Improve SCPS mixing time to quadratic dimension dependence

Establish an [?]O(d^2) mixing-time bound for the Sub-Cauchy Projection Sampler, at least for multivariate Student's t distributions with fixed degrees of freedom [?]nu>2.

Background

The paper proves polynomial total-variation mixing-time upper bounds for the Sub-Cauchy Projection Sampler (SCPS) on fixed-[?]kappa heavy-tailed targets. In the heavier-tailed regime, the established bound is [?]O(d{2beta+2}) under a projection scale [?]R asymp dbeta, while curvature assumptions improve this to [?]O(d{2beta+1}).

The authors explicitly conjecture that the dependence can be reduced to quadratic in dimension for multivariate Student's t targets with fixed degrees of freedom exceeding two. This would improve the currently established dimension dependence and is left unresolved by the paper.

References

We conjecture that the dimension dependence of the SCPS mixing time can be improved to \mathcal{O}(d2), at least for multivariate Student’s t distributions with degrees of freedom fixed at \nu>2.

— Rapid Mixing of Stereographic MCMC for Heavy-Tailed Sampling  (2610.18513 - Milinanni et al., 13 Oct 2026) in Discussion and Future Directions, Section Discussion and Future Directions

Our current s-conductance analysis, however, imposes a restriction on \beta that requires \beta>\frac{1}{2}, as in \Cref{cor:SCPS_mixing_kappa_concave}. We believe this restriction is suboptimal and conjecture that the natural projection scale R\asymp\sqrt{d}, corresponding to \beta=\frac{1}{2}, is optimal. Extending the analysis to this scaling would yield the conjectured \mathcal{O}(d2) mixing-time bound. Establishing this improvement remains an important direction for future work.

— Rapid Mixing of Stereographic MCMC for Heavy-Tailed Sampling  (2610.18513 - Milinanni et al., 13 Oct 2026) in Discussion and Future Directions, Section Discussion and Future Directions