PART I: Dimensionality Reduction and Transforms

Many complex systems exhibit dominant low-dimensional patterns in the data, despite the rapidly increasing resolution of measurements and computations. Pattern extraction is related to finding coordinate transforms that simplify the system. Indeed, the rich history of mathematical physics is centered around coordinate transformations (e.g., spectral decompositions, the Fourier transform, generalized functions, etc.), although these techniques have largely been limited to simple idealized geometries and linear dynamics. The ability to derive data-driven transformations opens up opportunities to generalize these techniques to new research problems with more complex geometries and boundary conditions.

This part of the book will investigate two of the most powerful and ubiquitous algorithms for transforming and reducing data: the singular value decomposition (SVD) and the Fourier transform. The fact that data can be compressed in these transformed coordinate systems enables efficient sensing, and compact representations for modeling and control. Thus, the third chapter involves sparse sampling approaches to exploit this low-dimensional structure.

Many complex systems exhibit dominant low-dimensional patterns in the data, despite the rapidly increasing resolution of measurements and computations. Pattern extraction is related to finding coordinate transforms that simplify the system. Indeed, the rich history of mathematical physics is centered around coordinate transformations (e.g., spectral decompositions, the Fourier transform, generalized functions, etc.), although these techniques have largely been limited to simple idealized geometries and linear dynamics. The ability to derive data-driven transformations opens up opportunities to generalize these techniques to new research problems with more complex geometries and boundary conditions.

This part of the book will investigate two of the most powerful and ubiquitous algorithms for transforming and reducing data: the singular value decomposition (SVD) and the Fourier transform. The fact that data can be compressed in these transformed coordinate systems enables efficient sensing, and compact representations for modeling and control. Thus, the third chapter involves sparse sampling approaches to exploit this low-dimensional structure.