My talk in eRum 2020. I present the main tools that package volesti provides.

1. High-dimensional sampling and volume computation
Apostolos Chalkis
Dept. of Informatics & Telecommunications, University of Athens
GeomScale Project

2. package volesti: sampling and volume estimation
https://github.com/GeomScale/volume_approximation
open source, written in C++, R package in CRAN.
depends on Eigen, LPSolve, Boost.
currently 3 volume algorithms, 4 sampling algorithms.
main contributors: A. Chalkis and V. Fisikopoulos plus several
community contributors.

3. Problems
Given P a convex body in Rn:
compute the volume of P.
sample from a distribution π truncated in P.
minimize a convex function f in P (convex optimization).
compute the integral of a function f over P.
volesti provides randomized approximation algorithms based on
sampling from P with geometric random walks.

4. volume computation of a polytope P
First thoughts
Compute a triangulation of P and sum up the volumes of the
triangles.
Sample from a bounding box, count the number of points in P.
both approaches fail in high dimensions (curse of
dimensionality).

5. Multiphase Monte Carlo
There are several algorithms based on MMC.
The only alternative in R: package geometry (exact
computations).
An example:
#exact computation with package geometry
P = gen_rand_vpoly(20, 40)
geom_values = geometry::convhulln(P$V, options = ’FA’)
QH6082 qhull error (qh_memalloc): insufficient memory
to allocate 1329542232 bytes
#computation with volesti
time2 = system.time({ vol2 = volume(P) })
cat(time2[3], vol2)
14.524 2.68443e-07

6. Convex bodies
H-polytope : P = {x | Ax ≤ b, A ∈ Rm×n, b ∈ Rm}
V-polytope : P is the convex hull of a set of points in Rn
Z-polytope : Minkowski sum of k segments (projection of k-cube)
Spectrahedron : P = {x | A0 + x1A1 + · · · + xnAn 0},
where Ai : symmetric matrices, B 0: B is positive
semidefinite

7. volesti provide three volume algorithms
volesti is the first software that performs volume
computations in thousands of dimensions.
Algorithm
H-polytope
V- & Z-polytope
n ≤ 20 n > 20
Sequence of Balls [1]
CoolingGaussians [2]
CoolingBodies [3]
[1] I.Z. Emiris, V. Fisikopoulos, 2014
[2] S. Vempala, B. Cousins, 2014
[3] A. Chalkis, I.Z. Emiris, V. Fisikopoulos, 2019

8. volesti provides four random walks
random walk
uniform gaussian exponential
distribution distribution distribution
Ball walk
hit and run
Random directions
hit and run
Coordinate directions
Billiard walk
B
p
q
p
q
p q
p
q

9. Comparing mixing times
Uniform sampling from the hypercube [−1, 1]200 and projection
to R3.
Rows: Ball Walk, Coordinate Directions Hit and Run, Random
Directions Hit and Run, Billiard Walk.
Columns: walk length, {1, 50, 100, 150, 200}

10. Rounding a convex body
Example
Let P = [−1, 1] × [−100, 100]. We want to sample uniformly
distributed points from P.
Left we sample directly from P.
Right we round the body to obtain a higher quality sample.
volesti provides three rounding methods.

11. Application
Multivariate integration
Let I = P f (x)dx.
Sample N uniformly distributed points x1, . . . , xN from P and,
I ≈ vol(P)
1
N
N
i=1
f (xi ).
Example:
P = gen_rand_vpoly(n, 2 * n, seed = 127)
f = function(x) {sum(x^2) + (2 * x[1]^2 + x[2] + x[3])}
points = sample_points(P, random_walk =
list("walk" = "BiW", "walk_length" = 1),
n = 5000, seed = 5)
int = 0
for (i in 1:num_of_points) int = int + f(points[, i])
V = volume(P, settings = list("error" = 0.05))
I = (int * V) / 5000 })

12. Application
Multivariate integration
Comparison with exact computation: R packages geometry
(triangulation) and SimplicialCubature (integration over
each triangle).
dimension error exact time volesti time
5 0.05129854 0.41 3.095
10 0.08964482 2.945 12.01
15 0.01541695 471.479 33.256
20 – – 64.058
volesti can do general convex bodies (not only triangles).
Future work: improve accuracy with more advanced MC
integration methods such as importance sampling.

13. Application in finance
Crisis detection in stock markets
Data: 52 popular exchange traded funds (ETFs) and the US
central bank (FED)
https://stanford.edu/class/ee103/portfolio.html.
Approximation of the joint distribution (copula) between portfolio return
and volatility (risk) in a given time period.
Left, a copula that corresponds to normal period (07/03/2007 -
31/05/2007).
Right, a copula that corresponds to a crisis period
(18/12/2008 − 13/03/2009).

14. Application in finance
Crisis detection in stock markets
We define an indicator as the ratio between two masses (red /
blue).
If the indicator is ≥ 1 then the copula corresponds to a crises,
otherwise corresponds to a normal period.

15. Applications in finance
Crisis detection in stock markets
Cales, C. et. al - SoCG 2018

16. GeomScale/volesti on Google Summer of Code 2020
Mentoring organization
https://summerofcode.withgoogle.com/organizations/
5673184117915648/
3 student projects this year:
1. Sampling log-concave densities.
2. Convex optimization
3. Uniform sampling for Machine Learning.
Communication channels:
gitter.im/GeomScale/community,
geomscale-gsoc@googlegroups.com
website geomscale.github.io/

17. Thank you!