The Degree of SO ( n , ℂ ) $$\mathop{\mathrm{SO}}\nolimits (n, \mathbb{C})$$

Brandt, Madeline; Bruce, Juliette; Brysiewicz, Taylor; Krone, Robert; Robeva, Elina

doi:10.1007/978-1-4939-7486-3_11

Cited by 6 publications

(1 citation statement)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The degree of V is by definition the number of intersection points of the variety with dim V general hyperplanes, and we have that deg SO(2) = 2, deg SO(3) = 8 (see [7] for a derivation) and deg Q = 2.…”

Section: The Varieties Of Rotationsmentioning

confidence: 99%

On the Tightness of Semidefinite Relaxations for Rotation Estimation

et al. 2021

View full text Add to dashboard Cite

Why is it that semidefinite relaxations have been so successful in numerous applications in computer vision and robotics for solving non-convex optimization problems involving rotations? In studying the empirical performance, we note that there are few failure cases reported in the literature, in particular for estimation problems with a single rotation, motivating us to gain further theoretical understanding. A general framework based on tools from algebraic geometry is introduced for analyzing the power of semidefinite relaxations of problems with quadratic objective functions and rotational constraints. Applications include registration, hand–eye calibration, and rotation averaging. We characterize the extreme points and show that there exist failure cases for which the relaxation is not tight, even in the case of a single rotation. We also show that some problem classes are always tight given an appropriate parametrization. Our theoretical findings are accompanied with numerical simulations, providing further evidence and understanding of the results.

show abstract

Section: The Varieties Of Rotationsmentioning

confidence: 99%

On the Tightness of Semidefinite Relaxations for Rotation Estimation

et al. 2021

View full text Add to dashboard Cite

show abstract

Fitness, Apprenticeship, and Polynomials

Sturmfels

2017

Fields Institute Communications

View full text Add to dashboard Cite

This article discusses the design of the Apprenticeship Program at the Fields Institute, held 21 August-3 September 2016. Six themes from combinatorial algebraic geometry were selected for the two weeks: curves, surfaces, Grassmannians, convexity, abelian combinatorics, parameters and moduli. The activities were structured into fitness, research and scholarship. Combinatorics and concrete computations with polynomials (and theta functions) empowers young scholars in algebraic geometry, and it helps them to connect with the historic roots of their field. We illustrate our perspective for the threefold obtained by blowing up six points in P 3 .

show abstract

Coordinate-wise powers of algebraic varieties

2020

View full text Add to dashboard Cite

We introduce and study coordinate-wise powers of subvarieties of P n , i.e. varieties arising from raising all points in a given subvariety of P n to the r-th power, coordinate by coordinate. This corresponds to studying the image of a subvariety of P n under the quotient of P n by the action of the finite group Z n+1 r . We determine the degree of coordinate-wise powers and study their defining equations, in particular for hypersurfaces and linear spaces. Applying these results, we compute the degree of the variety of orthostochastic matrices and determine iterated dual and reciprocal varieties of power sum hypersurfaces. We also establish a link between coordinate-wise squares of linear spaces and the study of real symmetric matrices with a degenerate eigenspectrum.

show abstract

The Degree of SO ( n , ℂ ) $$\mathop{\mathrm{SO}}\nolimits (n, \mathbb{C})$$

Cited by 6 publications

References 9 publications

On the Tightness of Semidefinite Relaxations for Rotation Estimation

On the Tightness of Semidefinite Relaxations for Rotation Estimation

Fitness, Apprenticeship, and Polynomials

Coordinate-wise powers of algebraic varieties

Contact Info

Product

Resources

About