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4 results for Orbitopes

We study polar orbitopes, i.e. convex hulls of orbits of a polar representation of a compact Lie group. The face structure is studied by means of the gradient momentum map and it is shown that every face is exposed and is again a polar orbitope. Up to conjugation the faces are completely determined by the momentum poly…

2012-06-25abs ↗pdf ↗

We develop methods to learn dictionaries invariant under group symmetries, useful in cryo-EM and tracking.

problem Learning dictionaries invariant under group symmetries.
method Representation theory, non-abelian Fourier analysis, matrix orbitopes, alternating minimization.
result Effective dictionary learning for SO(3) symmetries with guarantees.

The study connects circle actions, trigonometric polynomials, and thickenings of metric spaces.

problem Understanding the topology and connectivity of thickenings of metric spaces.
method Combining circle actions, trigonometric polynomials, and geometric proofs.
result Optimal bounds on thickenings of the sphere and circle, proving zero points of trigonometric polynomials.

Geometrically describes Satake compactifications without root data.

problem Understanding Satake-Furstenberg compactifications and their properties.
method Analyzes the facial structure of polar orbitopes and constructs maps between compactifications.
result Constructs a map between Satake compactifications and polar orbitopes, proving surjectivity for a large class of measures.