Reducing Isotropy and Volume to KLS: Faster Rounding and Volume Algorithms
He Jia, Aditi Laddha, Yin Tat Lee, Santosh S. Vempala
TL;DR
This work advances high-dimensional convex-volume computation by combining a fast isotropic transformation for well-rounded bodies with an outer-loop strategy to handle general convex bodies, yielding a volume algorithm that runs in $\tilde{O}(n^{3.5}\psi^{2} + n^{3}/\varepsilon^{2})$ oracle queries and $O(n^{2})$ time per query. The core innovation is an $\tilde{O}(n^{3}\psi^{2})$-cost isotropization of a well-rounded body, applied iteratively to bring general bodies into near-isotropic position, after which the well-rounded-volume routine of Cousins–Vempala can be employed efficiently. When $\psi=\tilde{O}(1)$, the resulting bound becomes $\tilde{O}(n^{3.5} + n^{3}/\varepsilon^{2})$, improving on the 2003 LV algorithm; the paper also delivers a faster implementation for polytopes, achieving a time bound $\tilde{O}(m n^{c}/\varepsilon^{2})$ with $c<3.7$, thanks to fast matrix multiplication. The significance lies in pushing toward near-cubic dependence on dimension for volume estimation, exploiting refined rounding, spectral-covariance control, and efficient sampling to enhance practical scalability in high dimensions.
Abstract
We show that the volume of a convex body in $\mathbb{R}^{n}$ in the general membership oracle model can be computed to within relative error $\varepsilon$ using $\widetilde{O}(n^{3.5}ψ^{2} + n^3/\varepsilon^{2})$ oracle queries, where $ψ$ is the KLS constant. With the current bound of $ψ=\widetilde{O}(1)$, this gives an $\widetilde{O}(n^{3.5} + n^3/\varepsilon^{2})$ algorithm, improving on the Lovász-Vempala $\widetilde{O}(n^{4}/\varepsilon^{2})$ algorithm from 2003. The main new ingredient is an $\widetilde{O}(n^{3}ψ^{2})$ algorithm for isotropic transformation of a well-rounded convex body; we apply this iteratively to isotropicize a general convex body. Following this, we can apply the $\widetilde{O}(n^{3}/\varepsilon^{2})$ volume algorithm of Cousins and Vempala for well-rounded convex bodies. We also give an efficient implementation of the new algorithm for convex polytopes defined by $m$ inequalities in $\mathbb{R}^{n}$: polytope volume can be estimated in time $\widetilde{O}(mn^{c}/\varepsilon^{2})$ where $c<3.7$ depends on the current matrix multiplication exponent and improves on the previous best bound.