# Student Feature - Shuyang Ling

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My research interests are applied and computational harmonic analysis, compressive sensing, optimization, probability, and numerical analysis. In particular, I have been actively working on developing provably convergent and efficient algorithms for various nonlinear inverse problems and data analysis. My work is driven by real-world applications in signal processing, imaging science, wireless communication, and machine learning.

Applied and computational mathematics have always been my favorite research field. After finishing required coursework at UC Davis, I talked to Prof. Strohmer and started my research journey with him. I still remember our very first meeting when he introduced several papers on compressive sensing [3], phase retrieval [1], and diffusion maps [2]. Although I had heard about these popular topics, I had never spent much time digging deep into them. The reading experience was quite enjoyable and fruitful since this was a perfect opportunity to actively learn a lot of new knowledge and techniques which turned out very useful later. I also realized how probability, optimization, linear algebra, information theory, and harmonic analysis together played such powerful roles in the real-world scientific problems.

After several months’ literature readings, I finally arrived at my dissertation topic: self-calibration and nonlinear inverse problems. Here I would like to give a brief explanation: every time you want to acquire data with devices or instruments, calibration always comes first. Otherwise, the obtained data can be highly biased and inaccurate. The purpose of calibration is to reduce uncertainty and bias in the sensing process so that we get more accurate measurements and thus make more convincing conclusions. However, calibration sometimes can be a highly difficult, costly, or even impossible task to accomplish. Therefore, we want to understand under what circumstances one is able to calibrate the devices and estimate the desired signal simultaneously. In fact, this idea leads to a challenging nonlinear inverse problem which is far from being well understood theoretically and algorithmically. Let’s take a look at a concrete example: suppose you are solving a linear system y = Ax. But unfortunately, the design matrix A may not be completely known due to the lack of calibration and it depends on an unknown parameter θ. The mathematical question of self-calibration is to estimate θ and x jointly from y = A(θ)x, where θ corresponds to the unknown calibrating parameter and x is the object. The sensing matrix A(θ) may depend on θ in both linear and nonlinear ways. Various scientific problems can be written into this model, such as phase retrieval [1], cryo-electron microscopy [7], and blind deconvolution [4, 6]. My contribution lies in solving y = A(θ)x under different assumptions and one of the prominent examples is blind deconvolution, i.e., recovering two unknown functions from their convolution. In image processing, one function is called the blurring kernel or point spread function and the other is the original image; in wireless communication, blind deconvolution becomes a joint channel and signal estimation problem.

In July 2017, I was lucky to receive the SIAM Student Paper Prize because of my work on self-calibration and biconvex compressive sensing [5]. There I established a convex optimization-based framework to recover a sparse x and the unknown parameter θ even if this nonlinear system is underdetermined by exploiting the sparsity of x. Later I proposed a provably convergent nonconvex optimization method to solve the blind deconvolution problem [4] rapidly, reliably, and robustly.

In Sep 2017, I joined the Courant Institute of Mathematical Sciences and the Center for Data Sciences. Besides my ongoing projects on the self-calibration problem, I am also currently working on various problems in the mathematics of data sciences, such as k-means, spectral clustering, and the geometry of nonconvex optimization. I feel very grateful for the opportunities to talk to many brilliant mathematicians, computer scientists, engineers, and statisticians at New York University. I am looking forward to a more fruitful experience in the next few years.

References
[1] E. J. Candès, T. Strohmer, and V. Voroninski. Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming. Communications on Pure and Applied Mathematics, 66(8):1241–1274, 2013.
[2] R. R. Coifman and S. Lafon. Diffusion maps. Applied and Computational Harmonic Analysis, 21(1):5–30, 2006.
[3] S. Foucart and H. Rauhut. A Mathematical Introduction to Compressive Sensing. Springer, 2013.
[4] X. Li, S. Ling, T. Strohmer, and K. Wei. Rapid, robust, and reliable blind deconvolution via nonconvex optimization. arXiv preprint arXiv:1606.04933, 2016.
[5] S. Ling and T. Strohmer. Self-calibration and biconvex compressive sensing. Inverse Problems, 31(11):115002, 2015.
[6] S. Ling and T. Strohmer. Blind deconvolution meets blind demixing: Algorithms and performance bounds. IEEE Transactions on Information Theory, 2017.
[7] A. Singer, Z. Zhao, Y. Shkolnisky, and R. Hadani. Viewing angle classification of cryo-electron microscopy images using eigenvectors. SIAM Journal on Imaging Sciences, 4(2):723–759, 2011.

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