During the last few years recent developments in differential
geometry and algebraic topology have provided powerful tools for
the analysis of high-dimensional datasets. In particular, recent
methods for nonlinear dimensionality reduction were inspired by
fundamental concepts in differential geometry. In parallel developments,
applied topology has delivered new methods for computing homological
information of a point cloud data. In this context, an important task
is to understand the interaction of these novel tools with well-established
signal analysis methods such as wavelets or Fourier transforms.
In this talk, we first review relevant concepts before presenting a
general framework for signal analysis based on geometrical tools.
We also address some applications in signal separation and classification.
The talk is based on ongoing joint work with Armin Iske.