17 research outputs found
Plantinga-Vegter algorithm takes average polynomial time
We exhibit a condition-based analysis of the adaptive subdivision algorithm
due to Plantinga and Vegter. The first complexity analysis of the PV Algorithm
is due to Burr, Gao and Tsigaridas who proved a worst-case cost bound for degree plane curves with maximum
coefficient bit-size . This exponential bound, it was observed, is in
stark contrast with the good performance of the algorithm in practice. More in
line with this performance, we show that, with respect to a broad family of
measures, the expected time complexity of the PV Algorithm is bounded by
for real, degree , plane curves. We also exhibit a smoothed
analysis of the PV Algorithm that yields similar complexity estimates. To
obtain these results we combine robust probabilistic techniques coming from
geometric functional analysis with condition numbers and the continuous
amortization paradigm introduced by Burr, Krahmer and Yap. We hope this will
motivate a fruitful exchange of ideas between the different approaches to
numerical computation.Comment: 8 pages, correction of typo
Condition Numbers for the Cube. I: Univariate Polynomials and Hypersurfaces
The condition-based complexity analysis framework is one of the gems of
modern numerical algebraic geometry and theoretical computer science. One of
the challenges that it poses is to expand the currently limited range of random
polynomials that we can handle. Despite important recent progress, the
available tools cannot handle random sparse polynomials and Gaussian
polynomials, that is polynomials whose coefficients are i.i.d. Gaussian random
variables.
We initiate a condition-based complexity framework based on the norm of the
cube that is a step in this direction. We present this framework for real
hypersurfaces and univariate polynomials. We demonstrate its capabilities in
two problems, under very mild probabilistic assumptions. On the one hand, we
show that the average run-time of the Plantinga-Vegter algorithm is polynomial
in the degree for random sparse (alas a restricted sparseness structure)
polynomials and random Gaussian polynomials. On the other hand, we study the
size of the subdivision tree for Descartes' solver and run-time of the solver
by Jindal and Sagraloff (arXiv:1704.06979). In both cases, we provide a bound
that is polynomial in the size of the input (size of the support plus logarithm
of the degree) for not only on the average, but all higher moments.Comment: 34 pages. Version 1, conference version; from version 2, journal
versio
Condition Numbers for the Cube. I: Univariate Polynomials and Hypersurfaces
The condition-based complexity analysis framework is one of the gems of modern numerical algebraic geometry and theoretical computer science. One of the challenges that it poses is to expand the currently limited range of random polynomials that we can handle. Despite important recent progress, the available tools cannot handle random sparse polynomials and Gaussian polynomials, that is polynomials whose coefficients are i.i.d. Gaussian random variables.We initiate a condition-based complexity framework based on the norm of the cube that is a step in this direction. We present this framework for real hypersurfaces and univariate polynomials. We demonstrate its capabilities in two problems, under very mild probabilistic assumptions. On the one hand, we show that the average run-time of the Plantinga-Vegter algorithm is polynomial in the degree for random sparse (alas a restricted sparseness structure) polynomials and random Gaussian polynomials. On the other hand, we study the size of the subdivision tree for Descartes' solver and run-time of the solver by Jindal and Sagraloff (2017). In both cases, we provide a bound that is polynomial in the size of the input (size of the support plus logarithm of the degree) for not only on the average, but all higher moments.[This is the journal version of the conference paper with the same title.
Condition Numbers for the Cube. I: Univariate Polynomials and Hypersurfaces
International audienceThe condition-based complexity analysis framework is one of the gems of modern numerical algebraic geometry and theoretical computer science. One of the challenges that it poses is to expand the currently limited range of random polynomials that we can handle. Despite important recent progress, the available tools cannot handle random sparse polynomials and Gaussian polynomials, that is polynomials whose coefficients are i.i.d. Gaussian random variables. We initiate a condition-based complexity framework based on the norm of the cube, that is a step in this direction. We present this framework for real hypersurfaces. We demonstrate its capabilities by providing a new probabilistic complexity analysis for the Plantinga-Vegter algorithm, which covers both random sparse (alas a restricted sparseness structure) polynomials and random Gaussian polynomials. We present explicit results with structured random polynomials for problems with two or more dimensions. Additionally, we provide some estimates of the separation bound of a univariate polynomial in our current framework
Certified Approximation Algorithms for the Fermat Point and n-Ellipses
Given a set A of n points in ?^d with weight function w: A??_{> 0}, the Fermat distance function is ?(x): = ?_{a?A}w(a)?x-a?. A classic problem in facility location dating back to 1643, is to find the Fermat point x*, the point that minimizes the function ?. We consider the problem of computing a point x?* that is an ?-approximation of x* in the sense that ?x?*-x*? ?(x*) and d = 2. Finally, all our planar (d = 2) algorithms are implemented in order to experimentally evaluate them, using both synthetic as well as real world datasets. These experiments show the practicality of our techniques
Bivariate systems and topology of plane curves: algebraic and numerical methods
The work presented in this thesis belongs to the domain of non-linear computational geometry in lowdimension. More precisely it focuses on solving bivariate systems and computing the topology of curvesin the plane. When the input is given by polynomials, the natural tools come from computer algebra.Our contributions are algorithms proven efficient in a deterministic or a Las Vegas settings together witha practical efficient software for topology certified drawing of a plane algebraic curve. When the input isnot restricted to be polynomials but given by interval functions, we design algorithms based on certifiednumerical approches using subdivision and interval arithmetic. The input is then required to fulfill somegeneric assumptions and our algorithms are certified in the sense that they terminate if and only if theassumptions are satisfied.Le travail prĂ©sentĂ© dans cette thĂšse appartient au domaine de la gĂ©omĂ©trie computationnelle non linĂ©aireen petite dimension. Plus prĂ©cisĂ©ment, il se concentre sur la rĂ©solution de systĂšmes bivariĂ©s et le calcul dela topologie des courbes dans le plan. Lorsque lâentrĂ©e est donnĂ©e par des polynĂŽmes, les outils naturelsproviennent du calcul formel. Nos contributions sont des algorithmes dont lâefficacitĂ© a Ă©tĂ© prouvĂ©e dansun cadre dĂ©terministe ou Las Vegas, ainsi quâun logiciel efficace pour le dessin certifiĂ© de la topologiedâune courbe algĂ©brique plane. Lorsque les donnĂ©es dâentrĂ©e ne sont pas limitĂ©es aux polynĂŽmes maissont donnĂ©es par des fonctions dâintervalles, nous concevons des algorithmes basĂ©s sur des approchesnumĂ©riques certifiĂ©es utilisant la subdivision et lâarithmĂ©tique dâintervalles. LâentrĂ©e doit alors satisfairecertaines hypothĂšses gĂ©nĂ©riques et nos algorithmes sont certifiĂ©s dans le sens oĂč ils se terminent si etseulement si les hypothĂšses sont satisfaites
On the number of iterations of the DBA algorithm
The DTW Barycenter Averaging (DBA) algorithm is a widely used algorithm for
estimating the mean of a given set of point sequences. In this context, the
mean is defined as a point sequence that minimises the sum of dynamic time
warping distances (DTW). The algorithm is similar to the -means algorithm in
the sense that it alternately repeats two steps: (1) computing an optimal
assignment to the points of the current mean, and (2) computing an optimal mean
under the current assignment. The popularity of DBA can be attributed to the
fact that it works well in practice, despite any theoretical guarantees to be
known. In our paper, we aim to initiate a theoretical study of the number of
iterations that DBA performs until convergence. We assume the algorithm is
given sequences of points in and a parameter that
specifies the length of the mean sequence to be computed. We show that, in
contrast to its fast running time in practice, the number of iterations can be
exponential in in the worst case - even if the number of input sequences is
. We complement these findings with experiments on real-world data that
suggest this worst-case behaviour is likely degenerate. To better understand
the performance of the algorithm on non-degenerate input, we study DBA in the
model of smoothed analysis, upper-bounding the expected number of iterations in
the worst case under random perturbations of the input. Our smoothed upper
bound is polynomial in , and , and for constant , it is also
polynomial in . For our analysis, we adapt the set of techniques that were
developed for analysing -means and observe that this set of techniques is
not sufficient to obtain tight bounds for general