Zhiqiang Xu

 

     Email: xuzq (at) lsec.cc.ac.cn



My main research areas are approximation theory, computational harmonic analysis and numerical analysis. In particular, I am interested in spline functions, compressed sensing, frame theory and phase retrieval. For more details, see my Research topics  and Publications pages.

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
    Compressed sensing and related problems
    Frame theory and quantization
    Phase retrieval
    Multivariate splines and computer aided geometry design

 
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) 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

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 Signals， with Yang Wang.

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.

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.

 
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 386–395.

l    Almost Everywhere Matrix Recovery, with Yi Rong, Yang Wang.

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 matrices recovery, to appear in Appl. Comp. Harm. Anal., Available online 16 February 2017

 

 
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.

66. Performance bound of the intensity-based model for noisy phase retrieval, with Meng Huang.
65. The recovery of complex sparse signals from few phaseless measurements, with Xia Yu.
64. Almost Everywhere Generalized Phase Retrieval, with Meng Huang, Yi Rong and Yang Wang.
63. The minimizers of the $p$-frame potential, with Zili Xu.
62. Subset Selection for Matrices with Fixed Blocks, with Jiaxin Xie.
61. Solving a perturbed amplitude-based model for phase retrieval, with B.Gao, Yang Wang.
60.The estimation performance of nonlinear least squares for phase retrieval, with M. Huang, to appear in IEEE Transactions on Information Theory
59. ON DC BASED METHODS FOR PHASE RETRIEVAL, with Meng Huang, Ming-jun Lai and Abraham  Varghese,  Approximation Theory XVI, Nashville, 2019
58. Improved bounds for the RIP of Subsampled Circulant matrices, with Meng Huang, Yuxuan Pang, Sampling Theory in Signal and Image Processing, (2019), 1–8.

57. Solving Systems of Quadratic Equations via Exponential-type Gradient Descent Algorithm, with Meng Huang, Journal of Computational Mathematics, 2019.

56. Phase retrieval from the norms of affine transformations, with Meng Huang.

55. J.-F. Cai, Y. Rong, Y. Wang, and Z. Xu, Data Recovery on a Manifold from Linear Samples: Theory and Computation, Annals of Mathematical Sciences and Applications Volume 3 (2018) Number 1, pp. 337-365
54. Almost everywhere injectivity conditions for the matrix recovery problem, with Yi Rong, Yang Wang, to appear in Applied and Computational Harmonic Analysis
53. On generalizations of p-sets and their applications, with Heng Zhou, Numerical Mathematics: Theory, Methods and Applications, 12 (2019), pp. 453-466..
52. The minimal measurement number problem in  phase retrieval: a review of recent developments,  Journal of Mathematical Research with Applications, Jan, 2017, Vol 37, No.1, pp.40-46.
51. Phase retrieval by hyperplanes, with Sara Botelho-Andrade, Peter G. Casazza, Desai Cheng, John Haas, Tin T. Tran, Janet C. Tremain

Contemporary Mathematics Volume "Frames and Harmonic Analysis", 2017

 

50. A gradient-enhanced L1 approach for the recovery of sparse trigonometric polynomials, with Tao Zhou. Commun. Comput. Phys. Vol. 24, No. 1, pp. 286-308, July 2018
49. Improvement of the lower bound of the PCM quantization error for vectors in R^2, with Heng Zhou, Journal of approximation theory. Vol. 215, 2017, 1-12.
48.   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.
47. Phaseless recovery using the Gauss-Newton method with B. Gao,  IEEE Trans. Signal Processing. VOL. 65, NO. 22, NOVEMBER 15, 2017
46. Generalized phase retrieval : measurement number, matrix recovery and beyond with Yang Wang. Applied and Computational Harmonic Analysis, 2019, No.2, 423-446
45. The minimal measurement number for low-rank matrices recovery,  Appl. Comp. Harm. Anal., 2018

44. Stable Signal Recovery from Phaseless Measurements with Bing Gao, Yang Wang.  J Fourier Anal Appl (2016) 22: 787.

43. Robustness Properties of Dimensionality Reduction with Gaussian Random Matrices, with Bin Han,  SCIENCE CHINA Mathematics, 2017, 60: 1753-1778, doi: 10.1007/s11425-016-9018-x.

42.  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.

41. 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.

40. Compressed Sensing Matrices from Fourier Matrices, with G. Xu, IEEE Transactions on Information Theory，61(2015), 469-478.

39. The lower bound of the PCM quantization error in high dimension, with H. Zhou, Applied Computational Harmonic Analysis, 2015

38. The performance of orthogonal multi-matching pursuit under RIP , J. Comp. Math 33(2015), 495-516.

37. Multivariate discrete least-squares approximations with a new type of collocation grid, with T. Zhou, A. Narayan, SIAM J. Sci. Comput., 36(5), A2401–A2422., 2014.

36. On sparse interpolation and the design of deterministic interpolation points, with T. Zhou, SIAM J Sci. Comp.,Vol.36,  1752-1769, 2014.

35. Phase Retrieval for Sparse Signals， with Yang Wang, Applied Computational Harmonic Analysis, Vol 37,  531-544, 2014.

34. 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, Issue 4, pp 537-541

33. On B-spline framelets derived from the unitary extension principle, with Zuowei Shen, SIAM Journal on Mathematical Analysis, 45(11) 2013, 127-151.

32. The regularity of refinable functions, with Yang Wang,  Applied Computational Harmonic Analysis，Volume 34, Issue 1, Pages 1-162,  January 2013

31． Compressed sensing, A survey in Chinese,  Sci Sin Math, 2012, 42(9).

 

30．The Performance of PCM Quantization Under Tight Frame Representations, with Yang Wang, SIAM J. MATH. ANAL, Vol. 44, No. 4, pp. 2802–2823, 2012.

29．A remark about orthogonal matching pursuit algorithm, Advances Adaptive Data Analysis, 2012.

28.  Hierarchical zonotopal spaces, with O. Holtz and A. Ron, Trans. Amer. Math. Soc., Volume 364, Number 2, 2012.

27．Deterministic Sampling of Sparse Trigonometric Polynomials,  Journal of Complexity, Volume 27, Issue 2, April 2011, Pages 133-140.

26． Multivariate splines and polytopes,  Journal of Approximation Theory, Vol. 163, Issue 3, March 2011..

25． 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.

24．Subspace modelling for structured noise suppression, with Laura Rebollo-Neira, A. Plastino,   Physica A: Statistical Mechanics and its Applications, Volume 389, Issue 10, 15 May 2010, Pages 2030-2035.

23．Sagbi bases of Cox-Nagata rings, with Bernd Sturmfels, Journal of the European Mathematical Society, Volume 12, Issue 2, 2010.

22．Eulerian numbers:  a spline interpretation , With R. Wang, Y. Xu,  Journal of Mathematical Analysis and Applications, 370 (2010) 486–490.

21．Refinement equations and spline functions, with A. Dubickas,  Adv. Comp. Math. 32: 1-23, 2010.

20．Marginal Likelihood Integrals for Mixtures of Independence Models, with Shaowei Lin and Bernd Sturmfels,   Journal of Machine Learning Research, 10(Jul):1611--1631, 2009.

19．Multivariate F-splines and fractional box splines,  Journal of Fourier Analysis and Applications, 15:723-738, 2009.

18．Discrete Schemes for Gaussian Curvature and Their Convergence, with Guoliang Xu, Computers and Mathematics with Applications, 2009.

17．Application of multivariate splines to discrete mathematics, preprint.

16．An explicit formulation for two dimensional vector partition functions,  Contemporary Math : Integer points in polyhedra--Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, 2008  PDF

15．Let me tell you my favorite lattice-point problem, with  MATTHIAS BECK, BENJAMIN NILL, BRUCE REZNICK, CARLA SAVAGE and IVAN SOPRUNOV,   Contemporary Math: Integer points in polyhedra--Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, 2008.

14．Analytic and algebraic properties of canal surfaces, Journal of Computational and Applied Mathematics. 195(2006)(With Jiaguang Sun, etc). PDF

13．Multi-dimensional versions of a formula of Popoviciu, Science in China Series A. , 2006, PDF

12．A robust algorithm for finding the real intersection of three quadric surfaces, Computer Aided Geometric Design, Vol 22, Issue 6, 1 (2005), 515-530. .(With Xiaoshen Wang, etc.) PDF

11．Convergence analysis of discrete differential geometry operators over surfaces,  Lecture Notes in Computer Science, Mathematics of Surfaces XI, 2005.PDF

10．The Structural Characterization and Locally Supported Bases for Bivariate Super Splines,  with Ren-hong Wang, Journal of Computational Mathematics.6(2004). PS

9．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.PDF 

8．Discrete Truncated Power And Lattice Points In Rational Polytope, with Ren-Hong Wang, Journal of Computational and Applied Mathematics 159 (2003) 149-159.PDF

7．On the dimension for multivariate weak spline function spaces, with Ren-hong Wang,  Journal of mathematical research & Exposition Vol.22, No.1,7-12, February, 2002.

6．The instability degree in the dimension of spaces of bivariate spline, with Ren-hong Wang, Approx. Theory & its Appl., 18:1,2002,68-80. PS

5．Multivariate weak spline function space, with Ren-hong Wang,  Journal of Computational and Applied Mathematics, Vol.144, No.1-2, 2002.

4．Some properties about crossing cut partition,  with Ren-hong Wang, Numerical Mathematics A journal of Chinese Universities, 2001, No.4.

3．Reasonable layout of pollution sources in nearshore waters,  with Liu Shu-Zhen, Feng En-min, Liu Heng-kui, Acta Oceanologic Sinica, 2000, No.2.

2．Application of uniform design to computer experiments,  with Wai-Man Ho, Journal of the Chinese Statistical Association,2000,No.4.

1．Numerical simulation of tidal motion and parameter identification, with Liu Shu-Zhen Feng En-min, J. Dalian Uni. Tech., 1999, No.3.

Compressed sensing and random matrices, Spring, 2014, Chinese Academy of Sciences

Numerical Approximation, Autumn 2013, Chinese Academy of Sciences

Compressed sensing and random matrices, Spring, 2013, Chinese Academy of Sciences

Compressed sensing and random matrices, Spring, 2012, Chinese Academy of Sciences

Numerical Approximation, Autumn 2011, Chinese Academy of Sciences

Differential equations (Math 235),  Summer 2011, Michigan State University

Numerical Approximation, Autumn 2010, Chinese Academy of Sciences

Computational Harmonic Analysis, Spring 2010,  Chinese Academy of Sciences

Numerical Approximation, Autumn 2009, Chinese Academy of Sciences

Advanced Numerical Approximation, Autumn, 2005, Chinese Academy of  Sciences;

Design and analysis of algorithms, Spring,2004, Tsinghua University;

Design and analysis of algorithms, Spring,2005, Tsinghua University;

Freshman Seminar,  TA,  Autumn, 2003, Tsinghua University;

Freshman Seminar, TA, Autumn,2004, Tsinghua University.

 Working  Experience

 

l  Professor, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences (2014.3--)  

l  Associate Professor, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences (2009.3-2014.3) 

l  Assistant Professor, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences (2005.7-2009.3)   

l  Visiting Associate Professor, Michigan State University (2011.7-2011.8)         

l  Visiting Scholar,  National University of Singapore (2011.3)  

l  Research Fellow, TU Berlin, Germany (2007.7-2008.7)

l  Postdoc, Department of Computer Science, Tsinghua University (2003.8-2005.7).

l  Research Assistant,  Hong Kong Baptist University (2000.3-2000.6).

 

 

 

Education

 

l  Ph.D., 2003.7,   Department of Applied Mathematics,  Dalian University of Technology. 

l  Dissertation: Multivariate Spline, Piecewise Algebraic Curve and   Linear DIOPHANTINE Equations.

l  B.A. , 1998.7,   Department of Applied Mathematics, Dalian University of Technology .

l  1998.8,  Summer School, Nanking University, Differential Geometry

l  2001.8,  Summer School, Zhejiang University, Finite Element

 

 

Services

 

l  Associate Editors, Journal of Computational Mathematics,  (2016--)

l  Associate Editors, Numerical Mathematics: Theory, Methods and Applications, (2016--)

l  Guest Editor GDC2013 Special Issue of Science in China, Series A

l  Secretary-General, China Society for Computational Mathematics, (2015-2019)

 

 

Selected honors and awards

 

l  2017, Youth Innovation Award (China Society for Computational Mathematics )

l  2012, Outstanding teacher award (Academy of Mathematics and System Science, Chinese Academy of Sciences )

l  2011, Membership of Youth Innovation Promotion Association of Chinese Academy of Sciences

l  2010, LU Jiaxi Young Talent Award (Chinese Academy of Sciences)

l  2010, Chen Jingrun Award for Young Talents (Academy of Mathematics and System Science,  Chinese Academy of Sciences )

 

 

Grants

 

l  2017-2019, State key program of the major research plan of the national natural science foundation of China, The recovery of signals from incomplete measurements

l  2015-2017, Outstanding Youth Science Foundation, Computational Harmonic Analysis

l  2011-2014, National Natural Science Foundation of China, Encoding and decoding methods in compressed sensing

l  2009-2011, National Natural Science Foundation of China, An interdisciplinary research between splines and discrete geometry

l  2005-2007, National Natural Youth Science Foundation of China, Using splines to count lattice points in polytopes: An interdisciplinary research between approximation theory and discrete mathematics