Scale-invariant Regularizations for Sparse Signal and Low-rank Tensor Recovery

Abstract: Regularization plays a crucial role in addressing ill-posed problems by guiding solutions towards desired properties.  In this presentation, I will introduce the ratio of the L1 and L2 norms, denoted as L1/L2, which acts as a scale-invariant and parameter-free regularization method for approximating the elusive L0 norm.  Our theoretical analysis reveals a strong null space property (sNSP) and demonstrates that any sparse vector qualifies as a local minimizer of the L1/L2 model when a system matrix adheres to the sNSP condition.  Furthermore, we extend the L1/L2 model to low-rank tensor recovery by introducing the tensor nuclear norm over the Frobenius norm.  We show that local optimality can be ensured under an NSP-type condition.  To find the model solution, I will discuss advanced optimization techniques such as alternating direction method of multipliers and proximal gradient descent, highlighting the trade-offs among accuracy, computational efficiency, and convergence guarantees.  Throughout the presentation, we will explore various applications, including limited angle computed tomography reconstruction and video background modeling, showcasing the superior performance of our approach compared to state-of-the-art methods.

Speaker’s Bio:  Yifei Lou holds a joint position in the Department of Mathematics and the School of Data Science and Society (SDSS) at the University of North Carolina at Chapel Hill. She was on the faculty in the Mathematical Sciences Department at the University of Texas Dallas from 2014 to 2023, first as an Assistant Professor and then as an Associate Professor.  She received her Ph.D. in Applied Mathematics from the University of California Los Angeles in 2010.  After graduation, she was a postdoctoral fellow at the School of Electrical and Computer Engineering at Georgia Institute of Technology, followed by another postdoctoral training at the Department of Mathematics, University of California Irvine from 2012-2014.  Dr. Lou received the National Science Foundation CAREER Award in 2019.