Product of non-dense matrix problem

[yiyuezhuo]

Product of two Triangular matrix:

using Tracker
using LinearAlgebra

X = randn(3, 3)

function g1(x)
    y = UpperTriangular(x)
    sum(y * y)
end

Tracker.gradient(g1, X)

throw:

MethodError: Cannot `convert` an object of type Array{Float64,2} to an object of type UpperTriangular{Float64,Array{Float64,2}}
Closest candidates are:
  convert(::Type{var"#s826"} where var"#s826"<:UpperTriangular, !Matched::Union{UnitUpperTriangular, UpperTriangular}) at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.5/LinearAlgebra/src/special.jl:61
  convert(::Type{var"#s826"} where var"#s826"<:UpperTriangular, !Matched::Union{LinearAlgebra.AbstractTriangular, Bidiagonal, Diagonal}) at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.5/LinearAlgebra/src/special.jl:65
  convert(::Type{T}, !Matched::T) where T<:AbstractArray at abstractarray.jl:14
  ...

Stacktrace:
 [1] setproperty!(::Tracker.Tracked{UpperTriangular{Float64,Array{Float64,2}}}, ::Symbol, ::Array{Float64,2}) at ./Base.jl:34
 [2] back(::Tracker.Tracked{UpperTriangular{Float64,Array{Float64,2}}}, ::Array{Float64,2}, ::Bool) at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:53
 [3] #13 at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:38 [inlined]
 [4] foreach at ./abstractarray.jl:2010 [inlined]
 [5] back_(::Tracker.Call{Tracker.var"#521#522"{TrackedArray{…,UpperTriangular{Float64,Array{Float64,2}}},TrackedArray{…,UpperTriangular{Float64,Array{Float64,2}}}},Tuple{Tracker.Tracked{UpperTriangular{Float64,Array{Float64,2}}},Tracker.Tracked{UpperTriangular{Float64,Array{Float64,2}}}}}, ::Array{Float64,2}, ::Bool) at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:38
 [6] back(::Tracker.Tracked{UpperTriangular{Float64,Array{Float64,2}}}, ::Array{Float64,2}, ::Bool) at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:58
 [7] #13 at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:38 [inlined]
 [8] foreach at ./abstractarray.jl:2010 [inlined]
 [9] back_(::Tracker.Call{Tracker.var"#494#495"{TrackedArray{…,UpperTriangular{Float64,Array{Float64,2}}}},Tuple{Tracker.Tracked{UpperTriangular{Float64,Array{Float64,2}}}}}, ::Float64, ::Bool) at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:38
 [10] back(::Tracker.Tracked{Float64}, ::Int64, ::Bool) at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:58
 [11] #back!#15 at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:77 [inlined]
 [12] #back!#32 at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/lib/real.jl:16 [inlined]
 [13] back!(::Tracker.TrackedReal{Float64}) at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/lib/real.jl:14
 [14] gradient_(::Function, ::Array{Float64,2}) at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:4
 [15] #gradient#24 at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:164 [inlined]
 [16] gradient(::Function, ::Array{Float64,2}) at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:164
 [17] top-level scope at In[43]:1
 [18] include_string(::Function, ::Module, ::String, ::String) at ./loading.jl:1091

Product of two Triangular matrix (result is dense matrix):

function g2(x)
    y = UpperTriangular(x)
    sum(y' * y)
end

Tracker.gradient(g2, X)

throw:

MethodError: Cannot `convert` an object of type Base.ReshapedArray{Float64,2,Adjoint{Float64,Array{Float64,2}},Tuple{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64}}} to an object of type UpperTriangular{Float64,Array{Float64,2}}
Closest candidates are:
  convert(::Type{var"#s826"} where var"#s826"<:UpperTriangular, !Matched::Union{UnitUpperTriangular, UpperTriangular}) at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.5/LinearAlgebra/src/special.jl:61
  convert(::Type{var"#s826"} where var"#s826"<:UpperTriangular, !Matched::Union{LinearAlgebra.AbstractTriangular, Bidiagonal, Diagonal}) at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.5/LinearAlgebra/src/special.jl:65
  convert(::Type{T}, !Matched::T) where T<:AbstractArray at abstractarray.jl:14
  ...

Stacktrace:
 [1] setproperty!(::Tracker.Tracked{UpperTriangular{Float64,Array{Float64,2}}}, ::Symbol, ::Base.ReshapedArray{Float64,2,Adjoint{Float64,Array{Float64,2}},Tuple{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64}}}) at ./Base.jl:34
 [2] back(::Tracker.Tracked{UpperTriangular{Float64,Array{Float64,2}}}, ::Base.ReshapedArray{Float64,2,Adjoint{Float64,Array{Float64,2}},Tuple{Base.MultiplicativeInverses.SignedMultiplicativeInverse{Int64}}}, ::Bool) at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:53
 [3] #13 at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:38 [inlined]
 [4] foreach at ./abstractarray.jl:2010 [inlined]
 [5] back_(::Tracker.Call{Tracker.var"#392#393"{TrackedArray{…,UpperTriangular{Float64,Array{Float64,2}}}},Tuple{Tracker.Tracked{UpperTriangular{Float64,Array{Float64,2}}}}}, ::Array{Float64,2}, ::Bool) at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:38
 [6] back(::Tracker.Tracked{Adjoint{Float64,UpperTriangular{Float64,Array{Float64,2}}}}, ::Array{Float64,2}, ::Bool) at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:58
 [7] #13 at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:38 [inlined]
 [8] foreach at ./abstractarray.jl:2010 [inlined]
 [9] back_(::Tracker.Call{Tracker.var"#521#522"{TrackedArray{…,Adjoint{Float64,UpperTriangular{Float64,Array{Float64,2}}}},TrackedArray{…,UpperTriangular{Float64,Array{Float64,2}}}},Tuple{Tracker.Tracked{Adjoint{Float64,UpperTriangular{Float64,Array{Float64,2}}}},Tracker.Tracked{UpperTriangular{Float64,Array{Float64,2}}}}}, ::Array{Float64,2}, ::Bool) at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:38
 [10] back(::Tracker.Tracked{Array{Float64,2}}, ::Array{Float64,2}, ::Bool) at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:58
 [11] #13 at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:38 [inlined]
 [12] foreach at ./abstractarray.jl:2010 [inlined]
 [13] back_(::Tracker.Call{Tracker.var"#494#495"{TrackedArray{…,Array{Float64,2}}},Tuple{Tracker.Tracked{Array{Float64,2}}}}, ::Float64, ::Bool) at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:38
 [14] back(::Tracker.Tracked{Float64}, ::Int64, ::Bool) at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:58
 [15] #back!#15 at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:77 [inlined]
 [16] #back!#32 at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/lib/real.jl:16 [inlined]
 [17] back!(::Tracker.TrackedReal{Float64}) at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/lib/real.jl:14
 [18] gradient_(::Function, ::Array{Float64,2}) at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:4
 [19] #gradient#24 at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:164 [inlined]
 [20] gradient(::Function, ::Array{Float64,2}) at /home/yiyuezhuo/.julia/packages/Tracker/AMi2h/src/back.jl:164
 [21] top-level scope at In[45]:6
 [22] include_string(::Function, ::Module, ::String, ::String) at ./loading.jl:1091

While it seems that one can get gradient if just one argument is Triangular matrix:

function g3(x)
    y = UpperTriangular(x)
    sum(2*y' .+ 1)
end

Tracker.gradient(g3, X)

gives:

([2.0 2.0 2.0; 0.0 2.0 2.0; 0.0 0.0 2.0] (tracked),)