Fix remove_svd_gauge_dependence! for rank-deficient SVDs

[leburgel]

When some singular values are below rank_atol (r < length(S)), remove_svd_gauge_dependence! has two bugs:

• The degeneracy mask is built from all singular values, but gaugepart is only r × r, so the function throws a BoundsError.
• The columns of ΔU (rows of ΔVᴴ) beyond r are meant to be zeroed, but zero!(ΔU[:, (r + 1):end]) zeroes a copy. The first bug has kept this one from surfacing.

This PR builds the mask from Sdiag[1:r] and zeroes views. The full-rank case is unchanged.

Reproducer

using LinearAlgebra, Random
using MatrixAlgebraKit
using MatrixAlgebraKit: remove_svd_gauge_dependence!

Random.seed!(1)
m, n, r = 6, 5, 3
Q₁ = Matrix(qr(randn(ComplexF64, m, m)).Q)
Q₂ = Matrix(qr(randn(ComplexF64, n, n)).Q)
A = Q₁[:, 1:n] * Diagonal([1.0, 0.5, 0.25, 0.0, 0.0]) * Q₂' # rank 3
U, S, Vᴴ = svd_compact(A)
ΔU, ΔVᴴ = randn(ComplexF64, size(U)), randn(ComplexF64, size(Vᴴ))

ΔU, ΔVᴴ = remove_svd_gauge_dependence!(ΔU, ΔVᴴ, U, S, Vᴴ)
@show norm(ΔU[:, (r + 1):end]) norm(ΔVᴴ[(r + 1):end, :])

On main:

ERROR: LoadError: BoundsError: attempt to access 3×3 Matrix{ComplexF64} at index [5×5 BitMatrix]

With only the mask fixed, the rank-deficient part is not zeroed:

norm(ΔU[:, r + 1:end]) = 3.595426447598924
norm(ΔVᴴ[r + 1:end, :]) = 3.181565844025091

With this PR:

norm(ΔU[:, r + 1:end]) = 0.0
norm(ΔVᴴ[r + 1:end, :]) = 0.0

I have a small regression test I could add, but that would be another instance of testing pullback-related things directly like the test added in #282, so I didn't add it right away. Let me know if this would be good to add anyway.

[lkdvos]

Except for the views, looks good to me!

(I am guessing @Jutho will oppose to the @view instead of directly using view, so that would be my only other suggestion 😉 )

[Jutho]

This was quite stupid indeed 😄 .

[Jutho]

We could have had an S₁ = view(Sdiag, 1:r) definition and use this here, but it's fine like this.

[codecov]

Codecov Report

❌ Patch coverage is 33.33333% with 2 lines in your changes missing coverage. Please review.

Files with missing lines	Patch %	Lines
src/pullbacks/svd.jl	33.33%	2 Missing ⚠️

Files with missing lines	Coverage Δ
src/pullbacks/svd.jl	93.36% <33.33%> (ø)

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