DeepReshapes.jl

Reshape arbitrarily nested structures of Tuples and Arrays in Julia
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Started In
November 2014

DeepReshapes

Extends reshape to arbitrarily nested structures of Tuples and Arrays, both in source and target. Also provides a deep flatten function that transforms these structures into a flat Vector.

As I am pretty new to Julia, before I consider registering this package, I would like a code review to know whether I am actually doing this "right". Please just have a look, and if you think this is useful and ready, open an issue or something like that.

Note that this only works on Julia 0.4 right now.

What?

Provides a deep_reshape function that can transform the structure of data:

A = [1 2; 3 4; 5 6]
b = [1, 2, 3, 4]
deep_reshape((A, b), (2, 5))
# => [1 5 4 1 3;
#     3 2 6 2 4]

deep_reshape([1:25], ((3, 3), (4, 4)))
# => ([ 1  4  7;
#       2  5  8;
#       3  6  9],
#     [10 14 18 22;
#      11 15 19 23;
#      12 16 20 24;
#      13 17 21 25])

α, β, c = deep_reshape([1.23, 2.34, 3, 4, 5], (Float64, Float64, (Int, 3)))
# => (1.23,2.34,[3,4,5])

This works on all (potentially nested) structures of Tuples and Arrays, as long as the actual scalar values contained within are Numbers (for now).

Why?

Say you want to optimize a non-linear function. Many optimization frameworks (NLopt, Optim) require you to supply cost and gradient functions and expect them to operate on plain Vector{Float64}s for that purpose. However, some algorithms are expressed more naturally in terms of more structured data.

Consider for example the popular [backpropagation algorithm] (http://ufldl.stanford.edu/wiki/index.php/Backpropagation_Algorithm) for training neural networks. The outline of the gradient calculation might look like this:

function cost_and_gradient!(
  W::Vector{Matrix{Float64}},  # weight matrices for each neuron layer
  b::Vector{Vector{Float64}},  # bias vectors for each neuron layer
  ∇W::Vector{Matrix{Float64}}, # vector to receive resulting weight gradients
  ∇b::Vector{Vector{Float64}}  # vector to receive resulting bias gradients
)
  # ...do feedforward and backpropagation, obtain some intermediate results
  # ...calculate gradients and fill the result vectors ∇W and ∇b
  # ...calculate and return the cost
end

For optimization, we cannot use this function directly, because the optimization package expects it to work on plain number vectors:

using NLopt

W, b = initialize_parameters()
# ...we need to flatten W, b to number vector θ

optimization = Opt(:LD_LBFGS, length(θ))
min_objective!(optimization, cost_and_gradient!) # <- we need to define this
result = optimize(optimization, θ)

Flattening the initial parameters is easy with flatten:

using DeepReshapes

θ = flatten(Float64, W, b)

As for cost_and_gradient!, we can simply wrap the original calculation with deep_reshapes:

function cost_and_gradient!(θ::Vector{Float64}, ∇θ::Vector{Float64})
  W, b = deep_reshape(θ, s) # <- s is a specification of the original structure
                            # which can be obtained by calling describe on the
                            # initial parameters before flattening them.

  # ...do the original calculation
  ∇θ[:] = flatten(Float64, ∇W, ∇b)
  # ... calculate and return the cost
end

How?

A deep reshape consists of two independent processes: a producer that recursively walks the input to produce scalar values, and a consumer that consumes these values and puts them into a new object according to a structure specification:

result = deep_reshape(input, specification)

Source

The input can be any object, but by default, the producer only descends into Arrays and Tuples and considers anything else to be a scalar:

deep_reshape([1:4], (2, 2)) # => Any[1 3; 2 4]
deep_reshape(1:4, (2, 2))   # => Any[1:4 (); () ()]

What objects to descend into can be overridden through the optional Deep argument:

deep_reshape(1:4, (2, 2), Deep = Range)

Any input of type Deep will be considered iterable and all contained values will be produced. Any other input will be considered scalar and produced directly, without further recursion.

Target

The produced scalars will then be assigned into the objects under construction according to the specification, which is of the following format:

  • An empty tuple () describes Any value.
  • A Type describes a value of that type.
  • A tuple of (Integer...) dimensions describes an Any[] with these dimensions.
  • A tuple (Type, Integer...) of a type and dimensions describes an Array with that element type and these dimensions.
  • Any other Tuple recursively describes a tuple, where each contained value describes an entry of the result.
  • An Array recursively describes an array in the same way.

For simple inputs (recursively) consisting only of Tuples, Arrays and scalars, describe() returns the corresponding specification:

s = describe(([1:10], [1 2; 3 4])) # => ((Int, 10), (Int, 2, 2))

These can be used directly as deep_reshape specifications:

nested = deep_reshape([1:14], s)
# => ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10],
#     [11 13;
#      12 14])

flatten and pack

There is also a convenience function flatten that can recursively flattens the input to a Vector, optionally with a fixed target type that the scalars are to be converted to:

flatten(nested)
# => Any[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]

flatten(Int, nested)
# => [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]

The very similar pack function returns both the flattened Vector and the original structure as defined by describe. This can be used to later reverse the flattening:

flattened, s = pack(Int, ([1:10], [1 2; 3 4]))
# => ([1,2,3,4,5,6,7,8,9,10,1,3,2,4],(((Int,10),(Int,2,2)),))

deep_reshape(flattened, s)
# => ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10],
#     [11 13;
#      12 14])