ActiveSubspaceMethods

ActiveSubspaceMethods.ADFunctionWrapper(fcn, d, backend; [init_space])

Wraps the evaluation of a pure Julia function fcn to use automatic differentiation.

Arguments

• fcn(z::AbstractVector)::Float64 function we want to reduce dimension of
• d::Int dimension of input
• backend::ADTypes.AbstractADType Backend for autodiff (e.g., ReverseDiff, Enzyme, etc.). Must be explicitly installed!
• init_space::Vector values to prepare the memory spaces for autodiff, default zeros(d).

ActiveSubspaces(input::AbstractActiveSubspacesInput)

Type to represent traditional Active Subspaces method [Russi, 2010].

GaussianizedUniformInputFunction([f0], f1!, bounds; [test_eval])

Take a function that we wish to minimize L2 error w.r.t. uniform measure and transform it to be w.r.t. GaussianizedUniformInputFunction

Arguments

• f0(z)::Float64 Returns function eval at z. Will default to using f1! if not provided
• f1!(grad, z)::Float64 Returns same function eval at z and puts function gradient at z into grad
• bounds::AbstractVector{<:NTuple{2}} A sequence of (lower, upper) bounds that is as long as the number of inputs.
• test_eval::Bool Whether to test function eval when constructing this functor, default true.

ModifiedActiveSubspaces(input::AbstractActiveSubspacesInput)

Type to represent Modified Active Subspaces method [Lee, 2019].

ASXX_bound(as::AbstractActiveSubspacesXX, U_perp::AbstractMatrix)

Evaluate the active subspaces++ bound using the perp subspace.

ActiveSubspacesXXManopt(input::AbstractActiveSubspacesInput)

Type to represent Active Subspaces ++, proposed in [Li et al., 2025]. Only usable when Manopt and Manifolds are installed.

GaussianizedUniformInputFunctionAD(f0, bounds, backend; [test_eval])

Equivalent to GaussianizedUniformInputFunction(f0, ADFunctionWrapper(f0, d, backend), bounds; test_eval)

MCActiveSubspacesInput(eval_grad_fcn!, d, N; [rand_fcn, rng, corrected])

Construct input for an active subspace method from a function (with inplace gradient) and Monte Carlo integration.

Arguments

• eval_grad_fcn!(grad, z) Returns real-valued function and puts gradient of this function in grad
• d::Int dimension
• N::Int number of samples
• rand_fcn(rng, d, N) function to sample random numbers using rng, default randn
• rng::AbstractRNG Random number generator
• corrected::Bool Use corrected Monte Carlo estimators downstream, default true

MC_variance(fcn_eval,d,N; [rand_fcn,rng])

Approximate the N sample variance of function fcn_eval with d inputs

QuadratureActiveSubspacesInput(eval_grad_fcn!, d, tensor_order, [quad_fcn1d, verbose])

Construct input for an active subspace method from a function (with inplace gradient) and Monte Carlo integration.

Arguments

• eval_grad_fcn!(grad, z) Returns real-valued function and puts gradient of this function in grad
• d::Int dimension
• tensor_order::Int number of one-dimensional quadrature points, will give tensor_order^d quadrature points
• quad_fcn1d(order)::Tuple{Vector,Vector} One-dimensional quadrature, defaults to normalized Gauss-Hermite
• verbose::Bool Whether to print the number of quadrature points.

QuadratureActiveSubspacesInput(eval_grad_fcn!, d, tensor_order, [quad_fcn1d, verbose])

Construct input for an active subspace method from a function (with inplace gradient) and Monte Carlo integration.

Arguments

• eval_grad_fcn!(grad, z) Returns real-valued function and puts gradient of this function in grad
• tensor_orders::NTuple{d,Int} number of one-dimensional quadrature points for each dimension, will give number of quad points as product of these orders
• quad_fcn1d(order)::Tuple{Vector,Vector} One-dimensional quadrature, defaults to normalized Gauss-Hermite
• verbose::Bool Whether to print the number of quadrature points.

create_Ur(U_perp, [initial_guess, rng])

Create a projector $U_r$ that projects onto orthogonal space to given $U_\perp$.

nested_MC_err(fcn_eval, U_r, U_perp, N_inner, N_outer; [rand_fcn, rng])

Calculates the conditional variance of subspaces using a pick-and-freeze nested MC method.

Arguments

• fcn_eval(z::Vector)::Float64 Evaluation of function taking in length-d vector
• U_r a size (d,r) orthogonal matrix
• U_perp a size (d,d-r) orthogonal matrix
• N_inner::Int number of inner Monte Carlo samples
• N_outer::Int number of outer Monte Carlo samples
• rand_fcn(rng,k)::Vector{Float64} sampler of random numbers, default randn
• rng::AbstractRNG random number generator

orthogonalize(U)

Return matrix with orthonormal columns but same range as U