Affected Repositories

• PyAutoArray (primary)

Branch Survey

Suggested branch: feature/matrix-free-interferometer-phase3

Implementation Steps

See the prompt's "What to build".

Key Files

• autoarray/inversion/inversion/interferometer/inversion_interferometer_matrix_free_util.py — phase-1 module to extend
• autoarray/inversion/regularization/matern_sparse.py — phase-2 factor R
• test_autoarray/inversion/inversion/interferometer/test_matrix_free_util.py

Matrix-free interferometer inversion, phase 3: log-evidence via exact sparse log-det + stochastic Lanczos quadrature

Type: feature
Target: PyAutoArray
Repos:

• PyAutoArray
  Difficulty: large
  Autonomy: supervised
  Priority: high
  Parent: active/matrix_free_interferometer_inversion.md
  Community: Matrix-free solution to make very large meshes more efficient for Interferometer .github#39
  Status: formalised
  Consequence: glance
  Witness: tests asserting (i) ln det P from the sparse LU equals numpy slogdet of the dense P to 1e-10; (ii) the SLQ estimate of ln det(F + H) on a small rectangular-mesh interferometer case lies within 0.5% of the dense slogdet for 10 probes / 40 steps with a fixed seed, and the estimate is deterministic for a fixed seed; (iii) the matrix-free log_evidence equals InversionInterferometerSparse.log_evidence (unconstrained, same MaternSparse) within the same tolerance on the ln det term and exactly on the other four terms
  Review-minutes: 6
  Unattended: ready
  Filed: 2026-10-21

Why

Phase 3 of PyAutoArray#638 (hub #39, @HRSAstro; reference section 5 log_det_A / log_evidence of the attached upstream_snippets.py). The evidence needs ln det(F + H) and ln det H without forming n×n matrices. ln det H is exact from the sparse factor (phase 2 MaternSparse). ln det(F + H) = ln det P + ln det(P⁻¹(F + H)) with P the Powell preconditioner (phase 1): the first term is exact from P's sparse LU, the second from stochastic Lanczos quadrature (Ubaru, Chen & Saad 2017) in the P inner product with probes v ~ N(0, P⁻¹) drawn via P = BᵀB, B = [√κ₀ M; √λ R], full reorthogonalisation, fixed seed so the estimate is smooth in λ. The contributor reports ~0.2% of ln Z agreement; Powell et al. stop at ln det P (1-20% off).

What to build

1. In inversion_interferometer_matrix_free_util.py: log_det_preconditioner_from(lu); log_det_curvature_reg_via_slq_from(operator, regularization_sparse, R_sparse, coefficient, n_probes=10, n_steps=40, seed=...) (Lanczos on P⁻¹A, P-orthonormal, full reorthogonalisation, Gauss quadrature on the tridiagonal eigenpairs; returns mean estimate and the per-probe spread); log_evidence_matrix_free_from(operator, regularization, coefficient, reconstruction, data_term, noise_normalization) assembling the five terms (chi² = dᵀN⁻¹d − 2 sᵀD + sᵀFs, regularization term, ln det(F+H), −ln det H, noise normalisation) with the same sign/0.5 conventions as AbstractInversion.log_evidence.
2. MaternSparse exposes its factor R (phase 2) for the probe draw.
3. Tests test_matrix_free_util.py: ln det P exact; SLQ within 0.5% of dense; deterministic seed; probe-count convergence (error shrinks from 2 → 10 probes); evidence parity with the dense path term by term. numpy/scipy only; nojax leg.

Acceptance

Witness passes; tolerance documented in the module docstring; #638 and hub thread updated at PR-open.