dyna3/engine-proto/gram-test/Engine.jl

module Engine

using LinearAlgebra
using SparseArrays
using Random

export rand_on_shell, Q, DescentHistory, realize_gram

# === guessing ===

sconh(t, u) = 0.5*(exp(t) + u*exp(-t))

function rand_on_sphere(rng::AbstractRNG, ::Type{T}, n) where T
  out = randn(rng, T, n)
  tries_left = 2
  while dot(out, out) < 1e-6 && tries_left > 0
    out = randn(rng, T, n)
    tries_left -= 1
  end
  normalize(out)
end

##[TO DO] write a test to confirm that the outputs are on the correct shells
function rand_on_shell(rng::AbstractRNG, shell::T) where T <: Number
  space_part = rand_on_sphere(rng, T, 4)
  rapidity = randn(rng, T)
  sig = sign(shell)
  [sconh(rapidity, sig)*space_part; sconh(rapidity, -sig)]
end

rand_on_shell(rng::AbstractRNG, shells::Array{T}) where T <: Number =
  hcat([rand_on_shell(rng, sh) for sh in shells]...)

rand_on_shell(shells::Array{<:Number}) = rand_on_shell(Random.default_rng(), shells)

# === elements ===

plane(normal, offset) = [normal; offset; offset]

function sphere(center, radius)
  dist_sq = dot(center, center)
  return [
    center / radius;
    0.5 * ((dist_sq - 1) / radius - radius);
    0.5 * ((dist_sq + 1) / radius - radius)
  ]
end

# === Gram matrix realization ===

# the Lorentz form
Q = diagm([1, 1, 1, 1, -1])

# the difference between the matrices `target` and `attempt`, projected onto the
# subspace of matrices whose entries vanish at each empty index of `target`
function proj_diff(target::SparseMatrixCSC{T, <:Any}, attempt::Matrix{T}) where T
  J, K, values = findnz(target)
  result = zeros(size(target)...)
  for (j, k, val) in zip(J, K, values)
    result[j, k] = val - attempt[j, k]
  end
  result
end

# a type for keeping track of gradient descent history
struct DescentHistory{T}
  scaled_loss::Array{T}
  slope::Array{T}
  stepsize::Array{T}
  backoff_steps::Array{Int64}
  
  function DescentHistory{T}(
    scaled_loss = Array{T}(undef, 0),
    slope = Array{T}(undef, 0),
    stepsize = Array{T}(undef, 0),
    backoff_steps = Int64[]
  ) where T
    new(scaled_loss, slope, stepsize, backoff_steps)
  end
end

# seek a matrix `L` for which `L'QL` matches the sparse matrix `gram` at every
# explicit entry of `gram`. use gradient descent starting from `guess`
function realize_gram(
  gram::SparseMatrixCSC{T, <:Any},
  guess::Matrix{T};
  scaled_tol = 1e-30,
  target_improvement = 0.5,
  init_stepsize = 1.0,
  backoff = 0.9,
  max_descent_steps = 600,
  max_backoff_steps = 110
) where T <: Number
  # start history
  history = DescentHistory{T}()
  
  # scale tolerance
  scale_adjustment = sqrt(T(nnz(gram)))
  tol = scale_adjustment * scaled_tol
  
  # initialize variables
  stepsize = init_stepsize
  L = copy(guess)
  
  # do gradient descent
  Δ_proj = proj_diff(gram, L'*Q*L)
  loss = norm(Δ_proj)
  for step in 1:max_descent_steps
    # stop if the loss is tolerably low
    if loss < tol
      break
    end
    
    # find negative gradient of loss function
    neg_grad = 4*Q*L*Δ_proj
    slope = norm(neg_grad)
    
    # store current position, loss, and slope
    L_last = L
    loss_last = loss
    push!(history.scaled_loss, loss / scale_adjustment)
    push!(history.slope, slope)
    
    # find a good step size using backtracking line search
    push!(history.stepsize, 0)
    push!(history.backoff_steps, max_backoff_steps)
    for backoff_steps in 0:max_backoff_steps
      history.stepsize[end] = stepsize
      L = L_last + stepsize * neg_grad
      Δ_proj = proj_diff(gram, L'*Q*L)
      loss = norm(Δ_proj)
      improvement = loss_last - loss
      if improvement >= target_improvement * stepsize * slope
        history.backoff_steps[end] = backoff_steps
        break
      end
      stepsize *= backoff
    end
  end
  
  # return the factorization and its history
  push!(history.scaled_loss, loss / scale_adjustment)
  L, history
end

end