4,640 research outputs found
Practical Polytope Volume Approximation
International audienceWe 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
Practical Volume Estimation by a New Annealing Schedule for Cooling Convex Bodies
We study the problem of estimating the volume of convex polytopes, focusing
on H- and V-polytopes, as well as zonotopes. Although a lot of effort is
devoted to practical algorithms for H-polytopes there is no such method for the
latter two representations. We propose a new, practical algorithm for all
representations, which is faster than existing methods. It relies on
Hit-and-Run sampling, and combines a new simulated annealing method with the
Multiphase Monte Carlo (MMC) approach. Our method introduces the following key
features to make it adaptive: (a) It defines a sequence of convex bodies in MMC
by introducing a new annealing schedule, whose length is shorter than in
previous methods with high probability, and the need of computing an enclosing
and an inscribed ball is removed; (b) It exploits statistical properties in
rejection-sampling and proposes a better empirical convergence criterion for
specifying each step; (c) For zonotopes, it may use a sequence of convex bodies
for MMC different than balls, where the chosen body adapts to the input. We
offer an open-source, optimized C++ implementation, and analyze its performance
to show that it outperforms state-of-the-art software for H-polytopes by
Cousins-Vempala (2016) and Emiris-Fisikopoulos (2018), while it undertakes
volume computations that were intractable until now, as it is the first
polynomial-time, practical method for V-polytopes and zonotopes that scales to
high dimensions (currently 100). We further focus on zonotopes, and
characterize them by their order (number of generators over dimension), because
this largely determines sampling complexity. We analyze a related application,
where we evaluate methods of zonotope approximation in engineering.Comment: 20 pages, 12 figures, 3 table
Uniform sampling of steady states in metabolic networks: heterogeneous scales and rounding
The uniform sampling of convex polytopes is an interesting computational
problem with many applications in inference from linear constraints, but the
performances of sampling algorithms can be affected by ill-conditioning. This
is the case of inferring the feasible steady states in models of metabolic
networks, since they can show heterogeneous time scales . In this work we focus
on rounding procedures based on building an ellipsoid that closely matches the
sampling space, that can be used to define an efficient hit-and-run (HR) Markov
Chain Monte Carlo. In this way the uniformity of the sampling of the convex
space of interest is rigorously guaranteed, at odds with non markovian methods.
We analyze and compare three rounding methods in order to sample the feasible
steady states of metabolic networks of three models of growing size up to
genomic scale. The first is based on principal component analysis (PCA), the
second on linear programming (LP) and finally we employ the lovasz ellipsoid
method (LEM). Our results show that a rounding procedure is mandatory for the
application of the HR in these inference problem and suggest that a combination
of LEM or LP with a subsequent PCA perform the best. We finally compare the
distributions of the HR with that of two heuristics based on the Artificially
Centered hit-and-run (ACHR), gpSampler and optGpSampler. They show a good
agreement with the results of the HR for the small network, while on genome
scale models present inconsistencies.Comment: Replacement with major revision
A probabilistic interpretation of set-membership filtering: application to polynomial systems through polytopic bounding
Set-membership estimation is usually formulated in the context of set-valued
calculus and no probabilistic calculations are necessary. In this paper, we
show that set-membership estimation can be equivalently formulated in the
probabilistic setting by employing sets of probability measures. Inference in
set-membership estimation is thus carried out by computing expectations with
respect to the updated set of probability measures P as in the probabilistic
case. In particular, it is shown that inference can be performed by solving a
particular semi-infinite linear programming problem, which is a special case of
the truncated moment problem in which only the zero-th order moment is known
(i.e., the support). By writing the dual of the above semi-infinite linear
programming problem, it is shown that, if the nonlinearities in the measurement
and process equations are polynomial and if the bounding sets for initial
state, process and measurement noises are described by polynomial inequalities,
then an approximation of this semi-infinite linear programming problem can
efficiently be obtained by using the theory of sum-of-squares polynomial
optimization. We then derive a smart greedy procedure to compute a polytopic
outer-approximation of the true membership-set, by computing the minimum-volume
polytope that outer-bounds the set that includes all the means computed with
respect to P
The inverse moment problem for convex polytopes: implementation aspects
We give a detailed technical report on the implementation of the algorithm
presented in Gravin et al. (Discrete & Computational Geometry'12) for
reconstructing an -vertex convex polytope in from the
knowledge of of its moments
Matrix representations for toric parametrizations
In this paper we show that a surface in P^3 parametrized over a 2-dimensional
toric variety T can be represented by a matrix of linear syzygies if the base
points are finite in number and form locally a complete intersection. This
constitutes a direct generalization of the corresponding result over P^2
established in [BJ03] and [BC05]. Exploiting the sparse structure of the
parametrization, we obtain significantly smaller matrices than in the
homogeneous case and the method becomes applicable to parametrizations for
which it previously failed. We also treat the important case T = P^1 x P^1 in
detail and give numerous examples.Comment: 20 page
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