Ludwig-Maximilians-Universität München / Mathematisches Institut der Universität München
In finite dimensional numerical linear algebra, randomized methods are a prominent approach for dealing with high dimensional data. Since the approximation properties in these methods do not depend on the dimension, it is natural to extend the randomization philosophy to infinite dimensional spaces. However, deriving fundamental properties in general Hilbert spaces is more theoretically involved. In this talk I will present a universal analysis of randomized discretizations of generic continuous frames.
We will consider signal processing tasks in which the signal is mapped via a continuous frame to a higher dimensional space, called phase space, processed there, and synthesized to an output signal. The stochastic discretization speeds up computations with respect to regular discretizations, since the number of samples required for a certain accuracy is proportional to the resolution of the discrete signal, and not to the dimension of phase space, which is typically higher. As an example application, I will show how to derive a high dimensional phase vocoder with the complexity of a low dimensional method.