New Theoretical Framework for Pointwise Complexity in Gaussian Fields
Researchers have proved a variance-aware pointwise majorizing-measure theorem for centered Gaussian processes that yields simultaneous high-probability envelopes for entire fields, not just scalar expectations. The work extends classical generic chaining theory by introducing a pointwise Fernique-Talagrand functional and connects it to Bayesian algorithmic lower bounds derived from the interactive Fano/data-processing principle. The results offer local-geometric certificates of pointwise complexity for fixed estimators in overparameterized settings where classical minimax theory is too coarse or oracle-dependent.
A preprint submitted to arXiv on June 6, 2026 by Yunbei Xu presents a variance-aware pointwise majorizing-measure theorem for centered Gaussian processes. While classical generic chaining characterizes the expected supremum of a Gaussian field as a single scalar, the new theorem provides a simultaneous high-probability envelope for the entire field, governed at each point by a pointwise Fernique-Talagrand functional that integrates a logarithmic small-ball probability term up to twice the local standard deviation. The paper also derives a Bayesian algorithmic lower envelope using the interactive Fano and data-processing inequality, expressing the lower bound through the exact ghost small-ball mass of a concrete estimator rather than worst-case covering numbers. In Gaussian location experiments, comparison decoders are used to translate Bayes location error into lower bounds on decision-aligned Gaussian ranges. A weighted-basis separation example is constructed to demonstrate that the usual Fano relaxation, the Bayesian algorithmic lower envelope, the pointwise Gaussian envelope, and the full-class minimax risk are all genuinely distinct quantities. Together, the results position algorithmic lower bounds as local-geometric tools for analyzing fixed estimators in overparameterized ambient classes, a regime where classical minimax theory struggles. The work spans probability theory, statistical mechanics, information theory, machine learning, and statistics theory.
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- arXiv cs.LGCenter
Pointwise Complexity for Gaussian Fields: Upper Envelopes, Algorithmic Lower Bounds, and Separation
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