# On the Sample Complexity of Learning for Blind Inverse Problems

arXiv: https://arxiv.org/abs/2512.23405 · HTML: https://mlsanity.com/paper/2512.23405/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 36 references that are in the map.
- First posted on arXiv 29 Dec 2025.

## Builds on (most cited first, 15 of 15)

- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Implicit Bias of Gradient Descent on Linear Convolutional Networks](https://mlsanity.com/paper/1806.00468/) (2018)
- [Parallel Diffusion Models of Operator and Image for Blind Inverse Problems](https://mlsanity.com/paper/2211.10656/) (2022)
- [Deep Learning Methods for Parallel Magnetic Resonance Image Reconstruction](https://mlsanity.com/paper/1904.01112/) (2019)
- [Fast Diffusion EM: a diffusion model for blind inverse problems with application to deconvolution](https://mlsanity.com/paper/2309.00287/) (2023)
- [Blind Image Restoration via Fast Diffusion Inversion](https://mlsanity.com/paper/2405.19572/) (2024)
- [Learning the optimal Tikhonov regularizer for inverse problems](https://mlsanity.com/paper/2106.06513/) (2021)
- [Kernel Diffusion: An Alternate Approach to Blind Deconvolution](https://mlsanity.com/paper/2312.02319/) (2023)
- [On Learning the Optimal Regularization Parameter in Inverse Problems](https://mlsanity.com/paper/2311.15845/) (2023)
- [BlindDiff: Empowering Degradation Modelling in Diffusion Models for Blind Image Super-Resolution](https://mlsanity.com/paper/2403.10211/) (2024)
- [Learning sparsity-promoting regularizers for linear inverse problems](https://mlsanity.com/paper/2412.16031/) (2024)
- [Why the noise model matters: A performance gap in learned regularization](https://mlsanity.com/paper/2510.12521/) (2025)

## Built on by (most cited first, 1 of 1)

- [A Comparative Study of MAP and LMMSE Estimators for Blind Inverse Problems](https://mlsanity.com/paper/2602.11814/) (2026)

## Abstract (arXiv)

Blind inverse problems arise in many experimental settings where both the signal of interest and the forward operator are (partially) unknown. In this context, methods developed for the non-blind case cannot be adapted in a straightforward manner due to identifiability issues and symmetric solutions inherent to the blind setting. Recently, data-driven approaches have been proposed to address such problems, demonstrating strong empirical performance and adaptability. However, these methods often lack interpretability and are not supported by theoretical guarantees, limiting their reliability in domains such as applied imaging where a blind approach often relates to a calibration of the acquisition device. In this work, we shed light on learning in blind inverse problems within the insightful framework of Linear Minimum Mean Square Estimators (LMMSEs). We provide a theoretical analysis, deriving closed-form expressions for optimal estimators and extending classical recovery results to the blind setting. In particular, we establish equivalences with tailored Tikhonov-regularized formulations, where the regularization structure depends explicitly on the distributions of the unknown signal, of the noise, and of the random forward operator. We also show how the reconstruction error converges as the noise and the randomness of the operator diminish when we use a source condition assumption. Furthermore, we derive finite-sample error bounds that characterize the performance of the learned estimators as a function of the noise level, problem conditioning, and number of available samples. These bounds explicitly quantify the impact of operator randomness and show explicitly the dependence of the associated convergence rates to this randomness factors. Finally, we validate our theoretical findings through illustrative exemplar numerical experiments that confirm the predicted convergence behavior.