AffineMaps

Implements affine transformations in a modular way.

abstract type AbstractAffineMap <: Function

Abstract type for affine maps.

An affine map f::AbstractAffineMap acts like f(x) == A * x .+ b or f(x) == A * (x .+ b) or their inverses, depending on the type of the map f.

A may be a Number, an AbstractMatrix{<:Number}, a LinearAlgebra.UniformScaling or any other multiplicative linear operator that supports A * x. The packages LinearMaps, LinearOperators and SciMLOperators provide such operators, for example. Inverse maps require support for A \ y, and logabsdet(Jacobian) computations are supported for Number, AbstractMatrix and UniformScaling operators.

b must broadcast over A * x (resp. x) without changing its shape: a scalar, an array of equal shape, or e.g. a vector to add column-wise to a matrix-valued A * x.

A matrix-valued x is treated as a batch of column-vector samples, so ChangesOfVariables.with_logabsdet_jacobian returns a 1 × size(x, 2) matrix of per-column values then.

Subtypes of AbstractAffineMap should implement/support the APIs of

• InverseFunctions.jl
• ChangesOfVariables.jl
• Functors.jl

struct Add

f = Add(b) has the behavior f(x) == x .+ f.b.

See AbstractAffineMap for more information.

struct AddMul

f = AddMul(b, A) has the behavior f(x) == f.A * (x .+ f.b).

See AbstractAffineMap for more information.

struct InvAddMul

f = InvAddMul(b, A) has the behavior f(x) == (f.A \ x) .- f.b. It is the inverse of AddMul(b, A).

See AbstractAffineMap for more information.

struct InvMul

f = InvMul(A) has the behavior f(x) == f.A \ x. It is the inverse of Mul(A).

See AbstractAffineMap for more information.

struct InvMulAdd

f = InvMulAdd(A, b) has the behavior f(x) == f.A \ (x .- f.b). It is the inverse of MulAdd(A, b).

See AbstractAffineMap for more information.

struct Mul

f = Mul(A) has the behavior f(x) == f.A * x.

See AbstractAffineMap for more information.

struct MulAdd

f = MulAdd(A, b) has the behavior f(x) == f.A * x .+ f.b.

See AbstractAffineMap for more information.

struct Subtract

f = Subtract(b) has the behavior f(x) == x .- f.b. It is the inverse of Add(b).

See AbstractAffineMap for more information.