Linear Maps

The construction of GMRFs involves various kinds of structured matrices. These structures may be leveraged in downstream computations to save compute and memory. But to make this possible, we need to actually keep track of these structures - which we achieve through diverse subtypes of LinearMap.

GMRFs.SymmetricBlockTridiagonalMap — Type

SymmetricBlockTridiagonalMap(
    diagonal_blocks::Tuple{LinearMap{T},Vararg{LinearMap{T},ND}},
    off_diagonal_blocks::Tuple{LinearMap{T},Vararg{LinearMap{T},NOD}},
)

A linear map representing a symmetric block tridiagonal matrix with diagonal blocks diagonal_blocks and lower off-diagonal blocks off_diagonal_blocks.

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GMRFs.SSMBidiagonalMap — Type

SSMBidiagonalMap{T}(
    A::LinearMap{T},
    B::LinearMap{T},
    C::LinearMap{T},
    N_t::Int,
)

Represents the block-bidiagonal map given by the (Nt) x (Nt - 1) sized block structure:

\[\begin{bmatrix} A & 0 & \cdots & 0 \\ B & C & \cdots & 0 \\ 0 & B & C & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & B \end{bmatrix}\]

which occurs as a square root in the discretization of GMRF-based state-space models. N_t is the total number of blocks along the rows.

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GMRFs.OuterProductMap — Type

OuterProductMap{T}(
    A::LinearMap{T},
    Q::LinearMap{T},
)

Represents the outer product A' Q A, without actually forming it in memory.

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GMRFs.LinearMapWithSqrt — Type

LinearMapWithSqrt{T}(
    A::LinearMap{T},
    A_sqrt::LinearMap{T},
)

A symmetric positive definite linear map A with known square root A_sqrt, i.e. A = A_sqrt * A_sqrt'. Behaves just like A, but taking the square root directly returns A_sqrt.

Arguments

A::LinearMap{T}: The linear map A.
A_sqrt::LinearMap{T}: The square root of A.

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GMRFs.CholeskySqrt — Type

CholeskySqrt{T}(cho::Union{Cholesky{T},SparseArrays.CHOLMOD.Factor{T}})

A linear map representing the square root obtained from a Cholesky factorization, i.e. for A = L * L', this map represents L.

Arguments

cho::Union{Cholesky{T},SparseArrays.CHOLMOD.Factor{T}}: The Cholesky factorization of a symmetric positive definite matrix.

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GMRFs.ZeroMap — Type

ZeroMap{T}(N::Int, M::Int)

A linear map that maps all vectors to the zero vector.

Arguments

N::Int: Output dimension
M::Int: Input dimension

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GMRFs.ADJacobianMap — Type

ADJacobianMap(f::Function, x₀::AbstractVector{T}, N_outputs::Int)

A linear map representing the Jacobian of f at x₀. Uses forward-mode AD in a matrix-free way, i.e. we do not actually store the Jacobian in memory and only compute JVPs.

Requires ForwardDiff.jl!

Arguments

f::Function: Function to differentiate.
x₀::AbstractVector{T}: Input vector at which to evaluate the Jacobian.
N_outputs::Int: Output dimension of f.

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GMRFs.ADJacobianAdjointMap — Type

ADJacobianAdjointMap{T}(f::Function, x₀::AbstractVector{T}, N_outputs::Int)

A linear map representing the adjoint of the Jacobian of f at x₀. Uses reverse-mode AD in a matrix-free way, i.e. we do not actually store the Jacobian in memory and only compute VJPs.

Requires Zygote.jl!

Arguments

f::Function: Function to differentiate.
x₀::AbstractVector{T}: Input vector at which to evaluate the Jacobian.
N_outputs::Int: Output dimension of f.

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