Testing and estimation in orthosymmetric Gaussian sequence model
Zeyu Jia, Yury Polyanskiy
TL;DR
This paper analyzes goodness-of-fit testing, density estimation, and likelihood-free hypothesis testing in the Gaussian sequence model with mean constrained to a convex, compact set $\Gamma$. It establishes that for orthosymmetric $\Gamma$, the GOF minimax complexity $n_{\mathrm{gof}}$ is bounded below by the square root of the estimation cost (up to polylog factors), and that this relation becomes tight under the additional quadratically convex (QCO) assumption. It then provides a complete LFHT characterization for $\ell_p$ bodies, showing a universal LFHT regime for $p\ge 2$ and an effective-dimension dependent regime for $p\le 2$, including finite- and infinite-dimensional extensions and concrete examples. A counterexample demonstrates that the two-sided GP conjecture can fail without extra structure, highlighting the necessity of orthosymmetry and convexity in the bounds. Collectively, the results connect estimation, testing, and likelihood-free testing via dimension- and structure-dependent rates, and illuminate how complexity scales with the geometry of $\Gamma$ in high-dimensional Gaussian models.
Abstract
We study the Gaussian sequence model, i.e. $X \sim N(\mathbfθ, I_\infty)$, where $\mathbfθ \in Γ\subset \ell_2$ is assumed to be convex and compact. We show that goodness-of-fit testing sample complexity is lower bounded by the square-root of the estimation complexity, whenever $Γ$ is orthosymmetric. This lower bound is tight when $Γ$ is also quadratically convex (as shown by [Donoho et al. 1990, Neykov 2023]). We also completely characterize likelihood-free hypothesis testing (LFHT) complexity for $\ell_p$-bodies, discovering new types of tradeoff between the numbers of simulation and observation samples, compared to the case of ellipsoids (p = 2) studied in [Gerber and Polyanskiy 2024].