Reducing Isotropy and Volume to KLS: Faster Rounding and Volume Algorithms

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 $\epsilon$ using $\tilde{O}(n^{3.5}{\psi }^2+n^3\mathrm{/}{\epsilon }^2)$ oracle queries, where $\psi$ is the KLS constant. With the current bound of $\psi =\tilde{O}(1)$, this gives an $\widetilde{O}(n^{3.5} + n^3/\varepsilon^{2})$algorithm,improvingontheLov\acute{a}sz-Vempala$\widetilde{O}(n^{4}/\varepsilon^{2})$ algorithm from 2003. The main new ingredient is an $\tilde{O}(n^3{\psi }^2)$ 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 $\tilde{O}(n^3\mathrm{/}{\epsilon }^2)$ 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 $\tilde{O}(m{n}^c\mathrm{/}{\epsilon }^2)$ where $c<3.7$ depends on the current matrix multiplication exponent and improves on the previous best bound.