The current implementation of LinearAlgebra.eigen does not support sensitivities.
This package adds a new function eigen, that wraps the original function, but returns an array of reals instead of complex numbers (this is necessary, because some AD-frameworks do not support complex numbers).
This eigen function is differentiable by every AD-framework with support for ChainRulesCore.jl and ForwardDiff.jl.
1. Open a Julia-REPL, switch to package mode using ], activate your preferred environment.
2. Install DifferentiableEigen.jl:
(@v1.6) pkg> add DifferentiableEigen3. If you want to check that everything works correctly, you can run the tests bundled with DifferentiableEigen.jl:
(@v1.6) pkg> test DifferentiableEigenimport DifferentiableEigen
import LinearAlgebra
import ForwardDiff
A = rand(3,3) # Random matrix 3x3
eigvals, eigvecs = LinearAlgebra.eigen(A) # This is the default eigen-function in Julia. Note, that eigenvalues and -vectors are complex numbers.
jac = ForwardDiff.jacobian((A) -> LinearAlgebra.eigen(A)[1], A) # That doesn't work!
eigvals, eigvecs = DifferentiableEigen.eigen(A) # This is the differentiable eigen-function. Note, that eigenvalues and -vectors are not complex numbers, but real arrays!
jac = ForwardDiff.jacobian((A) -> DifferentiableEigen.eigen(A)[1], A) # That does work! eigenvalue- and eigenvector-sensitvitiesThis package was motivated by this discourse thread. For now, there is no other (known) ready to use solution for differentiable eigenvalues and -vectors. If this changes, please feel free to open a PR or discussion.
The sensitivity formulas are picked from:
Michael B. Giles. 2008. An extended collection of matrix derivative results for forward and reverse mode algorithmic differentiation. PDF
Tobias Thummerer and Lars Mikelsons. 2023. Eigen-informed NeuralODEs: Dealing with stability and convergence issues of NeuralODEs. ArXiv. PDF