module = PairwiseDistance(p) creates a module that takes a table of two vectors as input and outputs the distance between them using the =p=-norm.

Example: 

mlp_l1=nn.PairwiseDistance(1)
mlp_l2=nn.PairwiseDistance(2)
x=lab.new(1,2,3) 
y=lab.new(4,5,6)
print(mlp_l1:forward({x,y}))
print(mlp_l2:forward({x,y}))
gives the output: 
 9
[torch.Tensor of dimension 1]

 5.1962
[torch.Tensor of dimension 1]

A more complicated example: 

-- imagine we have one network we are interested in, it is called "p1_mlp"
p1_mlp= nn.Sequential(); p1_mlp:add(nn.Linear(5,2))

-- But we want to push examples towards or away from each other
-- so we make another copy of it called p2_mlp
-- this *shares* the same weights via the set command, but has its own set of temporary gradient storage
-- that's why we create it again (so that the gradients of the pair don't wipe each other)
p2_mlp= nn.Sequential(); p2_mlp:add(nn.Linear(5,2))
p2_mlp:get(1).weight:set(p1_mlp:get(1).weight)
p2_mlp:get(1).bias:set(p1_mlp:get(1).bias)

-- we make a parallel table that takes a pair of examples as input. they both go through the same (cloned) mlp
prl = nn.ParallelTable()
prl:add(p1_mlp)
prl:add(p2_mlp)

-- now we define our top level network that takes this parallel table and computes the pairwise distance betweem
-- the pair of outputs
mlp= nn.Sequential()
mlp:add(prl)
mlp:add(nn.PairwiseDistance(1))

-- and a criterion for pushing together or pulling apart pairs
crit=nn.HingeEmbeddingCriterion(1)

-- lets make two example vectors
x=lab.rand(5)
y=lab.rand(5)


-- Use a typical generic gradient update function
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)
end

-- push the pair x and y together, notice how then the distance between them given
-- by  print(mlp:forward({x,y})[1]) gets smaller
for i=1,10 do
gradUpdate(mlp,{x,y},1,crit,0.01)
print(mlp:forward({x,y})[1])
end


-- pull apart the pair x and y, notice how then the distance between them given
-- by  print(mlp:forward({x,y})[1]) gets larger

for i=1,10 do
gradUpdate(mlp,{x,y},-1,crit,0.01)
print(mlp:forward({x,y})[1])
end