Develop practical algorithms for testing coarse PI structure

Develop practical algorithms for statistically testing whether a data-generating measure belongs to the class of coarse PI measures, based on empirical verification of the coarse doubling and coarse Poincaré inequalities.

Background

The paper proves that if a probability measure is coarse PI, then its empirical measure is also coarse PI with high probability. This stability result suggests a possible statistical test for the coarse PI hypothesis: one could evaluate the defining doubling and Poincaré conditions directly on observed data.

The unresolved part is the construction of practical, computationally implementable algorithms that carry out this testing procedure. The problem is included because the coarse PI condition is intended as a tractable weakening of the manifold hypothesis, particularly for data with singularities, boundaries, corners, or branching structures.

References

It is then possible to verify numerically whether $\mu_n$ is a coarse PI measure by verifying whether the conditions eq:mu_condition and eq:PI_h are satisfied: doing so would give a statistical test for the hypothesis $\mu\in PI_r$. We do not develop this here, leaving this problem (and in particular the development of practical algorithms) to further inquiry.

eq:mu_condition:

∀x∈X, ∀tr, μ(B(x,t))CDμ(B(x,t/2)).\forall x\in X,\ \forall t r,\ \mu(B(x,t)) C_D\mu(B(x,t/2)).

eq:PI_h:

∫B∣u−uB∣2dμCPIt2∫κBLipμ,r[u]2dμ.\int_B |u-u_B|^2d \mu C_{PI} t^2 \int_{\kappa B}Lip_{\mu,r}[u]^2 d \mu.

— Spectral stability of empirical metric-measure Laplacians  (2608.23150 - Divol, 24 Aug 2026) in Section 1, subsection “Coarse PI measures”