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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


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

l   Phase Retrieval for Sparse Signalswith Yang Wang.

l   A strong restricted isometry property, with an application to phaseless compressed sensing, with V. Voroninski, submitted for publication.

l   Robustness Properties of Dimensionality Reduction with Gaussian Random Matrices, with Bin Han.

l    On sparse interpolation and the design of deterministic interpolation points, with T. Zhou.

l   On the $\ell_1$-Norm Invariant Convex k-Sparse Decomposition of Signals with G. Xu

l   Compressed Sensing Matrices from Fourier Matrices, with G. Xu,.

l   The performance of orthogonal multi-matching pursuit under RIP ,

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 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.


Spectral set conjecture

Recently, I am interested in spectral set conjecture, which was from Fuglede (1974). The conjecture is related with wavelet and sampling theory. It was disproved in dimensions 3 and higher by  Tao, Kolountzakis, Matolcsi, Farkas and Mora. But it remains open in dimensions 1 and 2.

 


 

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.