Practical Polytope Volume Approximation

Abstract

We experimentally study the fundamental problem of computing the volume of a convex polytope given as an intersection of linear inequalities. We implement and evaluate practical randomized algorithms for accurately approximating the polytope's volume in high dimensions (e.g. one hundred). To carry out this efficiently we experimentally correlate the effect of parameters, such as random walk length and number of sample points, on accuracy and runtime. Moreover, we exploit the problem's geometry by implementing an iterative rounding procedure, computing partial generations of random points and designing fast polytope boundary oracles. Our publicly available code is significantly faster than exact computation and more accurate than existing approximation methods. We provide volume approximations for the Birkhoff polytopes B 11 ,. .. , B 15 , whereas exact methods have only computed that of B 10 .

On the probabilistic analysis of neural networks

“…Equation (1) reduces the problem of computing the probability of a network producing a specific decision D for a given input distribution to the quantification of the volume of polytopes. A variety of algorithms have been studied for this calculation, including both exact and approximate solutions [3,5,10]. Exact solutions.…”

“…Input spaces with higher dimensionality may challenge the scalability of our bounding method, requiring the use of statistical volume estimation methods. While state of the art statistical methods can scale on high-dimensional polytopes [5], their results are confidence intervals, which are only probabilistically correct.…”

Sampling the feasible sets of SDPs and volume approximation

“…Our approach is to consider the f i 's as a sequence of indicator functions of concentric balls centered in S, as in [20]. In particular, let f k and f 0 be the indicator functions of r B n and RB n respectively, while r B n ⊆ S ⊆ RB n and S i = (2 (k −i)/n r B n ) ∩ S for i = 0, .…”

“…The bulk of the theoretical studies are either for general convex bodies [14,18] or polytopes [33]. Practical algorithms and implementations exist only for polytopes [15,20]. Nevertheless, there are notable exceptions that consider algorithms for computing the volume of compact (basic) semi-algebraic sets.…”

Monte Carlo and Kinetic Monte Carlo Models for Deposition Processes: A Review of Recent Works

“…Monte Carlo (MC) and kinetic Monte Carlo (kMC) are widely used methods in many fields of science and engineering: From materials science and polymers properties [1], astrophysics and black holes mergers [2] to computational geometry and volume approximation [3]. Their popularity in materials science stems from their inherit ability to simulate the molecular level of materials seamlessly.…”