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Research interests My research interests are centered
around the approximation theory, computational harmonic analysis, numerical
analysis and discrete mathematics. Topics that I am working on
include
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: l 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
l Upper and lower bounds for matrix discrepancy, with Jiaxin Xie and Ziheng Zhu, J Fourier Anal Appl, 2022l Subset Selection for Matrices with Fixed Blocks, with Jiaxin Xie, Israel Journal of Mathematics, 2021l Asymptotically
Sharp Upper Bound for the Column Subset Selection Problem, with
J.-F. Cai and Zili Xu, International Mathematics Research Notices,
2024.
l 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 the field of multivariate approximation theory and are closely connected with both combinatorics and algebra. They play a pivotal role in the study of various mathematical concepts, such as the number of integer solutions of linear Diophantine equations, the total count of integer points in polytopes, and the volume of polytopes. Furthermore, box splines are also intimately linked with the concept of zonotopal spaces. Selected
papers: l Hierarchical zonotopal spaces, with O. Holtz and A. Ron, Trans. Amer. Math. Soc., Volume 364, Number 2, 2012 l Multivariate splines and polytopes, Journal of Approximation Theory, Vol. 163, Issue 3, March 2011. l Sagbi bases of Cox-Nagata
rings, with Bernd Sturmfels, Journal of the European Mathematical
Society, Volume 12, Issue 2, 2010. l
Eulerian
numbers: a spline interpretation ,With
R. Wang, Y. Xu, J. Mathematical Analysis and Applications, 370 (2010) 486–490. l
Multivariate
F-splines and fractional box splines,
Journal of Fourier Analysis and Applications,
15:723-738, 2009. l
Multi-dimensional versions of
a formula of Popoviciu, Science in China Series A. , 2006. l
Discrete Truncated
Power And Lattice Points In Rational Polytope , with
Ren-Hong Wang, Journal of Computational and Applied Mathematics 159 (2003)
149-159. l
Refinement equations and spline functions, with
A. Dubickas, Adv. Comp. Math. 32: 1-23, 2010. l Marginal
Likelihood Integrals for Mixtures of Independence Models, with
Shaowei Lin and Bernd Sturmfels,
Journal of Machine Learning Research, 10(Jul):1611--1631, 2009. l The regularity of refinable functions, with Yang Wang,
Applied 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 using only a limited number of measurement samples. In order to achieve this, it is necessary to construct an encoding matrix and a decoding algorithm with great care. Random matrices are often used in compressed sensing to construct the encoding matrix. However, I am particularly interested in the construction of deterministic encoding matrices. Additionally, I find the conjectures related to random matrices to be especially intriguing. Selected
papers: l 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. l Phase Retrieval for Sparse Signals， with Yang Wang, Applied Computational Harmonic Analysis, Vol 37, 531-544, 2014. l A
strong restricted isometry property, with an application to phaseless
compressed sensing, with V. Voroninski, Applied Computational
Harmonic Analysis, Volume
40, Issue 2, March 2016, Pages 386–395. l Robustness Properties of Dimensionality Reduction with Gaussian Random Matrices, with Bin Han, SCIENCE CHINA Mathematics, 2017, 60: 1753-1778, l
On sparse interpolation and the
design of deterministic interpolation points, with T. Zhou, SIAM
J Sci. Comp.,Vol.36, 1752-1769, 2014. l
On the $\ell_1$-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 l
Compressed Sensing Matrices from
Fourier Matrices, with G. Xu, IEEE Transactions on
Information Theory，61(2015), 469-478 l The performance of orthogonal multi-matching pursuit under RIP , J. Comp. Math 33(2015), 495-516. l
Compressed
sensing, A survey
in Chinese,
Sci Sin Math, 2012, 42(9). l
Deterministic
Sampling of Sparse Trigonometric Polynomials, Journal of
Complexity, Volume 27, Issue
2, April 2011, Pages 133-140. l A remark about orthogonal matching
pursuit algorithm, Advances Adaptive Data Analysis, 2013. Frame
theory and quantization
Frames are increasingly gaining popularity in applied mathematics, particularly in areas such as image processing and information theory. As for myself, I am particularly interested in B-spline wavelet frames and the construction of maximally equiangular frames. Moreover, I am studying how frames can be applied to quantization; a highly relevant area of research. Selected
papers: l The minimizers of the $p$-frame potential, with Zili Xu, Applied and Computational Harmonic Analysis, Volume 52, May 2021, Pages 366-379l The lower bound of the PCM quantization error in high dimension, with H. Zhou, Applied Computational Harmonic Analysis, 2015 l On B-spline framelets derived from the unitary extension principle, with Zuowei Shen, , 45(11) 2013, 127-151. SIAM Journal on Mathematical Analysis l 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. l
Adaptive
non-uniform B-spline dictionaries on a compact interval, With Laura
Rebollo-Neira, Signal Processing,
Phase retrieval
Reconstructing signals through phaseless observations is a significant area of research. On one hand, I apply algebraic geometry methods to investigate its mathematical foundations, especially concerning the minimum number of observations required. On the other hand, I am also keen on developing algorithms for phase recovery, and have already devised multiple solving techniques. Selected
papers: l A
strong restricted isometry property, with an application to phaseless
compressed sensing, with V. Voroninski, Applied Computational
Harmonic Analysis, Volume 40, Issue 2,
March 2016, Pages 386–395. l 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 l
Phase Retrieval From the Magnitudes of
Affine Linear Measurements with Bing Gao, Qiyu Sun and Yang
Wang, Advance in Applied
Mathematics, 93 (2018), 121–141. l Phaseless
recovery using the Gauss-Newton method with
B. Gao, IEEE Trans. Signal Processing. VOL. 65, NO. 22, NOVEMBER 15, 2017 l Generalized
phase retrieval : measurement number, matrix recovery and beyond
with Yang Wang. Applied and Computational Harmonic Analysis, Available online
21 September 2017 l The
minimal measurement number for low-rank matrix recovery, Appl.
Comp. Harm. Anal., 2018 l The estimation performance of nonlinear least squares for phase retrieval, with M. Huang, IEEE Transactions on Information Theory, Volume: 66, Issue: 12, Dec. 7967-7977, 2020. l Phase
retrieval from the norms of affine transformations, with
Meng Huang, Advance in Applied Mathematics, Volume 130,2021 l The
recovery of complex sparse signals from few phaseless measurements, with
Yu Xia, Applied and Computational Harmonic Analysis, Volume
50,2021. l 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 l Sparse
phase retrieval via Phaseliftoff,
with Yu Xia, IEEE Transactions on Signal Processing, Page(s): 2129 –
2143, Vol.69, 18,2021. Multivariate
splines and computer aided geometry design
I
worked on these topics before 2008. Selected
papers: l Discrete Schemes for Gaussian Curvature
and Their Convergence, with Guoliang Xu, Computers and Mathematics with
Applications, 2009. l Convergence analysis of discrete differential geometry
operators over surfaces, Lecture Notes in Computer Science,
Mathematics of Surfaces XI, 2005. l Analytic and algebraic properties of canal surfaces,
Journal of Computational and Applied Mathematics, 195(2006)(With Jiaguang
Sun, etc). l A robust algorithm for finding the real intersections of
three quadric surfaces, Computer Aided Geometric Design, Vol 22, Issue 6,
1 (2005), 515-530. l The Structural Characterization and Locally Supported Bases
for Bivariate Super Splines, with Ren-hong Wang, Journal of Computational
Mathematics.6(2004). l 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. l The instability degree in the dimension of spaces of bivariate spline, with Ren-hong Wang, Approx. Theory & its Appl., 18:1,2002,68-80. |