arXiv subjects

Evelyne Hubert

Publications and source records attributed to Evelyne Hubert.

8 recordsLinked to original sources

arXiv2023

Optimization of trigonometric polynomials with crystallographic symmetry and spectral bounds for set avoiding graphs

Trigonometric polynomials are usually defined on the lattice of integers.We consider the larger class of weight and root lattices with crystallographic symmetry.This article gives a new approach to minimize trigonometric polynomials, which are invariant under the associated reflection group.The invariance assumption allows us to rewrite the objective function in terms of generalized Chebyshev polynomials. The new objective function is defined on a compact basic semi-algebraic set, so that we can benefit from the rich theory of polynomial optimization.We present an algorithm to compute the minimum: Based on the Hol-Scherer Positivstellensatz, we impose matrix-sums of squares conditions on the objective function in the Chebyshev basis.The degree of the sums of squares is weighted, defined by the root system. Increasing the degree yields a converging Lasserre-type hierarchy of lower bounds.This builds a bridge between trigonometric and polynomial optimization, allowing us to compare with existing techniques.The chromatic number of a set avoiding graph in the Euclidean space is defined through an optimal coloring.It can be computed via a spectral bound by minimizing a trigonometric polynomial. If the to be avoided set has crystallographic symmetry, our method has a natural application.Specifically, we compute spectral bounds for the first time for boundaries of symmetric polytopes.For several cases, the problem has such a simplified form that we can give analytical proofs for sharp spectral bounds.In other cases, we certify the sharpness numerically.

math.AG↗

arXiv2022

Orbit spaces of Weyl groups acting on compact tori: a unified and explicit polynomial description

The Weyl group of a crystallographic root system has a nonlinear action on the compact torus. The orbit space of this action is a compact basic semi-algebraic set. We present a polynomial description of this set for the Weyl groups of type A, B, C, D and G. Our description is given through a polynomial matrix inequality. The novelty lies in an approach via Hermite quadratic forms and a closed formula for the matrix entries. The orbit space of the nonlinear Weyl group action is the orthogonality region of generalized Chebyshev polynomials. In this polynomial basis, we show that the matrices obtained for the five types follow the same, surprisingly simple pattern. This is applied to the optimization of trigonometric polynomials with crystallographic symmetries.

math.AG↗

arXiv2020

Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials

Sparse interpolation} refers to the exact recovery of a function as a short linear combination of basis functions from a limited number of evaluations. For multivariate functions, the case of the monomial basis is well studied, as is now the basis of exponential functions. Beyond the multivariate Chebyshev polynomial obtained as tensor products of univariate Chebyshev polynomials, the theory of root systems allows to define a variety of generalized multivariate Chebyshev polynomials that have connections to topics such as Fourier analysis and representations of Lie algebras. We present a deterministic algorithm to recover a function that is the linear combination of at most r such polynomials from the knowledge of r and an explicitly bounded number of evaluations of this function.

cs.SC↗

arXiv2018

Rational invariants of even ternary forms under the orthogonal group

In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group $\mathrm{O}_3$ on the space $\mathbb{R}[x,y,z]_{2d}$ of ternary forms of even degree $2d$. The construction relies on two key ingredients: On one hand, the Slice Lemma allows us to reduce the problem to dermining the invariants for the action on a subspace of the finite subgroup $\mathrm{B}_3$ of signed permutations. On the other hand, our construction relies in a fundamental way on specific bases of harmonic polynomials. These bases provide maps with prescribed $\mathrm{B}_3$-equivariance properties. Our explicit construction of these bases should be relevant well beyond the scope of this paper. The expression of the $\mathrm{B}_3$-invariants can then be given in a compact form as the composition of two equivariant maps. Instead of providing (cumbersome) explicit expressions for the $\mathrm{O}_3$-invariants, we provide efficient algorithms for their evaluation and rewriting. We also use the constructed $\mathrm{B}_3$-invariants to determine the $\mathrm{O}_3$-orbit locus and provide an algorithm for the inverse problem of finding an element in $\mathbb{R}[x,y,z]_{2d}$ with prescribed values for its invariants. These are the computational issues relevant in brain imaging.

cs.SC↗

arXiv2013

Lagrangian Curves in Affine Symplectic 4-space

Lagrangian curves in 4-space entertain intriguing relationships with second order deformation of plane curves under the special affine group and null curves in a 3-dimensional Lorentzian space form. We provide a natural affine symplectic frame for Lagrangian curves. It allows us to classify Lagrangrian curves with constant symplectic curvatures, to construct a class of Lagrangian tori and determine Lagrangian geodesics.

math.SG↗

arXiv2008

Differential invariants of a Lie group action: syzygies on a generating set

Given a group action, known by its infinitesimal generators, we exhibit a complete set of syzygies on a generating set of differential invariants. For that we elaborate on the reinterpretation of Cartan's moving frame by Fels and Olver (1999). This provides constructive tools for exploring algebras of differential invariants.

cs.SC↗

arXiv2007

Differential Invariants of Conformal and Projective Surfaces

We show that, for both the conformal and projective groups, all the differential invariants of a generic surface in three-dimensional space can be written as combinations of the invariant derivatives of a single differential invariant. The proof is based on the equivariant method of moving frames.

math.DG↗

arXiv2005

Rational, Replacement, and Local Invariants of a Group Action

The paper presents a new algorithmic construction of a finite generating set of rational invariants for the rational action of an algebraic group on the affine space. The construction provides an algebraic counterpart of the moving frame method in differential geometry. The generating set of rational invariants appears as the coefficients of a Groebner basis, reduction with respect to which allows to express a rational invariant in terms of the generators. The replacement invariants, introduced in the paper, are tuples of algebraic functions of the rational invariants. Any invariant, whether rational, algebraic or local, can be can be rewritten terms of replacement invariants by a simple substitution.

math.AC↗