det, logdet, and logabsdet rrules for SparseMatrixCSC

Project.toml

@@ -13,8 +13,10 @@ LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 RealDot = "c1ae055f-0cd5-4b69-90a6-9a35b1a98df9"
 SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
+SparseInverseSubset = "dc90abb0-5640-4711-901d-7e5b23a2fada"
 Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
 StructArrays = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
+SuiteSparse = "4607b0f0-06f3-5cda-b6b1-a6196a1729e9"
 
 [compat]
 Adapt = "3.4.0"

src/rulesets/LinearAlgebra/structured.jl

@@ -1,5 +1,6 @@
 # Structured matrices
 using LinearAlgebra: AbstractTriangular
+using SparseInverseSubset
 
 # Matrix wrapper types that we know are square and are thus potentially invertible. For
 # these we can use simpler definitions for `/` and `\`.

src/rulesets/SparseArrays/sparsematrix.jl

@@ -49,3 +49,91 @@ function rrule(::typeof(findnz), v::AbstractSparseVector)
 
     return (I, V), findnz_pullback
 end
+
+if VERSION < v"1.7"
+    #=
+    The method below for `logabsdet(F::UmfpackLU)` is required to calculate the (log) 
+    determinants of sparse matrices, but was not defined prior to Julia v1.7. In order
+    for the rrules for the determinants of sparse matrices below to work, they need to be
+    able to compute the primals as well, so this import from the future is included. For
+    more recent versions of Julia, this definition lives in:
+    julia/stdlib/SuiteSparse/src/umfpack.jl
+    =#
+    using SuiteSparse.UMFPACK: UmfpackLU
+
+    # compute the sign/parity of a permutation
+    function _signperm(p)
+        n = length(p)
+        result = 0
+        todo = trues(n)
+        while any(todo)
+            k = findfirst(todo)
+            todo[k] = false
+            result += 1 # increment element count
+            j = p[k]
+            while j != k
+                result += 1 # increment element count
+                todo[j] = false
+                j = p[j]
+            end
+            result += 1 # increment cycle count
+        end
+        return ifelse(isodd(result), -1, 1)
+    end
+
+    function LinearAlgebra.logabsdet(F::UmfpackLU{T, TI}) where {T<:Union{Float64,ComplexF64},TI<:Union{Int32, Int64}} 
+        n = checksquare(F)
+        issuccess(F) || return log(zero(real(T))), zero(T)
+        U = F.U
+        Rs = F.Rs
+        p = F.p
+        q = F.q
+        s = _signperm(p)*_signperm(q)*one(real(T))
+        P = one(T)
+        abs_det = zero(real(T))
+        @inbounds for i in 1:n
+            dg_ii = U[i, i] / Rs[i]
+            P *= sign(dg_ii)
+            abs_det += log(abs(dg_ii))
+        end
+        return abs_det, s * P
+    end
+end
+
+
+function rrule(::typeof(logabsdet), x::SparseMatrixCSC)
+    F = cholesky(x)
+    L, D, U, P = SparseInverseSubset.get_ldup(F)
+    Ω = logabsdet(D)
+    function logabsdet_pullback(ΔΩ)
+        (Δy, Δsigny) = ΔΩ
+        (_, signy) = Ω
+        f = signy' * Δsigny
+        imagf = f - real(f)
+        g = real(Δy) + imagf
+        Z, P = sparseinv(F, depermute=true)
+        ∂x = g * Z'
+        return (NoTangent(), ∂x)
+    end
+    return Ω, logabsdet_pullback
+end
+
+function rrule(::typeof(logdet), x::SparseMatrixCSC)
+    Ω = logdet(x)
+    function logdet_pullback(ΔΩ)
+        Z, p = sparseinv(x, depermute=true)
+        ∂x = ΔΩ * Z'
+        return (NoTangent(), ∂x)
+    end
+    return Ω, logdet_pullback
+end
+
+function rrule(::typeof(det), x::SparseMatrixCSC)
+    Ω = det(x)
+    function det_pullback(ΔΩ)
+        Z, _ = sparseinv(x, depermute=true)
+        ∂x = Z' * dot(Ω, ΔΩ)
+        return (NoTangent(), ∂x)
+    end
+    return Ω, det_pullback
+end

test/rulesets/SparseArrays/sparsematrix.jl

@@ -33,3 +33,13 @@ end
     V̄ = rand!(similar(V))
     test_rrule(findnz, v ⊢ dv, output_tangent=(zeros(length(I)), V̄))
 end
+
+@testset "[log[abs[det]]] SparseMatrixCSC" begin
+    ii = [1:5; 2; 4]
+    jj = [1:5; 4; 2]
+    x = [ones(5); 0.1; 0.1]
+    A = sparse(ii, jj, x)
+    test_rrule(logabsdet, A)
+    test_rrule(logdet, A)
+    test_rrule(det, A)
+end