Modul: MAT870 Zurich Colloquium in Applied and Computational Mathematics

## Low rank matrix recovery from group orbits

Dr. Richard Kueng talk

**Date:** 07.11.18 **Time:** 16.15 - 17.45 **Room:** ETH HG E 1.2

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 question have appeared under different names in many areas of mathematics. In coding theory and quantum information, it is the complex Clifford group; in time-frequency analysis the oscillator group; and in mathematical physics the metaplectic group. It affords one particularly small and highly structured orbit that includes and generalizes the discrete Fourier basis: While the Fourier vectors have coefficients of constant modulus and phases that depend linearly on their index, the vectors in said orbit have phases with a quadratic dependence. Our proof methods could be adapted to cover orbits of other groups.