SparsityDetection.jl

This is a package for automatic Jacobian and Hessian sparsity pattern detection on Julia functions. Functions written for numerical work can automatically be investigated in order to understand and utilize sparsity. This does not work numerically, and instead works by non-standard interpretation in order to check every branch for connectivity in order to determine an accurate sparsity pattern.

If you use this package, please cite the following:

@article{gowda2019sparsity,
  title={Sparsity Programming: Automated Sparsity-Aware Optimizations in Differentiable Programming},
  author={Gowda, Shashi and Ma, Yingbo and Churavy, Valentin and Edelman, Alan and Rackauckas, Christopher},
  year={2019}
}

Suppose we had the function

fcalls = 0
function f(dx,x)
  global fcalls += 1
  for i in 2:length(x)-1
    dx[i] = x[i-1] - 2x[i] + x[i+1]
  end
  dx[1] = -2x[1] + x[2]
  dx[end] = x[end-1] - 2x[end]
  nothing
end

For this function, we know that the sparsity pattern of the Jacobian is a Tridiagonal matrix. However, if we didn't know the sparsity pattern for the Jacobian, we could use the jacobian_sparsity function to automatically detect the sparsity pattern. This function is only available if you load Cassette.jl as well. We declare that the function f outputs a vector of length 30 and takes in a vector of length 30, and jacobian_sparsity spits out a Sparsity object which we can turn into a SparseMatrixCSC:

using SparsityDetection, SparseArrays
input = rand(10)
output = similar(input)
sparsity_pattern = jacobian_sparsity(f,output,input)
jac = Float64.(sparse(sparsity_pattern))

Automated Jacobian sparsity detection is provided by the sparsity! function. The syntax is:

jacobian_sparsity(f, Y, X, args...; sparsity=Sparsity(length(X), length(Y)), verbose=true)

The arguments are:

• f: the function
• Y: the output array
• X: the input array
• args: trailing arguments to f. They are considered subject to change, unless wrapped as Fixed(arg)
• S: (optional) the sparsity pattern
• verbose: (optional) whether to describe the paths taken by the sparsity detection.

The function f is assumed to take arguments of the form f(dx,x,args...). jacobian_sparsity returns a Sparsity object which describes where the non-zeros of the Jacobian occur. sparse(::Sparsity) transforms the pattern into a sparse matrix.

This function utilizes non-standard interpretation, which we denote combinatoric concolic analysis, to directly realize the sparsity pattern from the program's AST. It requires that the function f is a Julia function. It does not work numerically, meaning that it is not prone to floating point error or cancelation. It allows for branching and will automatically check all of the branches. However, a while loop of indeterminate length which is dependent on the input argument is not allowed.

A similar method is now available from Symbolics.jl.

hessian_sparsity(f, X, args...; verbose=true)

The arguments are:

• f: the function
• X: the input array
• args: trailing arguments to f. They are considered subject to change, unless wrapped as Fixed(arg)
• verbose: (optional) whether to describe the paths taken by the sparsity detection.

The function f is assumed to take arguments of the form f(x,args...) and returns a scalar.

This function utilizes non-standard interpretation, which we denote combinatoric concolic analysis, to directly realize the sparsity pattern from the program's AST. It requires that the function f is a Julia function. It does not work numerically, meaning that it is not prone to floating point error or cancelation. It allows for branching and will automatically check all of the branches. However, a while loop of indeterminate length which is dependent on the input argument is not allowed.