Statistical Mechanics of High-Dimensional Inference
This work addresses a foundational problem in machine learning and statistics for researchers dealing with large-scale datasets, offering new theoretical insights and practical algorithms, though it builds on existing statistical physics frameworks.
The paper tackles the fundamental limits of parameter inference in high-dimensional settings with finite measurement density, revealing that traditional methods like maximum likelihood and maximum-a-posteriori cannot achieve these limits, and proposes optimal algorithms that reduce data requirements by up to 20% compared to MAP.
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 $α= \frac{N}{P}\rightarrow \infty$. 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.