Simple Modules are used for various tasks like adapting Tensor methods and providing affine transformations :
- Parameterized Modules :
- Linear : a linear transformation ;
- SparseLinear : a linear transformation with sparse inputs ;
- Add : adds a bias term to the incoming data ;
- Mul : multiply a single scalar factor to the incoming data ;
- CMul : a component-wise multiplication to the incoming data ;
- CDiv : a component-wise division to the incoming data ;
- Euclidean : the euclidean distance of the input to
kmean centers ; - WeightedEuclidean : similar to Euclidean, but additionally learns a diagonal covariance matrix ;
- Modules that adapt basic Tensor methods :
- Modules that adapt mathematical Tensor methods :
- Max : a max operation over a given dimension ;
- Min : a min operation over a given dimension ;
- Mean : a mean operation over a given dimension ;
- Sum : a sum operation over a given dimension ;
- Exp : an element-wise exp operation ;
- Abs : an element-wise abs operation ;
- Power : an element-wise pow operation ;
- Square : an element-wise square operation ;
- Sqrt : an element-wise sqrt operation ;
- Normalize : normalizes the input to have unit
L_pnorm ; - MM : matrix-matrix multiplication (also supports batches of matrices) ;
- Miscellaneous Modules :
- BatchNormalization - mean/std normalization over the mini-batch inputs (with an optional affine transform) ;
- Identity : forward input as-is to output (useful with ParallelTable);
- Dropout : masks parts of the
inputusing binary samples from a bernoulli distribution ; - SpatialDropout : Same as Dropout but for spatial inputs where adjacent pixels are strongly correlated ;
- Padding : adds padding to a dimension ;
- L1Penalty : adds an L1 penalty to an input (for sparsity);
module = nn.Linear(inputDimension, outputDimension)Applies a linear transformation to the incoming data, i.e. y = Ax + b. The input tensor given in forward(input) must be either a vector (1D tensor) or matrix (2D tensor). If the input is a matrix, then each row is assumed to be an input sample of given batch.
You can create a layer in the following way:
module = nn.Linear(10, 5) -- 10 inputs, 5 outputsUsually this would be added to a network of some kind, e.g.:
mlp = nn.Sequential()
mlp:add(module)The weights and biases (A and b) can be viewed with:
print(module.weight)
print(module.bias)The gradients for these weights can be seen with:
print(module.gradWeight)
print(module.gradBias)As usual with nn modules, applying the linear transformation is performed with:
x = torch.Tensor(10) -- 10 inputs
y = module:forward(x)module = nn.SparseLinear(inputDimension, outputDimension)Applies a linear transformation to the incoming sparse data, i.e. y = Ax + b. The input tensor given in forward(input) must be a sparse vector represented as 2D tensor of the form torch.Tensor(N, 2) where the pairs represent indices and values.
The SparseLinear layer is useful when the number of input dimensions is very large and the input data is sparse.
You can create a sparse linear layer in the following way:
module = nn.SparseLinear(10000, 2) -- 10000 inputs, 2 outputsThe sparse linear module may be used as part of a larger network, and apart from the form of the input, SparseLinear operates in exactly the same way as the Linear layer.
A sparse input vector may be created as so...
x = torch.Tensor({ {1, 0.1}, {2, 0.3}, {10, 0.3}, {31, 0.2} })
print(x)
1.0000 0.1000
2.0000 0.3000
10.0000 0.3000
31.0000 0.2000
[torch.Tensor of dimension 4x2]The first column contains indices, the second column contains values in a a vector where all other elements are zeros. The indices should not exceed the stated dimensions of the input to the layer (10000 in the example).
module = nn.Dropout(p)During training, Dropout masks parts of the input using binary samples from a bernoulli distribution.
Each input element has a probability of p of being dropped, i.e having its commensurate output element be zero. This has proven an effective technique for regularization and preventing the co-adaptation of neurons (see Hinton et al. 2012).
Furthermore, the ouputs are scaled by a factor of 1/(1-p) during training. This allows the input to be simply forwarded as-is during evaluation.
In this example, we demonstrate how the call to forward samples different outputs to dropout (the zeros) given the same input:
module = nn.Dropout()
> x = torch.Tensor{{1, 2, 3, 4}, {5, 6, 7, 8}}
> module:forward(x)
2 0 0 8
10 0 14 0
[torch.DoubleTensor of dimension 2x4]
> module:forward(x)
0 0 6 0
10 0 0 0
[torch.DoubleTensor of dimension 2x4]Backward drops out the gradients at the same location:
> module:forward(x)
0 4 0 0
10 12 0 16
[torch.DoubleTensor of dimension 2x4]
> module:backward(x, x:clone():fill(1))
0 2 0 0
2 2 0 2
[torch.DoubleTensor of dimension 2x4]In both cases the gradOutput and input are scaled by 1/(1-p), which in this case is 2.
During evaluation, Dropout does nothing more than forward the input such that all elements of the input are considered.
> module:evaluate()
> module:forward(x)
1 2 3 4
5 6 7 8
[torch.DoubleTensor of dimension 2x4]We can return to training our model by first calling Module:training():
> module:training()
> return module:forward(x)
2 4 6 0
0 0 0 16
[torch.DoubleTensor of dimension 2x4]When used, Dropout should normally be applied to the input of parameterized Modules like Linear or SpatialConvolution. A p of 0.5 (the default) is usually okay for hidden layers. Dropout can sometimes be used successfully on the dataset inputs with a p around 0.2. It sometimes works best following Transfer Modules like ReLU. All this depends a great deal on the dataset so its up to the user to try different combinations.
module = nn.SpatialDropout(p)
This version performs the same function as nn.Dropout, however it assumes the 2 right-most dimensions of the input are spatial, performs one Bernoulli trial per output feature when training, and extends this dropout value across the entire feature map.
As described in the paper "Efficient Object Localization Using Convolutional Networks" (http://arxiv.org/abs/1411.4280), if adjacent pixels within feature maps are strongly correlated (as is normally the case in early convolution layers) then iid dropout will not regularize the activations and will otherwise just result in an effective learning rate decrease. In this case, nn.SpatialDropout will help promote independence between feature maps and should be used instead.
nn.SpatialDropout accepts 3D or 4D inputs. If the input is 3D than a layout of (features x height x width) is assumed and for 4D (batch x features x height x width) is assumed.
module = Abs()m = nn.Abs()
ii = torch.linspace(-5, 5)
oo = m:forward(ii)
go = torch.ones(100)
gi = m:backward(ii, go)
gnuplot.plot({'f(x)', ii, oo, '+-'}, {'df/dx', ii, gi, '+-'})
gnuplot.grid(true)
module = nn.Add(inputDimension, scalar)Applies a bias term to the incoming data, i.e. yi = x_i + b_i, or if scalar = true then uses a single bias term, yi = x_i + b.
Example:
y = torch.Tensor(5)
mlp = nn.Sequential()
mlp:add(nn.Add(5))
function gradUpdate(mlp, x, y, criterion, learningRate)
local pred = mlp:forward(x)
local err = criterion:forward(pred, y)
local gradCriterion = criterion:backward(pred, y)
mlp:zeroGradParameters()
mlp:backward(x, gradCriterion)
mlp:updateParameters(learningRate)
return err
end
for i = 1, 10000 do
x = torch.rand(5)
y:copy(x);
for i = 1, 5 do y[i] = y[i] + i; end
err = gradUpdate(mlp, x, y, nn.MSECriterion(), 0.01)
end
print(mlp:get(1).bias)gives the output:
1.0000
2.0000
3.0000
4.0000
5.0000
[torch.Tensor of dimension 5]i.e. the network successfully learns the input x has been shifted to produce the output y.
module = nn.Mul()Applies a single scaling factor to the incoming data, i.e. y = w x, where w is a scalar.
Example:
y = torch.Tensor(5)
mlp = nn.Sequential()
mlp:add(nn.Mul())
function gradUpdate(mlp, x, y, criterion, learningRate)
local pred = mlp:forward(x)
local err = criterion:forward(pred, y)
local gradCriterion = criterion:backward(pred, y)
mlp:zeroGradParameters()
mlp:backward(x, gradCriterion)
mlp:updateParameters(learningRate)
return err
end
for i = 1, 10000 do
x = torch.rand(5)
y:copy(x)
y:mul(math.pi)
err = gradUpdate(mlp, x, y, nn.MSECriterion(), 0.01)
end
print(mlp:get(1).weight)gives the output:
3.1416
[torch.Tensor of dimension 1]i.e. the network successfully learns the input x has been scaled by pi.
module = nn.CMul(size)Applies a component-wise multiplication to the incoming data, i.e. y_i = w_i * x_i. Argument size can be one or many numbers (sizes) or a torch.LongStorage. For example, nn.CMul(3,4,5) is equivalent to nn.CMul(torch.LongStorage{3,4,5}).
Example:
mlp = nn.Sequential()
mlp:add(nn.CMul(5))
y = torch.Tensor(5)
sc = torch.Tensor(5)
for i = 1, 5 do sc[i] = i; end -- scale input with this
function gradUpdate(mlp, x, y, criterion, learningRate)
local pred = mlp:forward(x)
local err = criterion:forward(pred, y)
local gradCriterion = criterion:backward(pred, y)
mlp:zeroGradParameters()
mlp:backward(x, gradCriterion)
mlp:updateParameters(learningRate)
return err
end
for i = 1, 10000 do
x = torch.rand(5)
y:copy(x)
y:cmul(sc)
err = gradUpdate(mlp, x, y, nn.MSECriterion(), 0.01)
end
print(mlp:get(1).weight)gives the output:
1.0000
2.0000
3.0000
4.0000
5.0000
[torch.Tensor of dimension 5]i.e. the network successfully learns the input x has been scaled by those scaling factors to produce the output y.
module = nn.Max(dimension)Applies a max operation over dimension dimension.
Hence, if an nxpxq Tensor was given as input, and dimension = 2 then an nxq matrix would be output.
module = nn.Min(dimension)Applies a min operation over dimension dimension.
Hence, if an nxpxq Tensor was given as input, and dimension = 2 then an nxq matrix would be output.
module = nn.Mean(dimension)Applies a mean operation over dimension dimension.
Hence, if an nxpxq Tensor was given as input, and dimension = 2 then an nxq matrix would be output.
module = nn.Sum(dimension)Applies a sum operation over dimension dimension.
Hence, if an nxpxq Tensor was given as input, and dimension = 2 then an nxq matrix would be output.
module = nn.Euclidean(inputSize,outputSize)Outputs the Euclidean distance of the input to outputSize centers, i.e. this layer has the weights w_j, for j = 1,..,outputSize, where w_j are vectors of dimension inputSize.
The distance y_j between center j and input x is formulated as y_j = || w_j - x ||.
module = nn.WeightedEuclidean(inputSize,outputSize)This module is similar to Euclidean, but additionally learns a separate diagonal covariance matrix across the features of the input space for each center.
In other words, for each of the outputSize centers w_j, there is a diagonal covariance matrices c_j, for j = 1,..,outputSize, where c_j are stored as vectors of size inputSize.
The distance y_j between center j and input x is formulated as y_j = || c_j * (w_j - x) ||.
module = nn.Identity()Creates a module that returns whatever is input to it as output. This is useful when combined with the module ParallelTable in case you do not wish to do anything to one of the input Tensors.
Example:
mlp = nn.Identity()
print(mlp:forward(torch.ones(5, 2)))gives the output:
1 1
1 1
1 1
1 1
1 1
[torch.Tensor of dimension 5x2]Here is a more useful example, where one can implement a network which also computes a Criterion using this module:
pred_mlp = nn.Sequential() -- A network that makes predictions given x.
pred_mlp:add(nn.Linear(5, 4))
pred_mlp:add(nn.Linear(4, 3))
xy_mlp = nn.ParallelTable() -- A network for predictions and for keeping the
xy_mlp:add(pred_mlp) -- true label for comparison with a criterion
xy_mlp:add(nn.Identity()) -- by forwarding both x and y through the network.
mlp = nn.Sequential() -- The main network that takes both x and y.
mlp:add(xy_mlp) -- It feeds x and y to parallel networks;
cr = nn.MSECriterion()
cr_wrap = nn.CriterionTable(cr)
mlp:add(cr_wrap) -- and then applies the criterion.
for i = 1, 100 do -- Do a few training iterations
x = torch.ones(5) -- Make input features.
y = torch.Tensor(3)
y:copy(x:narrow(1,1,3)) -- Make output label.
err = mlp:forward{x,y} -- Forward both input and output.
print(err) -- Print error from criterion.
mlp:zeroGradParameters() -- Do backprop...
mlp:backward({x, y})
mlp:updateParameters(0.05)
endmodule = nn.Copy(inputType, outputType, [forceCopy, dontCast])This layer copies the input to output with type casting from input type from inputType to outputType. Unless forceCopy is true, when the first two arguments are the same, the input isn't copied, only transfered as the output. The default forceCopy is false.
When dontCast is true, a call to nn.Copy:type(type) will not cast the module's output and gradInput Tensors to the new type. The default is false.
module = nn.Narrow(dimension, offset, length)Narrow is application of narrow operation in a module.
module = nn.Replicate(nFeature [, dim, ndim])This class creates an output where the input is replicated nFeature times along dimension dim (default 1).
There is no memory allocation or memory copy in this module.
It sets the stride along the dimth dimension to zero.
When provided, ndim should specify the number of non-batch dimensions.
This allows the module to replicate the same non-batch dimension dim for both batch and non-batch inputs.
> x = torch.linspace(1, 5, 5)
1
2
3
4
5
[torch.DoubleTensor of dimension 5]
> m = nn.Replicate(3)
> o = m:forward(x)
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
[torch.DoubleTensor of dimension 3x5]
> x:fill(13)
13
13
13
13
13
[torch.DoubleTensor of dimension 5]
> print(o)
13 13 13 13 13
13 13 13 13 13
13 13 13 13 13
[torch.DoubleTensor of dimension 3x5]module = nn.Reshape(dimension1, dimension2, ... [, batchMode])Reshapes an nxpxqx.. Tensor into a dimension1xdimension2x... Tensor, taking the elements column-wise.
The optional last argument batchMode, when true forces the first dimension of the input to be considered the batch dimension, and thus keep its size fixed. This is necessary when dealing with batch sizes of one. When false, it forces the entire input (including the first dimension) to be reshaped to the input size. Default batchMode=nil, which means that the module considers inputs with more elements than the produce of provided sizes, i.e. dimension1xdimension2x..., to be batches.
Example:
> x = torch.Tensor(4,4)
> for i = 1, 4 do
> for j = 1, 4 do
> x[i][j] = (i-1)*4+j
> end
> end
> print(x)
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
[torch.Tensor of dimension 4x4]
> print(nn.Reshape(2,8):forward(x))
1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16
[torch.Tensor of dimension 2x8]
> print(nn.Reshape(8,2):forward(x))
1 2
3 4
5 6
7 8
9 10
11 12
13 14
15 16
[torch.Tensor of dimension 8x2]
> print(nn.Reshape(16):forward(x))
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
[torch.Tensor of dimension 16]
> y = torch.Tensor(1, 4):fill(0)
> print(y)
0 0 0 0
[torch.DoubleTensor of dimension 1x4]
> print(nn.Reshape(4):forward(y))
0 0 0 0
[torch.DoubleTensor of dimension 1x4]
> print(nn.Reshape(4, false):forward(y))
0
0
0
0
[torch.DoubleTensor of dimension 4]
module = nn.View(sizes)This module creates a new view of the input tensor using the sizes passed to the constructor. The parameter sizes can either be a LongStorage or numbers.
The method setNumInputDims() allows to specify the expected number of dimensions of the inputs of the modules. This makes it possible to use minibatch inputs when using a size -1 for one of the dimensions.
Example 1:
> x = torch.Tensor(4, 4)
> for i = 1, 4 do
> for j = 1, 4 do
> x[i][j] = (i-1)*4+j
> end
> end
> print(x)
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
[torch.Tensor of dimension 4x4]
> print(nn.View(2, 8):forward(x))
1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16
[torch.DoubleTensor of dimension 2x8]
> print(nn.View(torch.LongStorage{8,2}):forward(x))
1 2
3 4
5 6
7 8
9 10
11 12
13 14
15 16
[torch.DoubleTensor of dimension 8x2]
> print(nn.View(16):forward(x))
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
[torch.DoubleTensor of dimension 16]Example 2:
> input = torch.Tensor(2, 3)
> minibatch = torch.Tensor(5, 2, 3)
> m = nn.View(-1):setNumInputDims(2)
> print(#m:forward(input))
6
[torch.LongStorage of size 2]
> print(#m:forward(minibatch))
5
6
[torch.LongStorage of size 2]module = nn.Select(dim, index)Selects a dimension and index of a nxpxqx.. Tensor.
Example:
mlp = nn.Sequential()
mlp:add(nn.Select(1, 3))
x = torch.randn(10, 5)
print(x)
print(mlp:forward(x))gives the output:
0.9720 -0.0836 0.0831 -0.2059 -0.0871
0.8750 -2.0432 -0.1295 -2.3932 0.8168
0.0369 1.1633 0.6483 1.2862 0.6596
0.1667 -0.5704 -0.7303 0.3697 -2.2941
0.4794 2.0636 0.3502 0.3560 -0.5500
-0.1898 -1.1547 0.1145 -1.1399 0.1711
-1.5130 1.4445 0.2356 -0.5393 -0.6222
-0.6587 0.4314 1.1916 -1.4509 1.9400
0.2733 1.0911 0.7667 0.4002 0.1646
0.5804 -0.5333 1.1621 1.5683 -0.1978
[torch.Tensor of dimension 10x5]
0.0369
1.1633
0.6483
1.2862
0.6596
[torch.Tensor of dimension 5]This can be used in conjunction with Concat to emulate the behavior of Parallel, or to select various parts of an input Tensor to perform operations on. Here is a fairly complicated example:
mlp = nn.Sequential()
c = nn.Concat(2)
for i = 1, 10 do
local t = nn.Sequential()
t:add(nn.Select(1, i))
t:add(nn.Linear(3, 2))
t:add(nn.Reshape(2, 1))
c:add(t)
end
mlp:add(c)
pred = mlp:forward(torch.randn(10, 3))
print(pred)
for i = 1, 10000 do -- Train for a few iterations
x = torch.randn(10, 3)
y = torch.ones(2, 10)
pred = mlp:forward(x)
criterion = nn.MSECriterion()
err = criterion:forward(pred, y)
gradCriterion = criterion:backward(pred, y)
mlp:zeroGradParameters()
mlp:backward(x, gradCriterion)
mlp:updateParameters(0.01)
print(err)
endmodule = nn.Exp()Applies the exp function element-wise to the input Tensor, thus outputting a Tensor of the same dimension.
ii = torch.linspace(-2, 2)
m = nn.Exp()
oo = m:forward(ii)
go = torch.ones(100)
gi = m:backward(ii,go)
gnuplot.plot({'f(x)', ii, oo, '+-'}, {'df/dx', ii, gi, '+-'})
gnuplot.grid(true)
module = nn.Square()Takes the square of each element.
ii = torch.linspace(-5, 5)
m = nn.Square()
oo = m:forward(ii)
go = torch.ones(100)
gi = m:backward(ii, go)
gnuplot.plot({'f(x)', ii, oo, '+-'}, {'df/dx', ii, gi, '+-'})
gnuplot.grid(true)
module = nn.Sqrt()Takes the square root of each element.
ii = torch.linspace(0, 5)
m = nn.Sqrt()
oo = m:forward(ii)
go = torch.ones(100)
gi = m:backward(ii, go)
gnuplot.plot({'f(x)', ii, oo, '+-'}, {'df/dx', ii, gi, '+-'})
gnuplot.grid(true)
module = nn.Power(p)Raises each element to its p-th power.
ii = torch.linspace(0, 2)
m = nn.Power(1.25)
oo = m:forward(ii)
go = torch.ones(100)
gi = m:backward(ii, go)
gnuplot.plot({'f(x)', ii, oo, '+-'}, {'df/dx', ii, gi, '+-'})
gnuplot.grid(true)
module = nn.Normalize(p, [eps])Normalizes the input Tensor to have unit L_p norm. The smoothing parameter eps prevents division by zero when the input contains all zero elements (default = 1e-10).
Input can be 1D or 2D (in which case it's considered as in batch mode)
A = torch.randn(3, 5)
m = nn.Normalize(2)
B = m:forward(A) -- B is also 3 x 5
-- take the L2 norm over the second axis:
print(torch.norm(B, 2, 2)) -- norms is [1, 1, 1]module = nn.MM(transA, transB)Performs multiplications on one or more pairs of matrices. If transA is set, the first matrix is transposed before multiplication. If transB is set, the second matrix is transposed before multiplication. By default, the matrices do not get transposed.
The module also accepts 3D inputs which are interpreted as batches of matrices. When using batches, the first input matrix should be of size b x m x n and the second input matrix should be of size b x n x p (assuming transA and transB are not set).
model = nn.MM()
A = torch.randn(b, m, n)
B = torch.randn(b, n, p)
C = model:forward({A, B}) -- C will be of size `b x m x p`module = nn.BatchNormalization(N [, eps] [, momentum] [,affine])where N is the dimensionality of input
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
During training, this layer keeps a running estimate of its computed mean and std. The running sum is kept with a default momentum of 0.1 (unless over-ridden) During evaluation, this running mean/std is used for normalization.
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:
x - mean(x)
y = ----------------------------- * gamma + beta
standard-deviation(x) + epswhere the mean and standard-deviation are calculated per-dimension over the mini-batches and where gamma and beta are learnable parameter vectors of size N (where N is the input size).
The learning of gamma and beta is optional.
The module only accepts 2D inputs.
-- with learnable parameters
model = nn.BatchNormalization(m)
A = torch.randn(b, m)
C = model:forward(A) -- C will be of size `b x m`
-- without learnable parameters
model = nn.BatchNormalization(m, nil, nil, false)
A = torch.randn(b, m)
C = model:forward(A) -- C will be of size `b x m`module = nn.Padding(dim, pad [, nInputDim, value])
This module adds pad units of padding to dimension dim of the input.
If pad is negative, padding is added to the left, otherwise, it is added to the right of the dimension. When nInputDim is provided, inputs larger than that value will be considered batches where the actual dim to be padded will
be dimension dim + 1. When value is provide, the padding will be filled with that value. The default value is zero.
Example 1:
module = nn.Padding(1, 2, 1, -1) --pad right x2
module:forward(torch.randn(3)) --non-batch input
0.2008
0.4848
-1.0783
-1.0000
-1.0000
[torch.DoubleTensor of dimension 5]Example 2:
module = nn.Padding(1, -2, 1, -1) --pad left x2
module:forward(torch.randn(2, 3)) --batch input
-1.0000 -1.0000 1.0203 0.2704 -1.6164
-1.0000 -1.0000 -0.2219 -0.6529 -1.9218
[torch.DoubleTensor of dimension 2x5]penalty = nn.L1Penalty(L1weight, sizeAverage)L1Penalty is an inline module that in its forward propagation copies the input Tensor directly to the output, and computes an L1 loss of the latent state (input) and stores it in the module's loss field.
During backward propagation: gradInput = gradOutput + gradLoss.
This module can be used in autoencoder architectures to apply L1 losses to internal latent state without having to use Identity and parallel containers to carry the internal code to an output criterion.
Example (sparse autoencoder, note: decoder should be normalized):
encoder = nn.Sequential()
encoder:add(nn.Linear(3, 128))
encoder:add(nn.Threshold())
decoder = nn.Linear(128, 3)
autoencoder = nn.Sequential()
autoencoder:add(encoder)
autoencoder:add(nn.L1Penalty(l1weight))
autoencoder:add(decoder)
criterion = nn.MSECriterion() -- To measure reconstruction error
-- ...