Sparsity of a mathematical object refers to its representability in terms of a low number of degrees of freedom. This concept is commonly used in vectors or matrices with few nonzero entries or sparsely connected graphs, but also appears more generally in approximations of functions by selection of few elements from a given dictionary, in locally refined grids, or in matrices and tensors of low rank. Such low-dimensional parameterizations of objects in high-dimensional spaces play a central role not only in many problems of signal processing and machine learning, but also in partial differential equations (PDEs). These problems and their solutions commonly exhibit singular structures, that is, localized features that represent a concentration of information, such as singularities, shock fronts, or sharp material interfaces. Structural features of this kind are closely connected to notions of sparsity, and their parsimonious description is often the key both to understanding the properties of PDEs and to their efficient numerical solution.

These notions of sparsity and singular structures, and how they interact in a wide variety of topics, are at the core of our activities in CRC 1481. In signal and image processing, compressive sensing is a well-known approach for identifying sparse structures. The extraction of sparse features is also an important tool in machine learning, where the mathematical understanding of the performance of neural networks is closely tied to questions of optimization methods and gradient flows resulting from different choices of metrics. Certain concepts of sparsity are of interest also in the context of challenging nonlinear and singular PDE problems, such as the study of nucleation of magnetic skyrmions, vortex tracking in the Gross-Pitaevskii equation, as well as in inverse problems regularized with singular energies and in kinetic equations. Both low-rank approximations and neural networks are used for PDEs with many parameters, but at the same time are used to recover high-dimensional functions. In such recovery tasks, also symmetry properties and algebraic structures come into play.

The CRC aims to forge connections between areas that lead to new insights from the transfer of mathematical methods. The center combines the expertise of mathematicians in various different fields such as analysis, probability theory, numerical analysis, machine learning, optimization, and algebra.