AI · 8 min read · April 21, 2026
Theory for learning blind inverse problems with finite samples
Researchers establish sample complexity bounds and optimal estimators for blind inverse problems using linear minimum mean square estimation framework.
Source: arxiv/cs.LG · Nathan Buskulic, Luca Calatroni, Lorenzo Rosasco, Silvia Villa · open original ↗ ↗
New theoretical framework quantifies how many samples are needed to learn blind inverse problems where both signal and operator are unknown.
- — Blind inverse problems lack ground truth for the forward operator, creating identifiability and symmetry challenges absent in standard settings.
- — Data-driven methods show empirical promise but offer no theoretical guarantees, limiting adoption in calibration-critical imaging applications.
- — Linear minimum mean square estimators (LMMSEs) provide closed-form optimal solutions with explicit dependence on signal, noise, and operator distributions.
- — Finite-sample error bounds connect convergence rates directly to noise level, problem conditioning, operator randomness, and training sample count.
- — Tikhonov regularization structure adapts automatically based on unknown signal and operator statistics, improving interpretability over black-box approaches.
- — Reconstruction error decreases predictably as noise and operator randomness diminish, validated by numerical experiments matching theoretical predictions.
- — Source condition assumptions enable explicit convergence rate analysis, bridging classical recovery theory with the blind setting.
Frequently asked
- A blind inverse problem requires recovering a signal when both the signal and the forward operator (measurement process) are unknown. This is harder than standard inverse problems because you cannot use known operator properties to guide recovery, and multiple different signal-operator pairs may produce identical measurements, creating ambiguity. The paper addresses this by deriving theoretical bounds on how many samples are needed to resolve this ambiguity.