Characterize test-point placement and saturation on blind hyperplanes

Determine a placement theory or saturation rate in the number of Barankin test points that characterizes where the achievable finite-perturbation bound saturates on the blind hyperplane for nonlinear observables such as stochastic efficiency.

Background

The paper applies the finite-order Barankin bound by selecting perturbation test points and reports substantial recovery of the variance of stochastic efficiency, including approximately 75% on a blind hyperplane where all linear-response bounds vanish. However, the authors use empirical rules—matching pursuit, joint optimization, and dictionary selection—and none is shown to be optimal or universally dominant.

The paper explicitly identifies the unresolved issue as both the absence of a general placement theory or convergence rate in the number of test points and the lack of knowledge about the limiting achievable bound on the blind hyperplane. Resolving this would turn numerical recovery percentages into theoretically justified predictions.

References

Test-point placement is unsolved. We give three rules, none of which dominates (Table~\ref{tab:select}). In the Gaussian example of Section~\ref{sec:diffusion} the question has a complete answer, three points, but only because the response there is a single mode of the kernel. Appendix~\ref{sec:placement} measures what that costs (about $1.2$ percentage points of standard deviation across random designs) and where the edge-tilt family saturates, but the extrapolated ceiling there is a fit and not a theorem, and we still do not know where the achievable bound saturates on the blind hyperplane.

— Blind directions of linear response and the limits of finite-perturbation bounds on efficiency fluctuations  (2609.37940 - Farih, 29 Sep 2026) in Section ‘Discussion’, subsection ‘What is bounded’

And our dictionary rule discretises Marzetta's continuum formulation eq:marzetta rather than solving it; the integral equation for the optimal weight function is the natural thing to race against matching pursuit, and remains untried thirty years after it was posed.

— Blind directions of linear response and the limits of finite-perturbation bounds on efficiency fluctuations  (2609.37940 - Farih, 29 Sep 2026) in Section ‘Discussion’, subsection ‘What is bounded’