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LERE: Learning-Based Low-Rank Matrix Recovery with Rank Estimation
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Proposes LERE, a learning-based low-rank matrix recovery method that estimates rank via a new Gerschgorin-disk decision rule (R_GDE) and recovers RPCA low-rank components using random row/column sub-matrix sampling with a 17-iteration FRMNN.
Problem
Low-rank matrix recovery (LRMR) requires a known rank, and existing rank estimators fail on ill-conditioned matrices; existing LRMR optimizers are also parameter-sensitive and prone to local optima.
Approach
A new heuristic rank decision rule (R_GDE) that uses descending Gerschgorin disk centers and a unified adjustment factor combining center and radius information, plus a learned, rank-driven sub-matrix recovery scheme (LERE) using random row/column sampling and a 17-iteration FRMNN, applied to RPCA.
Key results
- R_GDE achieves 100% rank estimation accuracy on most synthetic settings where GDE and RANK fail at large condition numbers (e.g., GDE 0% at kappa=30 for n1=3000).
- LERE reaches reconstruction errors around 1e-8 to 1e-7 on synthetic RPCA with n2=3000, outperforming ARE-RPCA, ADW-RPCA, AccAltProj, RPCA-HQF and LRPCA in most settings.
- On HumanBody2 foreground-background separation, LERE achieves PSNR 37.64 and MSSSIM 0.997, above the best baseline RPCA-HQF (36.34 PSNR).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use R_GDE instead of RANK for rank estimation | Rank estimation · Synthetic matrices (n2=3000, α=0.4, κ=30) | rank estimation accuracy: 77 -> 100 | Helped |
| Use R_GDE instead of GDE for rank estimation | Rank estimation · Synthetic matrices (n2=3000, α=0.4, κ=30) | rank estimation accuracy: 0 -> 100 | Helped |
| Use LERE instead of LRPCA† (200 iterations) | RPCA · Synthetic (n2=500, κ=10, α=0.5) | Lrec: 6.1E-3 -> 4.8E-8 | Helped |
| Use LERE instead of LRPCA† (200 iterations) | RPCA · Synthetic (n2=500, κ=10, α=0.5) | Srec: 2.0E-2 -> 1.6E-6 | Helped |
| Use LERE instead of RPCA-HQF | Foreground-background separation · HumanBody2 (one frame) | PSNR: 36.34 -> 37.64 | Helped |
| Use LERE instead of RPCA-HQF | Foreground-background separation · HumanBody2 (one frame) | MSSSIM: 0.995 -> 0.997 | Helped |
| Increase LRPCA iterations from 17 to 200 | LRPCA · RPCA synthetic datasets | reconstruction error | Helped |
Surprising findings
- Rank estimators GDE and RANK fail on ill-conditioned matrices, with GDE dropping to 0% accuracy at kappa=30 in some settings.
Limitations
- Evaluated only on RPCA as an instance of LRMR; other LRMR tasks not tested.
- Relies on a learned 17-iteration FRMNN whose training data and convergence guarantees depend on incoherence and sparsity assumptions.
- Future work noted on extending to tensor recovery, implying matrix-only scope.
Experimental setup
Datasets: synthetic low-rank plus sparse matrices SBI dataset LDR dataset SuiteSparse Database
Benchmarks: synthetic RPCA matrices SuiteSparse Database SBI HumanBody2 LDR dataset
Metrics: rank estimation accuracy relative reconstruction error (Lrec, Srec) AGE pEPs pCEPS MSSSIM PSNR CQM
Baselines: ARE-RPCA ADW-RPCA AccAltProj ScaledGD RPCA-HQF LRPCA GDE RANK
Builds on: GDE RANK LRPCA ScaledGD FRMNN
Closest papers by meaning
- Rank Overspecified Robust Matrix Recovery: Subgradient Method and Exact Recovery (NeurIPS 2021 · similarity 0.87)
- A Unified Variance Reduction-Based Framework for Nonconvex Low-Rank Matrix Recovery (ICML 2017 · similarity 0.86)
- RGNMR: A Gauss-Newton method for robust matrix completion with theoretical guarantees (NeurIPS 2025 · similarity 0.86)
- Guarantees of a Preconditioned Subgradient Algorithm for Overparameterized Asymmetric Low-rank Matrix Recovery (ICML 2025 · similarity 0.85)
- StelLA: Subspace Learning in Low-rank Adaptation using Stiefel Manifold (NeurIPS 2025 · similarity 0.85)
- Efficient Learning with Sine-Activated Low-Rank Matrices (ICLR 2025 · similarity 0.84)
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