Colorado State University

My research takes an interdisciplinary approach to harmonic analysis with applications in image processing and data analysis. The modern world is powered by a deluge of data, and much of human progress hinges on developing better methods to process this data. Many powerful data processing algorithms rely on a well-designed representation system, and we can leverage beautiful mathematics to design such systems. Much of my research concerns the design of representation systems involving rich symmetries and combinatorial structures that allow one to extract critical features from various classes of data.

Representation systems play a crucial role in both low complexity models (where they are known as dictionaries) and redundant transforms (where they are known as frames). By and large, my research is guided by two fundamental questions along these lines:

• How can low complexity models be used to develop tools and analyze success in data analysis and machine learning?
• How can algebraic, geometric, and combinatorial methods be leveraged to solve open problems in frame theory and quantum information theory?

Low complexity models include mathematical tools like sparsity, low-rank assumptions, and regularization via generative models.  As an example, sparse representation refers to the case where each element in a class of data can be well approximated by a linear combination of a few elements of a certain representation system (i.e., a set of building blocks).  When this happens, it is a sign that the building blocks chosen are in some sense optimal for the data  and allows one to use simplified methods to analyze the data. I am interested in how low complexity models can be used to better understand neural networks and image processing techniques and how they can be used to create customized tools for feature extraction and signal & image processing.

Frames are redundant systems in Hilbert spaces which have reconstruction properties akin to orthonormal bases. In finite dimensions, one often seeks frames which represent line configurations which are optimally geometrically spread apart and are in some cases also equivalent to special measurement systems in quantum information theory.  In recent years, more has been discovered about the deep connections between such frames and mathematical fields like algebraic combinatorics, real algebraic geometry, and algebraic number theory.  My aim is to attack open problems in finite frame theory using these tools. I am also interested in characterizing frames in L²(G), where G is a locally compact abelian group, and using representation theory to generate new transforms in L2(R^d).