28 research outputs found
Data-Discriminants of Likelihood Equations
Maximum likelihood estimation (MLE) is a fundamental computational problem in
statistics. The problem is to maximize the likelihood function with respect to
given data on a statistical model. An algebraic approach to this problem is to
solve a very structured parameterized polynomial system called likelihood
equations. For general choices of data, the number of complex solutions to the
likelihood equations is finite and called the ML-degree of the model. The only
solutions to the likelihood equations that are statistically meaningful are the
real/positive solutions. However, the number of real/positive solutions is not
characterized by the ML-degree. We use discriminants to classify data according
to the number of real/positive solutions of the likelihood equations. We call
these discriminants data-discriminants (DD). We develop a probabilistic
algorithm for computing DDs. Experimental results show that, for the benchmarks
we have tried, the probabilistic algorithm is more efficient than the standard
elimination algorithm. Based on the computational results, we discuss the real
root classification problem for the 3 by 3 symmetric matrix~model.Comment: 2 table
Computing the Real Isolated Points of an Algebraic Hypersurface
Let be the field of real numbers. We consider the problem of
computing the real isolated points of a real algebraic set in
given as the vanishing set of a polynomial system. This problem plays an
important role for studying rigidity properties of mechanism in material
designs. In this paper, we design an algorithm which solves this problem. It is
based on the computations of critical points as well as roadmaps for answering
connectivity queries in real algebraic sets. This leads to a probabilistic
algorithm of complexity for computing the real isolated
points of real algebraic hypersurfaces of degree . It allows us to solve in
practice instances which are out of reach of the state-of-the-art.Comment: Conference paper ISSAC 202
Extended abstract for: Solving Rupertâs problem algorithmically
International audienc
A Direttissimo Algorithm for Equidimensional Decomposition
We describe a recursive algorithm that decomposes an algebraic set into
locally closed equidimensional sets, i.e. sets which each have irreducible
components of the same dimension. At the core of this algorithm, we combine
ideas from the theory of triangular sets, a.k.a. regular chains, with Gr\"obner
bases to encode and work with locally closed algebraic sets. Equipped with
this, our algorithm avoids projections of the algebraic sets that are
decomposed and certain genericity assumptions frequently made when decomposing
polynomial systems, such as assumptions about Noether position. This makes it
produce fine decompositions on more structured systems where ensuring
genericity assumptions often destroys the structure of the system at hand.
Practical experiments demonstrate its efficiency compared to state-of-the-art
implementations
VariĂ©tĂ©s bipolaires et rĂ©solution dâune Ă©quation polynomiale rĂ©elle
In previous work we designed an efficient procedure that finds an algebraic sample point for each connected component of a smooth real complete intersection variety. This procedure exploits geometric properties of generic polar varieties and its complexity is intrinsic with respect to the problem. In the present paper we introduce a natural construction that allows to tackle the case of a nonâsmooth real hypersurface by means of a reduction to a smooth complete intersection.Nous avons dĂ©crit prĂ©cĂ©demment un algorithme efficace qui exhibe un point reprĂ©sentatif (algĂ©brique) par composante connexe dâune intersection complĂšte rĂ©elle lisse. Ce processus est basĂ© sur lâexploitation des propriĂ©tĂ©s gĂ©omĂ©triques des variĂ©tĂ©s polaires gĂ©nĂ©riques et sa complexitĂ© est intrinsĂšque au problĂšme. Nous introduisons ici une construction naturelle nous permettant de traiter le cas dâune hypersurface singuliĂšre par rĂ©duction Ă une situation intersection complĂšte lisse
Workspace, Joint space and Singularities of a family of Delta-Like Robot
International audienceThis paper presents the workspace, the joint space and the singularities of a family of delta-like parallel robots by using algebraic tools. The different functions of SIROPA library are introduced, which is used to induce an estimation about the complexity in representing the singularities in the workspace and the joint space. A Groebner based elimination is used to compute the singularities of the manipulator and a Cylindrical Algebraic Decomposition algorithm is used to study the workspace and the joint space. From these algebraic objects, we propose some certified three-dimensional plotting describing the shape of workspace and of the joint space which will help the engineers or researchers to decide the most suited configuration of the manipulator they should use for a given task. Also, the different parameters associated with the complexity of the serial and parallel singularities are tabulated, which further enhance the selection of the different configuration of the manipulator by comparing the complexity of the singularity equations
On the complexity of computing real radicals of polynomial systems
International audienceLet f= (f1, ..., fs) be a sequence of polynomials in Q[X1,...,Xn] of maximal degree D and Vâ Cn be the algebraic set defined by f and r be its dimension. The real radical re associated to f is the largest ideal which defines the real trace of V . When V is smooth, we show that re , has a finite set of generators with degrees bounded by V. Moreover, we present a probabilistic algorithm of complexity (snDn )O(1) to compute the minimal primes of re . When V is not smooth, we give a probabilistic algorithm of complexity sO(1) (nD)O(nr2r) to compute rational parametrizations for all irreducible components of the real algebraic set V â© Rn. Experiments are given to show the efficiency of our approaches