Research Interests

My research interests are centered around approximation theory, computational harmonic analysis, numerical analysis, and discrete mathematics. Topics that I am working on include:

• Interlacing Polynomials
• Box Splines and Algebra
• Compressed Sensing and Related Problems
• Frame Theory and Quantization
• Phase Retrieval
• Multivariate Splines and Computer-Aided Geometric Design

Interlacing Polynomials

The interlacing polynomial method has been successfully used to solve the Kadison-Singer problem. On one hand, we further develop the interlacing polynomial method itself. On the other hand, we use this method to investigate multiple problems in data science, such as subset selection problems.

Selected papers:

• Improved bounds in Weaver's KS_r conjecture for high rank positive semidefinite matrices, with Zili Xu, Ziheng Zhu, Journal of Functional Analysis, Volume 285, Issue 4, 2023.
• Upper and lower bounds for matrix discrepancy, with Jiaxin Xie and Ziheng Zhu, J. Fourier Anal. Appl., 2022.
• Subset Selection for Matrices with Fixed Blocks, with Jiaxin Xie, Israel Journal of Mathematics, 2021.
• Asymptotically Sharp Upper Bound for the Column Subset Selection Problem, with J.-F. Cai and Zili Xu, International Mathematics Research Notices, 2024.
• Interlacing Polynomial Method for Matrix Approximation via Generalized Column and Row Selection, with J.-F. Cai and Zili Xu.

Box Splines and Algebra

Box splines are powerful tools in multivariate approximation theory and are closely connected with combinatorics and algebra. They play a pivotal role in the study of integer solutions of linear Diophantine equations, integer point counts in polytopes, polytope volumes, and zonotopal spaces.

Selected papers:

• Hierarchical zonotopal spaces, with O. Holtz and A. Ron, Trans. Amer. Math. Soc., Volume 364, Number 2, 2012.
• Multivariate splines and polytopes, Journal of Approximation Theory, Vol. 163, Issue 3, March 2011.
• Sagbi bases of Cox-Nagata rings, with Bernd Sturmfels, Journal of the European Mathematical Society, Volume 12, Issue 2, 2010.
• Eulerian numbers: a spline interpretation, with R. Wang, Y. Xu, J. Math. Anal. Appl., 370 (2010) 486–490.
• Multivariate F-splines and fractional box splines, Journal of Fourier Analysis and Applications, 15:723–738, 2009.
• Multi-dimensional versions of a formula of Popoviciu, Science in China Series A, 2006.
• Discrete Truncated Power And Lattice Points In Rational Polytope, with Ren-Hong Wang, Journal of Computational and Applied Mathematics 159 (2003) 149–159.
• Refinement equations and spline functions, with A. Dubickas, Adv. Comp. Math. 32: 1–23, 2010.
• Marginal Likelihood Integrals for Mixtures of Independence Models, with Shaowei Lin and Bernd Sturmfels, Journal of Machine Learning Research, 10(Jul):1611–1631, 2009.
• The regularity of refinable functions, with Yang Wang, Applied and Computational Harmonic Analysis, Volume 34, Issue 1, Pages 1–162, January 2013.

Compressed Sensing and Related Problems

It is possible to reconstruct a sparse signal accurately from a limited number of measurements. I am particularly interested in deterministic encoding matrices and theoretical conjectures related to random matrices.

Selected papers:

• One-Bit Compressed Sensing by Greedy Algorithms, with Wenhui Liu, D. Gong, Numerical Mathematics: Theory, Methods and Applications, Vol. 9, No. 2, pp. 169–184, 2016.
• Phase Retrieval for Sparse Signals, with Yang Wang, Applied and Computational Harmonic Analysis, Vol 37, 531–544, 2014.
• A strong restricted isometry property, with an application to phaseless compressed sensing, with V. Voroninski, Applied and Computational Harmonic Analysis, Volume 40, Issue 2, March 2016, Pages 386–395.
• Robustness Properties of Dimensionality Reduction with Gaussian Random Matrices, with Bin Han, SCIENCE CHINA Mathematics, 2017, 60: 1753–1778.
• On sparse interpolation and the design of deterministic interpolation points, with T. Zhou, SIAM J. Sci. Comp., Vol.36, 1752–1769, 2014.
• On the ℓ₁-Norm Invariant Convex k-Sparse Decomposition of Signals, with G. Xu, Journal of the Operations Research Society of China, December 2013, Volume 1, pp 537–541.
• Compressed Sensing Matrices from Fourier Matrices, with G. Xu, IEEE Transactions on Information Theory, 61(2015), 469–478.
• The performance of orthogonal multi-matching pursuit under RIP, J. Comp. Math 33(2015), 495–516.
• Compressed sensing: A survey, Sci Sin Math, 2012, 42(9).
• Deterministic Sampling of Sparse Trigonometric Polynomials, Journal of Complexity, Volume 27, Issue 2, April 2011, Pages 133–140.
• A remark about orthogonal matching pursuit algorithm, Advances in Adaptive Data Analysis, 2013.

Frame Theory and Quantization

Frames are widely used in applied mathematics, especially in image processing and information theory. My interests include B-spline wavelet frames, maximally equiangular frames, and applications to quantization.

Selected papers:

• The minimizers of the p-frame potential, with Zili Xu, Applied and Computational Harmonic Analysis, Volume 52, May 2021, Pages 366–379.
• The lower bound of the PCM quantization error in high dimension, with H. Zhou, Applied and Computational Harmonic Analysis, 2015.
• On B-spline framelets derived from the unitary extension principle, with Zuowei Shen, SIAM Journal on Mathematical Analysis, 45(11) 2013, 127–151.
• The Performance of PCM Quantization Under Tight Frame Representations, with Yang Wang, SIAM Journal on Mathematical Analysis, Vol. 44, No. 4, pp. 2802–2823, 2012.
• Adaptive non-uniform B-spline dictionaries on a compact interval, with Laura Rebollo-Neira, Signal Processing, Volume 90, Issue 7, July 2010, Pages 2308–2313.

Phase Retrieval

Reconstructing signals from phaseless observations is a fundamental problem. I apply algebraic geometry to study its theoretical limits and develop practical recovery algorithms.

Selected papers:

• A strong restricted isometry property, with an application to phaseless compressed sensing, with V. Voroninski, Applied and Computational Harmonic Analysis, Volume 40, Issue 2, March 2016, Pages 386–395.
• Almost everywhere injectivity conditions for the matrix recovery problem, with Yi Rong, Yang Wang, Applied and Computational Harmonic Analysis, Vol.50, January 2021, Pages 386–400.
• Phase Retrieval From the Magnitudes of Affine Linear Measurements, with Bing Gao, Qiyu Sun and Yang Wang, Advances in Applied Mathematics, 93 (2018), 121–141.
• Phaseless recovery using the Gauss-Newton method, with B. Gao, IEEE Trans. Signal Processing, Vol. 65, No. 22, November 15, 2017.
• Generalized phase retrieval: measurement number, matrix recovery and beyond, with Yang Wang, Applied and Computational Harmonic Analysis, available online 21 September 2017.
• The minimal measurement number for low-rank matrix recovery, Appl. Comp. Harm. Anal., 2018.
• The estimation performance of nonlinear least squares for phase retrieval, with M. Huang, IEEE Transactions on Information Theory, Volume 66, Issue 12, Dec. 2020, 7967–7977.
• Phase retrieval from the norms of affine transformations, with Meng Huang, Advances in Applied Mathematics, Volume 130, 2021.
• The recovery of complex sparse signals from few phaseless measurements, with Yu Xia, Applied and Computational Harmonic Analysis, Volume 50, 2021.
• Almost Everywhere Generalized Phase Retrieval, with Meng Huang, Yi Rong and Yang Wang, Applied and Computational Harmonic Analysis, Vol. 50, January 2021, Pages 16–33.
• Sparse phase retrieval via Phaseliftoff, with Yu Xia, IEEE Transactions on Signal Processing, Vol.69, 2021, Pages 2129–2143.
• Stability in Phase Retrieval: Characterizing Condition Numbers and the Optimal Vector Set, with Yu Xia and Zili Xu, Mathematics of Computation, 2025.

Multivariate Splines and Computer-Aided Geometric Design

I worked on these topics before 2008.

Selected papers:

• Discrete Schemes for Gaussian Curvature and Their Convergence, with Guoliang Xu, Computers and Mathematics with Applications, 2009.
• Convergence analysis of discrete differential geometry operators over surfaces, Lecture Notes in Computer Science, Mathematics of Surfaces XI, 2005.
• Analytic and algebraic properties of canal surfaces, Journal of Computational and Applied Mathematics, 195(2006) (with Jiaguang Sun, etc.).
• A robust algorithm for finding the real intersections of three quadric surfaces, Computer Aided Geometric Design, Vol 22, Issue 6, 2005, 515–530 (with Xiaoshen Wang, etc.).
• The Structural Characterization and Locally Supported Bases for Bivariate Super Splines, with Ren-hong Wang, Journal of Computational Mathematics, 6(2004).
• The Estimation of the Bezout Number of Piecewise Algebraic Curve, with Ren-hong Wang, Science in China Series A, 2003, Vol.46, No.5, 710–717.
• The instability degree in the dimension of spaces of bivariate spline, with Ren-hong Wang, Approx. Theory & its Appl., 18:1, 2002, 68–80.