Crystallization of Random Matrix Orbits

Random Homoclinic Orbits

We introduce random homoclinic points and orbits for random dynamical systems with hyperbolic stationary orbits and investigate their meaning for irregular behaviour in form of a stochastic version of the Birkhoo-Smale Theorem. In particular we show that transversal random homoclinic points cause a conjugacy of the random dynamical system in a part of the phase space to a random shift system.

Quantum Orbits of R-matrix Type

Given a simple Lie algebra g, we consider the orbits in g * which are of R-matrix type, i.e., which possess a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the so-called R-matrix bracket. We call an algebra quantizing the latter bracket a quantum orbit of R-matrix type. We describe some orbits of this type explicitly and we construct a quantization of the whole Poisson pe...

\random" Random Matrix Products

Low rank matrix recovery from Clifford orbits

We prove that low-rank matrices can be recovered efficiently from a small number of measurements that are sampled from orbits of a certain matrix group. As a special case, our theory makes statements about the phase retrieval problem. Here, the task is to recover a vector given only the amplitudes of its inner product with a small number of vectors from an orbit. Variants of the group in questi...

Matrix multiplication algorithms from group orbits

We show how to construct highly symmetric algorithms for matrix multiplication. In particular, we consider algorithms which decompose the matrix multiplication tensor into a sum of rank-1 tensors, where the decomposition itself consists of orbits under some finite group action. We show how to use the representation theory of the corresponding group to derive simple constraints on the decomposit...