Talk by Daniel Kressner (EPFL) скачать в хорошем качестве

Talk by Daniel Kressner (EPFL)

Randomized joint diagonalization By a basic linear algebra result, a family of two or more commuting symmetric matrices has a common eigenvector basis and can thus be jointly diagonalized. Perhaps surprisingly, the development of a robust numerical algorithm for effecting such a joint diagonalization is by no means trivial. To start with, roundoff error or other forms of error will inevitably destroy the commutativity assumption. In turn, one can at best hope to find an orthogonal matrix that nearly diagonalizes every matrix. Most existing methods use optimization techniques to tackle this problem. In this talk, we propose a novel randomized method that addresses the joint diagonalization problem via the solution of one or a few standard eigenvalue problems. Unlike existing optimization-based methods, our algorithm is trivial to implement and leverages existing high-quality linear algebra software packages. We analyze its robustness and prove that the backward error of the computed orthogonal joint diagonalizer will be asymptotically small with high probability. The algorithm can be further improved by deflation techniques. Through numerical experiments on synthetic and real-world data, we show that our algorithm reaches state-of-the-art performance. This talk is based on joint work with Haoze He.