4,313 research outputs found
Almost-Uniform Sampling of Points on High-Dimensional Algebraic Varieties
We consider the problem of uniform sampling of points on an algebraic
variety. Specifically, we develop a randomized algorithm that, given a small
set of multivariate polynomials over a sufficiently large finite field,
produces a common zero of the polynomials almost uniformly at random. The
statistical distance between the output distribution of the algorithm and the
uniform distribution on the set of common zeros is polynomially small in the
field size, and the running time of the algorithm is polynomial in the
description of the polynomials and their degrees provided that the number of
the polynomials is a constant
Learning Algebraic Varieties from Samples
We seek to determine a real algebraic variety from a fixed finite subset of
points. Existing methods are studied and new methods are developed. Our focus
lies on aspects of topology and algebraic geometry, such as dimension and
defining polynomials. All algorithms are tested on a range of datasets and made
available in a Julia package
Smooth Parametrizations in Dynamics, Analysis, Diophantine and Computational Geometry
Smooth parametrization consists in a subdivision of the mathematical objects
under consideration into simple pieces, and then parametric representation of
each piece, while keeping control of high order derivatives. The main goal of
the present paper is to provide a short overview of some results and open
problems on smooth parametrization and its applications in several apparently
rather separated domains: Smooth Dynamics, Diophantine Geometry, Approximation
Theory, and Computational Geometry.
The structure of the results, open problems, and conjectures in each of these
domains shows in many cases a remarkable similarity, which we try to stress.
Sometimes this similarity can be easily explained, sometimes the reasons remain
somewhat obscure, and it motivates some natural questions discussed in the
paper. We present also some new results, stressing interconnection between
various types and various applications of smooth parametrization
Likelihood Geometry
We study the critical points of monomial functions over an algebraic subset
of the probability simplex. The number of critical points on the Zariski
closure is a topological invariant of that embedded projective variety, known
as its maximum likelihood degree. We present an introduction to this theory and
its statistical motivations. Many favorite objects from combinatorial algebraic
geometry are featured: toric varieties, A-discriminants, hyperplane
arrangements, Grassmannians, and determinantal varieties. Several new results
are included, especially on the likelihood correspondence and its bidegree.
These notes were written for the second author's lectures at the CIME-CIRM
summer course on Combinatorial Algebraic Geometry at Levico Terme in June 2013.Comment: 45 pages; minor changes and addition
A PSPACE Construction of a Hitting Set for the Closure of Small Algebraic Circuits
In this paper we study the complexity of constructing a hitting set for the
closure of VP, the class of polynomials that can be infinitesimally
approximated by polynomials that are computed by polynomial sized algebraic
circuits, over the real or complex numbers. Specifically, we show that there is
a PSPACE algorithm that given n,s,r in unary outputs a set of n-tuples over the
rationals of size poly(n,s,r), with poly(n,s,r) bit complexity, that hits all
n-variate polynomials of degree-r that are the limit of size-s algebraic
circuits. Previously it was known that a random set of this size is a hitting
set, but a construction that is certified to work was only known in EXPSPACE
(or EXPH assuming the generalized Riemann hypothesis). As a corollary we get
that a host of other algebraic problems such as Noether Normalization Lemma,
can also be solved in PSPACE deterministically, where earlier only randomized
algorithms and EXPSPACE algorithms (or EXPH assuming the generalized Riemann
hypothesis) were known.
The proof relies on the new notion of a robust hitting set which is a set of
inputs such that any nonzero polynomial that can be computed by a polynomial
size algebraic circuit, evaluates to a not too small value on at least one
element of the set. Proving the existence of such a robust hitting set is the
main technical difficulty in the proof.
Our proof uses anti-concentration results for polynomials, basic tools from
algebraic geometry and the existential theory of the reals
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