Topological features in signal processing using frame theory and persistent homology
Mijail Guillemard · The Journal of the Acoustical Society of America · 2017
We present some interactions between frame theory and persistent homology as a new way to construct classification mechanisms in signal processing. On the one hand, frame theory generalizes basic ideas from time-frequency analysis including aspects of short term Fourier transformations and wavelet theory. On the other hand, persistent homology provides new algorithms applying concepts from algebraic topology to data analysis. The question of finding adequate sparse representations of data can be seen from several points of view, including dimensionality reduction and modern developments in neural networks. Persistent homology, as a topic in topological data analysis, presents alternative mechanisms for finding adequate sparse representations of data. We explain some interactions between these tools with applications to the analysis of acoustic signals.