# Statistical Mechanics of High-Dimensional Inference

arXiv: https://arxiv.org/abs/1601.04650 · HTML: https://mlsanity.com/paper/1601.04650/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jan 2016.

## Built on by (most cited first, 13 of 13)

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- [Contrasting random and learned features in deep Bayesian linear regression](https://mlsanity.com/paper/2203.00573/) (2022)
- [Features are fate: a theory of transfer learning in high-dimensional regression](https://mlsanity.com/paper/2410.08194/) (2024)
- [The price of ignorance: how much does it cost to forget noise structure in low-rank matrix estimation?](https://mlsanity.com/paper/2205.10009/) (2022)
- [Kernel Methods and Multi-layer Perceptrons Learn Linear Models in High Dimensions](https://mlsanity.com/paper/2201.08082/) (2022)
- [Risk and cross validation in ridge regression with correlated samples](https://mlsanity.com/paper/2408.04607/) (2024)
- [Performance of Bayesian linear regression in a model with mismatch](https://mlsanity.com/paper/2107.06936/) (2021)
- [Ising Model Selection Using ℓ1-Regularized Linear Regression: A Statistical Mechanics Analysis](https://mlsanity.com/paper/2102.03988/) (2021)
- [Soft-margin classification of object manifolds](https://mlsanity.com/paper/2203.07040/) (2022)
- [Optimal Implicit Bias in Linear Regression](https://mlsanity.com/paper/2506.17187/) (2025)

## Abstract (arXiv)

To model modern large-scale datasets, we need efficient algorithms to infer a set of P unknown model parameters from N noisy measurements. What are fundamental limits on the accuracy of parameter inference, given finite signal-to-noise ratios, limited measurements, prior information, and computational tractability requirements? How can we combine prior information with measurements to achieve these limits? Classical statistics gives incisive answers to these questions as the measurement density α = fracNPrightarrow ∞. However, these classical results are not relevant to modern high-dimensional inference problems, which instead occur at finite α. We formulate and analyze high-dimensional inference as a problem in the statistical physics of quenched disorder. Our analysis uncovers fundamental limits on the accuracy of inference in high dimensions, and reveals that widely cherished inference algorithms like maximum likelihood (ML) and maximum-a posteriori (MAP) inference cannot achieve these limits. We further find optimal, computationally tractable algorithms that can achieve these limits. Intriguingly, in high dimensions, these optimal algorithms become computationally simpler than MAP and ML, while still outperforming them. For example, such optimal algorithms can lead to as much as a 20% reduction in the amount of data to achieve the same performance relative to MAP. Moreover, our analysis reveals simple relations between optimal high dimensional inference and low dimensional scalar Bayesian inference, insights into the nature of generalization and predictive power in high dimensions, information theoretic limits on compressed sensing, phase transitions in quadratic inference, and connections to central mathematical objects in convex optimization theory and random matrix theory.