Prerequisites: Fourier transform, linear algebra, neural networks, numerical analysis, and familiarity with Python and/or PyTorch.
Harmonic analysis has been a wellspring of research ideas with applications across mathematics, engineering, and the sciences. It has flourished, evolved, and deepened with continued research and exploration. It enjoys intricate and fundamental relationships to other key modern disciplines, such as signal and image processing, machine learning, and data science. Some of the recent topics covered by this area include compressed sensing, dimension reduction, phaseless reconstruction, or spectral estimation. However, more recently, connections to machine learning and deep neural networks have emerged as some of the most intriguing developments in applied mathematics.
The major goal of this summer project is to gain an understanding of the novel numerical approaches for classical data representation problems. Students will begin by investigating both traditional machine learning techniques for feature extraction and deep learning methods. They will analyze the dependence of these implementations on the amount of used training data as well as its quality, and the available computational resources. In the next step, students will optimize their performance, by learning how to modify and adapt the algorithms, based on state-of-the-art examples of scattering networks or cortical transforms, or by exploring their own ideas. Resulting methods will be applied to audio-processing problems, including matching, demixing, and reconstruction.