Rewrite svd_rev to reduce allocations

src/rulesets/LinearAlgebra/factorization.jl

@@ -243,7 +243,7 @@ function rrule(::typeof(getproperty), F::T, x::Symbol) where T <: SVD
     return getproperty(F, x), getproperty_svd_pullback
 end
 
-# When not `ZeroTangent`s expect `Ū::AbstractMatrix, s̄::AbstractVector, V̄::AbstractMatrix`
+# When not `ZeroTangent`s expect `Ū::AbstractMatrix, s̄::AbstractVector, V̄t::AbstractMatrix`
 function svd_rev(USV::SVD, Ū, s̄, V̄t)
     # Note: assuming a thin factorization, i.e. svd(A, full=false), which is the default
     U = USV.U
@@ -252,25 +252,24 @@ function svd_rev(USV::SVD, Ū, s̄, V̄t)
 
     k = length(s)
     T = eltype(s)
-    F = T[i == j ? 1 : inv(@inbounds s[j]^2 - s[i]^2) for i = 1:k, j = 1:k]
-
-    UtŪ = U'*Ū
-    V̄tV = V̄t*Vt'
-
-    FUᵀŪS = F .* (UtŪ .- UtŪ') .* s'
-    SFVᵀV̄ = F .* (V̄tV' .- V̄tV) .* s
-
-    S = Diagonal(s)
-    S̄ = s̄ isa AbstractZero ? s̄ : Diagonal(s̄)
-    
-    if size(Vt,1) == size(Vt,2) 
+    UtŪ = U' * Ū
+    V̄tV = V̄t * Vt'
+    M = @inbounds T[
+        if i == j
+            s̄[i]
+        else
+            (s[j] * (UtŪ[i, j] - UtŪ[j, i]) + s[i] * (V̄tV[j, i] - V̄tV[i, j])) /
+            (s[j]^2 - s[i]^2)
+        end for i in 1:k, j in 1:k
+    ]
+
+    if size(Vt, 1) == size(Vt, 2)
         # V is square, VVᵀ = I and therefore V̄ᵀ - V̄ᵀVVᵀ = 0
-        Ā = (U * (FUᵀŪS + S̄ + SFVᵀV̄) + ((Ū .- U * UtŪ) / S)) * Vt
-    else 
+        Ā = (U * M .+ ((Ū .- U * UtŪ) ./ s')) * Vt
+    else
         # If V is not square then U is, so UUᵀ == I and Ū - UUᵀŪ = 0
-        Ā = U * ((FUᵀŪS + S̄ + SFVᵀV̄) * Vt + (S \ (V̄t .- V̄tV * Vt)))
+        Ā = U * (M * Vt .+ ((V̄t .- V̄tV * Vt) ./ s))
     end
-
     return Ā
 end