# Phase Diagram and Approximate Message Passing for Blind Calibration and Dictionary Learning

arXiv: https://arxiv.org/abs/1301.5898 · HTML: https://mlsanity.com/paper/1301.5898/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jan 2013.

## Built on by (most cited first, 6 of 6)

- [Statistical limits of dictionary learning: random matrix theory and the spectral replica method](https://mlsanity.com/paper/2109.06610/) (2021)
- [Statistical mechanics of extensive-width Bayesian neural networks near interpolation](https://mlsanity.com/paper/2505.24849/) (2025)
- [On the phase diagram of extensive-rank symmetric matrix denoising beyond rotational invariance](https://mlsanity.com/paper/2411.01974/) (2024)
- [Automatic Hyperparameter Tuning in Sparse Matrix Factorization](https://mlsanity.com/paper/2305.10114/) (2023)
- [Optimal generalisation and learning transition in extensive-width shallow neural networks near interpolation](https://mlsanity.com/paper/2501.18530/) (2025)
- [Statistical physics of deep learning: Optimal learning of a multi-layer perceptron near interpolation](https://mlsanity.com/paper/2510.24616/) (2025)

## Abstract (arXiv)

We consider dictionary learning and blind calibration for signals and matrices created from a random ensemble. We study the mean-squared error in the limit of large signal dimension using the replica method and unveil the appearance of phase transitions delimiting impossible, possible-but-hard and possible inference regions. We also introduce an approximate message passing algorithm that asymptotically matches the theoretical performance, and show through numerical tests that it performs very well, for the calibration problem, for tractable system sizes.