Simple varieties for limited precision points

Abstract

We present a symbolic-numeric approach for the analysis of a given set of noisy data, represented as a finite set X of limited precision points. Starting from X and a permitted tolerance ε on its coordinates, our method automatically determines a low degree monic polynomial whose associated variety passes close to each point of X by less than the given tolerance ε.Proof. Letᾱ j = α j (ē) and f = t − t j ∈Oᾱ j t j . We observe that f i (e i ) = t(p i (e i )) + t j ∈Oᾱ j t j (p i (e i )) is a polynomial function of e i and its Jacobian J f i (e i ) ∈ Mat 1×n (R) is a Lipschitz function inB(ē i , r i ). We prove that

Gradient Boosts the Approximate Vanishing Ideal

“…Both of these works use the gradient for purposes that are totally different from ours. The closest work to ours is (Fassino and Torrente 2013), which proposes an algorithm to compute an approximate vanishing polynomial of low degree based on the geometrical distance using the gradient. However, their algorithm does not compute a basis set but only provide a single approximate vanishing polynomial.…”

An Euclidean norm based criterion to assess robots’ 2D path-following performance

“…An interesting class of recently developed algorithms relies on tools from Numerical Commutative Algebra [17,2,10,6,7]. For all these algorithms the input is a set of points possibly in n-dimensions and the output is a polynomial f in n-variables whose zero locus (which is a curve, or a surface, or more generally an algebraic variety) gives an approximation for the input points and can be interpreted as an implicit polynomial regression model [12,Ch 2].…”

“…For all these algorithms the input is a set of points possibly in n-dimensions and the output is a polynomial f in n-variables whose zero locus (which is a curve, or a surface, or more generally an algebraic variety) gives an approximation for the input points and can be interpreted as an implicit polynomial regression model [12,Ch 2]. The algorithm presented in [7] called Low-degree Polynomial Algorithm (LPA) is particularly interesting for our purposes because it returns a "simple" polynomial f whose zero locus "almost" contains the points. The "simplicity" of a polynomial f is measured by its total degree whereas a point is said to be "almost" contained in the zero locus of f if the ε 1 -ball (w.r.t.…”

“…In the LPA the ∞-norm is used, and ε 1 represents the maximum error of the coordinates of a point. For further details on the algebraic perspective see [7].…”

Almost vanishing polynomials and an application to the Hough transform

“…In the rest of the section we discuss some illustrative examples, in which our approach is effectively used to compute the accumulator function, which is the core of the recognition algorithm based on the Hough transform. In examples 6.3 and 6.4 below, our outputs are compared with the results obtained by using well-established pattern recognition techniques for the detection of curves in images (see [1,Sections 6,7] and also [9,Sections 4,5]). Our aim is simply to show that our approach may be successfully used, in a complementary way, in this context too.…”