Optimal Errors and Phase Transitions in High-Dimensional Generalized Linear Models

High-dimensional inference has become central in modern data science, with applications spanning neural networks, compressed sensing, and coding theory. Early work focused on Gaussian linear models, where tools from random matrix theory provided sharp results. However, many real-world problems involve nonlinearities and complex noise, making analysis more challenging. The replica method from statistical physics offered heuristic predictions for these models, suggesting phase transitions and capacity limits, but lacked rigorous proof. Recent advances introduced the adaptive interpolation technique, enabling rigorous validation of these predictions. This paper builds on these developments, extending the analysis to general nonlinear models with arbitrary priors and output channels, thus filling a crucial theoretical gap.

The main challenge lies in rigorously characterizing the mutual information and optimal errors in high-dimensional nonlinear models, where traditional tools falter. Existing heuristic methods, such as the replica approach, provided conjectures but lacked formal proof. Additionally, understanding the performance limits of practical algorithms like GAMP, especially in non-Gaussian and nonlinear settings, remains unresolved. Clarifying these aspects is vital for designing efficient inference algorithms and understanding the fundamental learnability limits, particularly in the presence of noise, sparsity, and complex activation functions.

The paper introduces an innovative combination of adaptive interpolation and replica symmetry analysis to rigorously derive the free entropy formula for general GLMs. This approach confirms the physics-based predictions and extends their validity beyond Gaussian assumptions. The analysis of the GAMP algorithm via state evolution equations provides a precise characterization of its optimality regions, revealing sharp phase transitions. This dual theoretical and algorithmic perspective offers a comprehensive understanding of the inference landscape, unifying previously disparate results and establishing new rigorous benchmarks.

• �� Define the probabilistic model with random measurement matrix Φ and prior P0. • Employ adaptive interpolation to connect the complex model to a tractable reference, ensuring rigorous bounds. • Derive the variational formula for free entropy as an extremum over auxiliary parameters q and r. • Analyze the posterior overlap to compute the MMSE and generalization error. • Use state evolution equations to track GAMP’s performance, identifying fixed points corresponding to optimal inference. • Establish phase transition boundaries by comparing the extremizers of the potential function with SE fixed points.

Simulations involve generating synthetic signals with specified sparsity and noise levels, applying random measurement matrices, and running GAMP alongside Bayesian estimators. Numerical integration computes the potential functions, verifying the theoretical phase diagrams. The experiments test various activation functions and noise models, confirming the predicted thresholds for successful recovery and GAMP optimality. Results show high agreement with theory, demonstrating the robustness of the formulas across different settings and validating the phase transition predictions.

The derivation of the replica-symmetric free entropy formula (equation (3)) confirms longstanding physics conjectures. The posterior overlap converges to a unique limit, enabling exact calculation of the Bayes MMSE and generalization error. The state evolution analysis precisely delineates regions where GAMP achieves the Bayes-optimal performance, identifying sharp phase transitions. These results unify the understanding of inference limits across diverse models, providing rigorous benchmarks for algorithmic performance and theoretical capacity.

The findings directly impact compressed sensing, neural network training, and error-correcting code design by clarifying the fundamental limits of signal recovery and classification. Practitioners can use the phase diagrams to determine the feasibility of reconstruction under given measurement ratios and noise conditions. The theoretical guarantees guide the development of new algorithms that approach optimality, fostering advances in machine learning, communications, and data science. Long-term, these insights could inform the design of deep learning architectures with provable performance bounds.

The analysis assumes iid measurement matrices, limiting applicability to structured or correlated data matrices common in practice. Results are asymptotic, relying on the limit of large system size, so finite-sample deviations are not characterized. The models focus on smooth activation functions and well-behaved noise, leaving out heavy-tailed or discontinuous cases. Extending the framework to more complex neural architectures and real-world datasets remains a future challenge, requiring further theoretical and computational work.