Pointwise Complexity for Gaussian Fields: Upper Envelopes, Algorithmic Lower Bounds, and Separation (opens in new tab)

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 $\mu$, the envelope at $x$ is governed by a pointwise Fernique-Talagrand functional \[\Phi_\mu(x):=\int_0^{4\sigma(x)}\sqrt{\log\frac{1}{\mu(B_d(x,\vareps...