Pointwise Complexity for Gaussian Fields: Upper Envelopes, Algorithmic Lower Bounds, and Separation
We prove a variance-aware pointwise majorizing-measure theorem for centered Gaussian processes. Classical generic chaining characterizes the scalar quantity $\mathbb E\sup_{x\in T}X_x$; the theorem here gives a simultaneous high-probability envelope for the entire field. For an ambient prior $μ$, the envelope at $x$ is governed by a pointwise Fernique-Talagrand functional \[Φ_μ(x):=\int_0^{4σ(x)}\sqrt{\log\frac{1}{μ(B_d(x,\varepsilon))}}\,d\varepsilon,\] together with the corresponding Gaussian tail term. The theorem provides a reusable field-level refinement of classical generic chaining and a Gaussian-process counterpart of pointwise empirical-process bounds for deep neural networks. We...
| Authors | Yunbei Xu |
|---|---|
| Primary category | math.PR |
| Published | 6 June 2026 |
| Last updated at source | 2 July 2026 |
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