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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

Selected papers:

 

l   Improved bounds in Weaver's KS_r conjecture for high rank positive semidefinite matrices, with Zili Xu, Ziheng Zhu.

l  Upper and lower bounds for matrix discrepancy, with Jiaxin Xie and Ziheng Zhu

l  Subset Selection for Matrices with Fixed Blockswith Jiaxin Xie, Israel Journal of Mathematics, (2021) https://doi.org/10.1007/s11856-021-2171-8.

Box splines and algebra

Box splines are tools in multivariate approximation theory. It is also related with combinatorics and algebra.

In particular, it plays a key role in investigating the number of integer solutions of linear Diophantine equations,the number of integer points in polytopes and the volume of  polytopes. Moreover, zonotopal space also arises from box splines.

 

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) 486490.

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 AnalysisVolume 34, Issue 1, Pages 1-162,  January 2013.

 

Compressed sensing and related problems

One can recover the sparse signal from a few measurement Samples. To do that, we need construct encoding matrix and decoding algorithm carefully. Random matrixes play a key role in constructing encoding matrix in compressed sensing.  I am interested in constructing the deterministic encoding matrix. I also like some conjectures related with random matrix.

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 386395.

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 Signalswith 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 Theory61(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 become more and more popular in applied mathematics, in particular in image process, information theory etc. I am interested in B-spline wavelet frames and construct maximally equiangular frames. And also apply frames to quantization.

Selected papers:

l  The minimizers of the $p$-frame potential, with Zili Xu, Applied and Computational Harmonic Analysis, Volume 52, May 2021, Pages 366-379 

l  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, SIAM Journal on Mathematical Analysis, 45(11) 2013, 127-151.

l  The Performance of PCM Quantization Under Tight Frame Representations, with Yang Wang, SIAM J. MATH. ANAL, 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, Volume 90, Issue 7, July 2010, Pages 2308-2313.

 

Phase retrieval

 

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 386395.

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 this 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. .(With Xiaoshen Wang, etc.)

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.