# The TAP free energy for high-dimensional linear regression

arXiv: https://arxiv.org/abs/2203.07539 · HTML: https://mlsanity.com/paper/2203.07539/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 34 references that are in the map.
- First posted on arXiv 14 Mar 2022.

## Builds on (most cited first, 5 of 5)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Optimal Errors and Phase Transitions in High-Dimensional Generalized Linear Models](https://mlsanity.com/paper/1708.03395/) (2017)
- [Local convexity of the TAP free energy and AMP convergence for Z2-synchronization](https://mlsanity.com/paper/2106.11428/) (2021)
- [An Instability in Variational Inference for Topic Models](https://mlsanity.com/paper/1802.00568/) (2018)
- [Variational Inference in high-dimensional linear regression](https://mlsanity.com/paper/2104.12232/) (2021)

## Built on by (most cited first, 3 of 3)

- [High-dimensional scaling limits and fluctuations of online least-squares SGD with smooth covariance](https://mlsanity.com/paper/2304.00707/) (2023)
- [A Mean Field Approach to Empirical Bayes Estimation in High-dimensional Linear Regression](https://mlsanity.com/paper/2309.16843/) (2023)
- [TAP Accuracy Below the Fluctuation Scale and Universal Posterior Geometry in Spherical Linear Models](https://mlsanity.com/paper/2609.20577/) (2026)

## Abstract (arXiv)

We derive a variational representation for the log-normalizing constant of the posterior distribution in Bayesian linear regression with a uniform spherical prior and an i.i.d. Gaussian design. We work under the "proportional" asymptotic regime, where the number of observations and the number of features grow at a proportional rate. This rigorously establishes the Thouless-Anderson-Palmer (TAP) approximation arising from spin glass theory, and proves a conjecture of Krzakala et. al. (2014) in the special case of the spherical prior.