Mathematics and its applications are two aspects of my research on the mathematical foundations of data science, including approximation theory, wavelet theory, image processing, compressed sensing, and machine learning. I enjoy developing mathematics for applications, by applications, and of applications. Working alongside my collaborators, I have made several contributions to this field.
I started with approximation theory. Some early and notable projects included the algebra of box spline spaces, interpolations by box splines, and constructions of interpolatory subdivision schemes and wavelets using box splines. From this, I progressed to wavelet theory and its applications.
In the theory of wavelets, we developed a duality analysis for frames that provides a comprehensive understanding and lays a foundation for Gabor frames, wavelet frames, and frames in Hilbert spaces. The duality analysis undergirds the duality principle for Gabor frames, the unitary extension principle, and the oblique extension principle for wavelet frames. These principles make wavelet and Gabor frame constructions that adapt to applications transparent and easy to implement.
The success in applications of wavelet frames depends on the capability of choosing a sparse approximation of the targeted function from a frame system. We developed algorithms for image restorations that apply thresholding operators iteratively to derive a sparse representation of images in the wavelet frame domain. The convergence analysis of these iterative thresholding algorithms led to the formulation of sparsity-based balanced models for frames, taking the analysis and synthesis models as special ones (survey papers), together with the error estimation of these models for noisy data fitting. This sparse representation of the targeted images allows our formulation of efficient and robust tools to analyze and process images for various applications. One of our successful applications is image deblurring with an unknown blurring kernel.
The sparsity-based image restorations using wavelet frames spurred three exciting projects. In the first project, the ideas of iterative thresholding in the sparse domain led to the iterative singular value thresholding algorithm for low-rank matrix completion. Our applications of this algorithm in medical imaging and computer vision include cine cone-beam CT reconstruction, video restoration, and surveillance video analysis. In the second project, we developed a mathematical theory that bridges the spline wavelet frame-based model to the total variation and nonlinear evolution partial differential equation-based models for image processing, leading to a deeper understanding of both approaches and deriving a few new models. In the third project, we learned frame systems adapted to the targeted image via a data driven approach to obtain a better sparse approximation for image restorations and classifications.
This data driven frame approach for image restorations and classifications extended my research into machine learning. In particular, we have established a robust mathematical framework to elucidate the expressive capabilities of deep neural networks by viewing them as approximators of diverse functions through function composition. This framework culminates in the identification of their universal approximation properties and the characterization of their approximation prowess in relation to the number of neurons, spanning various architectural configurations within deep neural networks.
We also designed a family of simple architectures of neural networks that approximate functions with arbitrary accuracy by a fixed-size network.