Reducing Isotropy and Volume to KLS: An O(n^3ψ^2) Volume Algorithm

08/05/2020
by   He Jia, et al.
0

We show that the the volume of a convex body in ℝ^n in the general membership oracle model can be computed with O(n^3ψ^2/ε^2) oracle queries, where ψ is the KLS constant (O suppresses polylogarithmic terms. O^*suppresses dependence on error parameters as well as polylogarithmic terms.). With the current bound of ψ≲ n^1/4, this gives an O(n^3.5/ε^2) algorithm, the first general improvement on the Lovász-Vempala O(n^4/ε^2) algorithm from 2003. The main new ingredient is an O(n^3ψ^2) algorithm for isotropic transformation, following which we can apply the O(n^3/ε^2) volume algorithm of Cousins and Vempala for well-rounded convex bodies. A positive resolution of the KLS conjecture would imply an O(n^3/ϵ^2) volume algorithm. We also give an efficient implementation of the new algorithm for convex polytopes defined by m inequalities in ℝ^n: polytope volume can be estimated in time O(mn^c/ε^2) where c<3.7 depends on the current matrix multiplication exponent and improves on the the previous best bound.

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset