Abstract: Although ``big data'' is ubiquitous in data science, one often faces challenges of ``small data,''
as the amount of data that can be taken or transmitted is limited by technical or economic constraints.
To retrieve useful information from the insufficient amount of data, additional assumptions on the signal
of interest are required, e.g. sparsity (having only a few non-zero elements). Conventional methods favor
incoherent systems, in which any two measurements are as little correlated as possible. In reality, however,
many problems are coherent. I will present two nonconvex approaches: one is the difference of the $L_1$
and $L_2$ norms and the other is the ratio of the two. The difference model $L_1$-$L_2$ works particularly
well for the coherent case, while $L_1/L_2$ is a scale-invariant metric that works better when underlying
signals have large fluctuations in non-zero values. Various numerical experiments have demonstrated a
dvantages of the proposed methods over the state-of-the-art. Applications, ranging from MRI reconstruction
to super-resolution and low-rank approximation, will be discussed.
Bio: Yifei Lou has been an Assistant Professor in the Mathematical Sciences Department, University of
Texas Dallas, since 2014. She received her Ph.D. in Applied Math from the University of California Los
Angeles (UCLA) in 2010. After graduation, she joined the School of Electrical and Computer Engineering,
Georgia Institute of Technology as a postdoctoral fellow, working on medical imaging applications. She
was a postdoc at Department of Mathematics, University of California Irvine from 2012-2014.
Her research interests include compressive sensing and its applications, image analysis (medical imaging, hyperspectral, imaging through turbulence), and (nonconvex) optimization algorithms.