Subspace conditional Poincaré inequality for uniform product measures

Establish the subspace conditional Poincaré inequality for general finite-rank orthogonal projectors under infinite-dimensional uniform product measures, thereby converting active-subspace tail-energy bounds for affine uniform elliptic PDEs into ridge-reconstruction error bounds.

Background

The paper derives algebraic decay bounds for the population active-subspace tail energy associated with affine uniform elliptic PDEs and obtains finite-sample recovery guarantees for the empirical active subspace. To translate these gradient-projection estimates into errors for the corresponding ideal ridge approximations, the analysis requires a subspace conditional Poincaré inequality of the form ∥h−E[h∣GV]∥Lμ22≤CP(ΠV)∥(I−ΠV)∇h∥Lμ22\|h-\mathbb E[h\mid\mathcal G_V]\|_{L^2_\mu}^2\leq C_P(\Pi_V)\|(I-\Pi_V)\nabla h\|_{L^2_\mu}^2.

For Gaussian product measures, the paper proves this inequality with a dimension- and projector-independent constant. For uniform product measures, conditioning on arbitrary linear combinations couples the coordinates, so the conditional measures are uniform on affine sections of the cube rather than product measures. The paper therefore leaves the required inequality unresolved, and consequently does not obtain a ridge-reconstruction error bound for the affine uniform model.

References

Recall, however, that we cannot convert this active subspace tail energy bound into a ridge-reconstruction error bound. Establishing the required subspace conditional Poincaré inequality for uniform product measures is delicate and beyond the scope of this work; see Remark~\ref{rem:uniform-product-poincare}.

— On the sample complexity of the active subspace method  (2609.08940 - Nobile et al., 8 Sep 2026) in Section 4.2, subsection “The uniform model,” final discussion before Section 5; see also Remark 2.1, “Uniform product measures”

The reconstruction depends on the truncation $L$ of the basis, for which we have no general error bound: too small an $L$ underestimates the covariance, while increasing $L$ too much also degrades the estimate.

— Reconstructing the information processing capacity of physical systems from noisy observations  (2609.11268 - Kotoku et al., 10 Sep 2026) in Section 7, Conclusion