Mathematics and its applications are two aspects of my research. I believe that theoretical research should be driven by applications, and applied research should be guided by theory, and I enjoy doing research in this manner. This belief shapes my research in data science which is focused on its mathematical foundations and applications, especially in areas of approximation theory, wavelet theory, image processing, compressed sensing, and machine learning. Together with my collaborators, I have made several contributions in these areas.
My research began in approximation theory, especially, the algebra of box spline spaces, interpolations by box splines, and constructions of multivariable interpolatory subdivision schemes, pre-wavelets, orthogonal and biorthogonal wavelets using box splines. From this, I progressed to wavelet theory and its applications. For wavelet theory, we developed a duality analysis for wavelet frames, Gabor frames, and, more generally, frames in Hilbert spaces. The duality analysis undergirds three mathematical principles: the duality principle for Gabor frames, the unitary extension principle, and the oblique extension principle for wavelet frames. The duality analysis and these principles provide a comprehensive understanding and lay a foundation of Gabor frames, wavelet frames, and frames in Hilbert spaces,
Applications of wavelets are based on the basic idea of the ability to choose adaptively and flexibly a sparse approximation of functions from a unified family of representers via multiresolution analysis based non-linear approximation. The unitary extension principle and the oblique extension principle make the construction of wavelet frames that adapt to applications transparent and simple to implement. This allows the formulation of efficient and robust tools to analyze and process images from various applications. We developed algorithms for image restorations that apply thresholding operators iteratively to derive a sparse representation of images. The convergence analysis of these iterative thresholding algorithms was given that led to the formulation of sparsity-based balanced models using spline wavelet frames. (see e.g. survey papers.) Such models and algorithms were then applied to various applications. One of such applications of ours was image deblurring with an unknown blurring kernel. An analysis of noisy data fitting by spline frame models through the lens of the curve, surface, or function approximation was provided. Moreover, 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 which led to deeper understandings of both approaches.
The sparsity-based approach of image recovery is relevant to the foundation of compressed sensing which resulted in my research expanding further, leading to our design of the iterative singular value thresholding algorithm for low-rank matrix completion. Some of our applications of this algorithm in medical imaging and computer vision included cine cone-beam CT reconstruction, robust video restoration, and surveillance video analysis. This iterative singular value thresholding algorithm has been used in various applications by many others.
Our work in image restoration and classification using data driven frames extended my research into machine learning and reconnected me with approximation theory. In particular, on the approximation theory for machine learning, we developed a mathematical foundation for the approximation of various functions via function composition through flow maps of dynamic systems. This in turn led to the establishment of the universal approximation properties for a variety of deep neural network architectures and characterizations of their approximation power in terms of the number of neurons. Most significantly, a family of simple activation functions was designed, whose corresponding neural networks can approximate an arbitrary continuous function with arbitrary accuracy by a fixed size network.