT + 1
owidth  = (width  - kW) / dW + 1
oheight = (height - 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.

 ### VolumetricMaxPooling ### 
module = nn.VolumetricMaxPooling(kT, kW, kH [, dT, dW, dH])

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.

 ### 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.