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Defining custom LinearMap types

In this section, we want to demonstrate on a simple, actually built-in, linear map type how to define custom LinearMap subtypes. First of all, LinearMap{T} is an extendable abstract type, where T denotes the eltype.

Basics

As an example, we want to define a map type whose objects correspond to lazy analogues of filled matrices. Naturally, we need to store the filled value λ and the size of the linear map.

using LinearMaps, LinearAlgebra

struct MyFillMap{T} <: LinearMaps.LinearMap{T}
    λ::T
    size::Dims{2}
    function MyFillMap(λ::T, dims::Dims{2}) where {T}
        all(≥(0), dims) || throw(ArgumentError("dims of MyFillMap must be non-negative"))
        promote_type(T, typeof(λ)) == T || throw(InexactError())
        return new{T}(λ, dims)
    end
end

By default, for any A::MyFillMap{T}, eltype(A) returns T. Upon application to a vector x and/or interaction with other LinearMap objects, we need to check consistent sizes.

Base.size(A::MyFillMap) = A.size

By a couple of defaults provided for all subtypes of LinearMap, we only need to define a LinearMaps._unsafe_mul! method to have a minimal, operational type. The (internal) function _unsafe_mul! is called by LinearAlgebra.mul!, constructors, and conversions and only needs to be concerned with the bare computing kernel. Dimension checking is done on the level of mul! etc. Factoring out dimension checking is done to minimise overhead caused by repetitive checking.

Note

Multiple dispatch at the _unsafe_mul! level happens via the second (the map type) and the third arguments (AbstractVector or AbstractMatrix, see the Application to matrices section below). For that reason, the output argument can remain type-unbound.

function LinearMaps._unsafe_mul!(y, A::MyFillMap, x::AbstractVector)
    return fill!(y, iszero(A.λ) ? zero(eltype(y)) : A.λ*sum(x))
end

Again, due to generic fallbacks the following now "just work":

  • out-of-place multiplication A*x,
  • in-place multiplication with vectors mul!(y, A, x),
  • in-place multiply-and-add with vectors mul!(y, A, x, α, β),
  • in-place multiplication and multiply-and-add with matrices mul!(Y, A, X, α, β),
  • conversion to a (sparse) matrix Matrix(A) and sparse(A),
  • complete slicing of columns (and rows if the adjoint action is defined).
A = MyFillMap(5.0, (3, 3)); x = ones(3); sum(x)
3.0
A * x
3-element Vector{Float64}:
 15.0
 15.0
 15.0
mul!(zeros(3), A, x)
3-element Vector{Float64}:
 15.0
 15.0
 15.0
mul!(ones(3), A, x, 2, 2)
3-element Vector{Float64}:
 32.0
 32.0
 32.0
mul!(ones(3,3), A, reshape(collect(1:9), 3, 3), 2, 2)
3×3 Matrix{Float64}:
 62.0  152.0  242.0
 62.0  152.0  242.0
 62.0  152.0  242.0

Multiply-and-add and the MulStyle trait

While the above function calls work out of the box due to generic fallbacks, the latter may be suboptimally implemented for your custom map type. Let's see some benchmarks.

using BenchmarkTools

@benchmark mul!($(zeros(3)), $A, $x)
BenchmarkTools.Trial: 10000 samples with 999 evaluations.
 Range (min … max):  8.333 ns … 33.827 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     8.354 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   8.420 ns ±  0.730 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

   █ ▄▄                                                       
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  8.33 ns        Histogram: frequency by time        8.72 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.
@benchmark mul!($(zeros(3)), $A, $x, $(rand()), $(rand()))
BenchmarkTools.Trial: 10000 samples with 991 evaluations.
 Range (min … max):  41.055 ns …  1.306 μs  ┊ GC (min … max): 0.00% … 92.74%
 Time  (median):     42.441 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   45.887 ns ± 48.499 ns  ┊ GC (mean ± σ):  4.13% ±  3.80%

  ▃▆█▆▅▂                                                      ▂
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  41.1 ns      Histogram: log(frequency) by time      73.7 ns <

 Memory estimate: 80 bytes, allocs estimate: 1.

The second benchmark indicates the allocation of an intermediate vector z which stores the result of A*x before it gets scaled and added to (the scaled) y = zeros(3). For that reason, it is beneficial to provide a custom "5-arg _unsafe_mul!" if you can avoid the allocation of an intermediate vector. To indicate that there exists an allocation-free implementation of multiply-and-add, you should set the MulStyle trait, whose default is ThreeArg(), to FiveArg().

LinearMaps.MulStyle(A::MyFillMap) = FiveArg()

function LinearMaps._unsafe_mul!(y, A::MyFillMap, x::AbstractVector, α, β)
    if iszero(α)
        !isone(β) && rmul!(y, β)
        return y
    else
        temp = A.λ * sum(x) * α
        if iszero(β)
            y .= temp
        elseif isone(β)
            y .+= temp
        else
            y .= y .* β .+ temp
        end
    end
    return y
end

With this function at hand, let's redo the benchmark.

@benchmark mul!($(zeros(3)), $A, $x, $(rand()), $(rand()))
BenchmarkTools.Trial: 10000 samples with 999 evaluations.
 Range (min … max):  9.567 ns … 34.117 ns  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     9.588 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   9.638 ns ±  0.598 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

   ▅██                                                        
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  9.57 ns        Histogram: frequency by time        10.1 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.

There you go, the allocation is gone and the computation time is significantly reduced.

Adjoints and transposes

Generically, taking the transpose (or the adjoint) of a (real, resp.) map wraps the linear map by a TransposeMap, taking the adjoint of a complex map wraps it by an AdjointMap.

typeof(A')
LinearMaps.TransposeMap{Float64, Main.MyFillMap{Float64}}

Not surprisingly, without further definitions, multiplying A' by x yields an error.

try A'x catch e println(e) end
ErrorException("transpose not implemented for 3×3 Main.MyFillMap{Float64}")

If the operator is symmetric or Hermitian, the transpose and the adjoint, respectively, of the linear map A is given by A itself. So let us define corresponding checks.

LinearAlgebra.issymmetric(A::MyFillMap) = A.size[1] == A.size[2]
LinearAlgebra.ishermitian(A::MyFillMap) = isreal(A.λ) && A.size[1] == A.size[2]
LinearAlgebra.isposdef(A::MyFillMap) = (size(A, 1) == size(A, 2) == 1 && isposdef(A.λ))
Base.:(==)(A::MyFillMap, B::MyFillMap) = A.λ == B.λ && A.size == B.size

These are used, for instance, in checking symmetry or positive definiteness of higher-order LinearMaps, like products or linear combinations of linear maps, or signal to iterative eigenproblem solvers that real eigenvalues are to be computed. Without these definitions, the first three functions would return false (by default), and the last one would fall back to ===.

With this at hand, we note that A above is symmetric, and we can compute

transpose(A)*x
3-element Vector{Float64}:
 15.0
 15.0
 15.0

This, however, does not work for nonsquare maps

try MyFillMap(5.0, (3, 4))' * ones(3) catch e println(e) end
ErrorException("transpose not implemented for 3×4 Main.MyFillMap{Float64}")

which require explicit adjoint/transpose handling, for which there exist two distinct paths.

Path 1: Generic, non-invariant LinearMap subtypes

The first option is to write LinearMaps._unsafe_mul! methods for the corresponding wrapped map types; for instance,

function LinearMaps._unsafe_mul!(
    y,
    transA::LinearMaps.TransposeMap{<:Any,<:MyFillMap},
    x::AbstractVector
)
    λ = transA.lmap.λ
    return fill!(y, iszero(λ) ? zero(eltype(y)) : transpose(λ)*sum(x))
end

Now, the adjoint multiplication works.

MyFillMap(5.0, (3, 4))' * ones(3)
4-element Vector{Float64}:
 15.0
 15.0
 15.0
 15.0

If you have set the MulStyle trait to FiveArg(), you should provide a corresponding 5-arg mul! method for LinearMaps.TransposeMap{<:Any,<:MyFillMap} and LinearMaps.AdjointMap{<:Any,<:MyFillMap}.

Path 2: Invariant LinearMap subtypes

Before we start, let us delete the previously defined method to make sure we use the following definitions.

Base.delete_method(
    first(methods(
        LinearMaps._unsafe_mul!,
        (Any, LinearMaps.TransposeMap{<:Any,<:MyFillMap}, AbstractVector))
    )
)

The second option is when your class of linear maps that are modelled by your custom LinearMap subtype are invariant under taking adjoints and transposes.

LinearAlgebra.adjoint(A::MyFillMap) = MyFillMap(adjoint(A.λ), reverse(A.size))
LinearAlgebra.transpose(A::MyFillMap) = MyFillMap(transpose(A.λ), reverse(A.size))

With such invariant definitions, i.e., the adjoint/transpose of a MyFillMap is again a MyFillMap, no further method definitions are required, and the entire functionality listed above just works for adjoints/transposes of your custom map type.

mul!(ones(3), A', x, 2, 2)
3-element Vector{Float64}:
 32.0
 32.0
 32.0
MyFillMap(5.0, (3, 4))' * ones(3)
4-element Vector{Float64}:
 15.0
 15.0
 15.0
 15.0

Now that we have defined the action of adjoints/transposes, the following right action on vectors is automatically defined:

ones(3)' * MyFillMap(5.0, (3, 4))
1×4 adjoint(::Vector{Float64}) with eltype Float64:
 15.0  15.0  15.0  15.0

and transpose(x) * A correspondingly, as well as in-place multiplication

mul!(similar(x)', x', A)
1×3 adjoint(::Vector{Float64}) with eltype Float64:
 15.0  15.0  15.0

and mul!(transpose(y), transpose(x), A).

Application to matrices

By default, applying a LinearMap A to a matrix X via A*X does not apply A to each column of X viewed as a vector, but interprets X as a linear map, wraps it as such and returns (A*X)::CompositeMap. Calling the in-place multiplication function mul!(Y, A, X) for matrices, however, does compute the columnwise action of A on X and stores the result in Y. In case there is a more efficient implementation for the matrix application, you can provide _unsafe_mul! methods with signature _unsafe_mul!(Y, A::MyFillMap, X::AbstractMatrix), and, depending on the chosen path to handle adjoints/transposes, corresponding methods for wrapped maps of type AdjointMap or TransposeMap, plus potentially corresponding 5-arg mul! methods. This may seem like a lot of methods to be implemented, but note that adding such methods is only necessary/recommended for increased performance.

Computing a matrix representation

In some cases, it might be necessary to compute a matrix representation of a LinearMap. This is essentially done via the [LinearMaps._unsafe_mul!(::Matrix,::LinearMap,::Number)](@ref) method, for which a generic fallback exists: it applies the LinearMap successively to the standard unit vectors.

F = MyFillMap(5, (100,100))
M = Matrix{eltype(F)}(undef, size(F))
@benchmark Matrix($F)
BenchmarkTools.Trial: 10000 samples with 7 evaluations.
 Range (min … max):  4.427 μs … 426.546 μs  ┊ GC (min … max): 0.00% … 8.98%
 Time  (median):     5.122 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   5.887 μs ±   6.431 μs  ┊ GC (mean ± σ):  7.16% ± 8.41%

  ▇█▄▂                                                        ▁
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  4.43 μs      Histogram: log(frequency) by time      40.9 μs <

 Memory estimate: 79.05 KiB, allocs estimate: 3.
@benchmark LinearMaps._unsafe_mul!($(Matrix{Int}(undef, (100,100))), $(MyFillMap(5, (100,100))), true)
BenchmarkTools.Trial: 10000 samples with 9 evaluations.
 Range (min … max):  2.863 μs …  52.442 μs  ┊ GC (min … max): 0.00% … 83.05%
 Time  (median):     2.946 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   3.092 μs ± 551.842 ns  ┊ GC (mean ± σ):  0.14% ±  0.83%

   ▅█            ▂▃                                            
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  2.86 μs         Histogram: frequency by time        4.22 μs <

 Memory estimate: 896 bytes, allocs estimate: 1.

If a more performant implementation exists, it is recommended to overwrite this method, for instance (as before, size checks need not be included here since they are handled by the corresponding LinearAlgebra.mul! method):

LinearMaps._unsafe_mul!(M, A::MyFillMap, s::Number) = fill!(M, A.λ*s)
@benchmark Matrix($F)
BenchmarkTools.Trial: 10000 samples with 9 evaluations.
 Range (min … max):  2.967 μs … 249.999 μs  ┊ GC (min … max): 0.00% … 10.21%
 Time  (median):     4.061 μs               ┊ GC (median):    0.00%
 Time  (mean ± σ):   4.680 μs ±   4.590 μs  ┊ GC (mean ± σ):  8.07% ±  9.21%

  ▂▇█▅▃▁                                                      ▂
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  2.97 μs      Histogram: log(frequency) by time      31.4 μs <

 Memory estimate: 78.17 KiB, allocs estimate: 2.
@benchmark LinearMaps._unsafe_mul!($(Matrix{Int}(undef, (100,100))), $(MyFillMap(5, (100,100))), true)
BenchmarkTools.Trial: 10000 samples with 57 evaluations.
 Range (min … max):  861.456 ns …  1.681 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     889.561 ns              ┊ GC (median):    0.00%
 Time  (mean ± σ):   892.642 ns ± 25.845 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

         ▃▆███▃                                                 
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  861 ns          Histogram: frequency by time         1.03 μs <

 Memory estimate: 0 bytes, allocs estimate: 0.

As one can see, the above runtimes are dominated by the allocation of the output matrix, but still overwriting the multiplication kernel yields a speed-up of about factor 3 for the matrix filling part.

Slicing

As usual, generic fallbacks for LinearMap slicing exist and are handled by the following method hierarchy, where at least one of I and J has to be a Colon:

Base.getindex(::LinearMap, I, J)
-> LinearMaps._getindex(::LinearMap, I, J)

The method Base.getindex checks the validity of the the requested indices and calls LinearMaps._getindex, which should be overloaded for custom LinearMaps subtypes. For instance:

@benchmark F[1,:]
BenchmarkTools.Trial: 10000 samples with 216 evaluations.
 Range (min … max):  360.810 ns …   8.333 μs  ┊ GC (min … max): 0.00% … 14.22%
 Time  (median):     391.616 ns               ┊ GC (median):    0.00%
 Time  (mean ± σ):   419.916 ns ± 179.861 ns  ┊ GC (mean ± σ):  2.42% ±  5.68%

  ▂▇█▆▃▁▁▁▁                                                     ▂
  █████████▇▇▇▅▅▅▄▄▅▄▅▄▅▆▃▆▆▇▇▇▇▆▆▃▃▁▁▁▁▃▁▄▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▆▇▇▆▆ █
  361 ns        Histogram: log(frequency) by time       1.16 μs <

 Memory estimate: 1.77 KiB, allocs estimate: 3.
LinearMaps._getindex(A::MyFillMap, ::Integer, J::Base.Slice) = fill(A.λ, axes(J))
@benchmark F[1,:]
BenchmarkTools.Trial: 10000 samples with 460 evaluations.
 Range (min … max):  230.191 ns …   4.876 μs  ┊ GC (min … max): 0.00% … 61.23%
 Time  (median):     244.654 ns               ┊ GC (median):    0.00%
 Time  (mean ± σ):   293.202 ns ± 159.641 ns  ┊ GC (mean ± σ):  2.27% ±  6.06%

  ▇█▃▂▁        ▁▁        ▁       ▁                              ▁
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  230 ns        Histogram: log(frequency) by time        954 ns <

 Memory estimate: 912 bytes, allocs estimate: 2.

Note that in Base.getindex Colons are converted to Base.Slice via Base.to_indices, thus the dispatch must be on Base.Slice rather than on Colon.

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