Ben Cousins and Santosh Vempala Georgia Tech
The Volume Problem •
Solution: Use Randomness! Reference Complexity Remarks • Dyer, Frieze, and Kannan First polynomial algorithm (‘89) Lovász and Simonovits (‘90) Isoperimetry, localization Applegate and Kannan (‘90) Logconcave sampling and integration Lovász (‘90) Ball walk Dyer and Frieze (‘91) Lovász and Simonovits (‘93) Kannan, Lovász, and Speedy walk, isotropy Simonovits (‘97) Lovász and Vempala (‘03) Annealing, hit-and-run Cousins and Vempala (‘15) Gaussian cooling
Compute Volume in Phases [LV03] •
Cooling with Gaussians • …
Computing the Ratios •
Metropolis Ball Walk •
KLS Conjecture •
Mixing Time of Ball Walk •
Gaussian Sampling and Isoperimetry •
Gaussian Cooling Algorithm • Even at this faster rate, variance of ratio estimator remains a constant!!!
Sketch of Analysis . . . . . . . . . . . .
•
Bounding the Variance •
Localization Lemma •
Further Theoretical Questions • Illustration of one step of hit-and-run.
MATLAB Implementation • Volume algorithm has been implemented in MATLAB, available for download from MATLAB File Exchange • “volume computation matlab” • Can compute volume of 100 dimensional polytopes in ~10 minutes.
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