Investigating how well geometric objects can be represented using bit streams
Whenever data is processed using computers, analog-to-digital (A/D) conversion, that is, representing the data using just 0s and 1s, is a crucial ingredient of the procedure. This project will mathematically study approaches to this problem for data with geometric structural constraints.
Many results in mathematical data analysis follow the paradigm of increasing efficiency by incorporating structural assumptions on the underlying data. The most prominent models today are sparsity models and manifold models in high dimensional data analysis. One main goal in these scenarios is a faithful analog to digital conversion, which not only requires a discretization, but also a digitalization step, often termed quantization, as only a finite number of bits can be processed.
A key aspect in finding suitable methods for digitally representing data is a careful balance between the resolution of this quantization step and the redundancy of the underlying discretization. Arguably the most popular class of suitable quantization schemes for highly redundant settings are so-called Sigma-Delta modulators. The underlying idea is to quantize recursively and then deduce recovery guarantees from the stability of the resulting discrete dynamical system. Variants are available for bandlimited signals, for frame representations, and for compressed sensing. For manifold models, however, there is very little quantization literature available. Project C02 aims to fill this gap, attempting a systematic study of quantization under manifold constraints.
The following three viewpoints shall be investigated:
- Functions on manifolds: This work package is devoted to quantized representations of functions whose domain is a manifold. We will study bandlimited functions and also discuss applications to digital halftoning on manifolds.
- Data lying on a manifold: We aim to study redundancies resulting from the fact that high dimensional data lies on a or close to a known manifold. We aim to incorporate quantization into the compressed sensing methodology for manifold models, as they have recently received attention in the literature.
Geometric and topological properties: Our question here is to what extent the manifold as a whole or its important geometric and topological properties can be recovered despite quantization. This project part will closely interact with project C04, which studies a related problem without quantization.