[RESEARCH HIGHLIGHTS]

Our research group is deeply engaged in a multifaceted exploration at the intersection of statistics, geometry, and real world data, seeking to unravel complex problems through a series of innovative projects.

 

At the heart of our work in the Interaction of Statistics and Geometry, we explore manifold fitting and estimation, developing state-of-the-art methods for constructing and evaluating smooth submanifolds for data set approximation, inspired by the geometric complexity of the Whitney problem. We extend our geometric investigation through the study of principal submanifolds and classification challenges on Riemannian manifolds, developing sophisticated statistical techniques to exploit the unique properties of manifold structures for data analysis, including the development of principal currents, submanifolds and nonlinear classifiers. Our research also ventures into the geometrical intricacies of Calabi-Yau manifolds, aiming to decode the geometry of their moduli spaces. We are also pioneering the development of graphical models and classification strategies in high-dimensional settings, introducing a novel two-stage classification approach that uses innovative thresholding and quadratic discriminant analysis to navigate the complexities of high-dimensional data. Apart from advancing theoretical understanding, these methods have been applied to the analysis of complex, real-world datasets, such as single-cell RNA sequencing and UK Biobank data, highlighting the practical impact of our work.

 

Under the umbrella of Statistical Inference for Inverse Problem with Singularity, our projects address pressing questions in brain imaging, using advanced techniques to investigate time-varying source distributions in magnetoencephalography (MEG) and to develop predictive behavioral models based on MEG/EEG data. At the same time, we are pioneering non-traditional methods to address the inverse problem in tomographic reconstruction, to revolutionize 3D particle reconstruction from 2D random projections in cryo-electron microscopy (cryo-EM). Our group's work is characterized by a relentless pursuit of knowledge at the intersection of disparate fields, driven by the ambition to provide groundbreaking solutions to some of the most challenging problems in science today.

 

 

[PUBLICATIONS]

Distinguishing Calabi-Yau Threefolds: Topology and Machine-learning (with Yang-Hui He and Shing-Tung Yau) (2024) [Manuscript]

 

Single-Cell Analysis via Manifold Fitting: A New Framework for RNA Clustering and Beyond (with Bingjie Li, Yukun Lu and Shing-Tung Yau) (2024) Proceedings of the National Academy of Sciences of the United States of America, Revised [Manuscript][Code]

 

Manifold Fitting with CycleGAN (with Jiaji Su and Shing-Tung Yau) (2023). Proceedings of the National Academy of Sciences of the United States of America, 121 (5) e2311436121. [PDF][Manuscript][Code][NUS News][Tsinghua News]

 

Uncertainty Quantification in Cryo-EM: A Framework for Continuous Variability (with Bertrand Sim) (2023). [Manuscript][Code]

 

Fitting Manifolds on Manifolds (2023). [Manuscript]

 

Manifold Fitting (with Jiaji Su, Bingjie Li and Shing-Tung Yau) (2023). [PDF][Manuscript][Code]

 

Hunting Principal Sub-manifolds: New Theories and Methods (with Bingjie Li, Do Tran and Zhenyue Zhang) (2023). [Manuscript][Code]

 

Random Fixed Boundary Flows (with Yuqing Xia and Zengyan Fan). (2023). Journal of the American Statistical Association, To appear. [PDF][Code]

 

A Statistical Approach of Estimating Adsorption Isotherm Parameters in Gradient Elution Preparative Liquid Chromatography (with Jiaji Su, Cheng Li and Ye Zhang) (2022). Annals of Applied Statistics, In press. [PDF][Manuscript][Code]

 

High Dimensional Quadratic Discriminant Analysis: Optimality and Phase Transitions (with Wanjie Wang and Jingjing Wu) (2021). [PDF][Manuscript][Code]

 

Manifold Fitting and Projection via Quadratic Approximation (with Zheng Zhai) (2021). [PDF][Manuscript][Code]

 

Manifold Fitting by Ridge Estimation: A Subspace-constrained Approach (with Zheng Zhai) (2020). [PDF][Manuscript][Code]

 

Manifold Fitting under Unbounded Noise (with Yuqing Xia) (2019). [PDF][Manuscript][Code]

 

Community Detection Based on the L∞ Convergence of Eigenvectors in DCBM (with Yan Liu et al.) (2019). [PDF][Manuscript]

 

Manifold Learning in Ambient Space (with Bingjie Li and Wee Chin Tan). (2019). [PDF][Manuscript]

 

Quantifying Time-Varying Sources in Magnetoencephalography – A Discrete Approach (with Zengyan Fan, Masahito Hayashi and William F. Eddy). (2020). Annals of Applied Statistics, 14, 1379-1408 [PDF][Supplementary Material]

 

Estimating the Rate Constant from Biosensor Data via an Adaptive Variational Bayesian Approach (with Ye Zhang, Patrik Forssen and Fornstedt Torgny). (2019). Annals of Applied Statistics, 13, 2011-2042 [PDF][Supplementary Material]

 

Principal Boundary on Riemannian Manifolds (with Zhenyue Zhang). (2020). Journal of the American Statistical Association, 115 1435-1448 [PDF] [Supplementary Material]

 

A Level Set Based Variational Principal Flow Method for Nonparametric Dimension Reduction on Riemannian Manifolds (with Hao Liu, Shingyu Leung and Tony F. Chan). (2017). SIAM Journal on Scientific Computing, 39, A1616–A1646 [PDF]

 

Principal Sub-manifolds (with Benjamin Eltzner and Tung Pham). (2016). [PDF]

 

Estimating the Number of Sources in Magnetoencephalography Using Spiked Population Eigenvalues (with Ye Zhang, Zhidong Bai and William F. Eddy). (2018). Journal of the American Statistical Association, 113 505-518 [PDF] [Supplementary Material]

 

Partial Correlation Screening for Estimating Large Precision Matrices, with Applications to Classification (with Shiqiong Huang, Jiashun Jin). (2016). Annals of Statistics, 44 (Peter Hall Memorial Issue) 5 2018-2057 [PDF][Manuscript]

 

A Statistical Approach to the Inverse Problem in Magnetoencephalography (with William F. Eddy). (2014). Annals of Applied Statistics 8 1119-1144 [PDF][Manuscript]

 

Principal Flows (with Victor Panaretos, Tung Pham). (2014). Journal of the American Statistical Association 109 424-436 [PDF]

 

Optimal Classification in Sparse Gaussian Graphic Model (with Jiashun Jin, Yingying Fan). (2013). Annals of Statistics 41 2537-2571 [PDF] [Supplementary Material]

 

A Comparison of the Lasso and Marginal Regression (with Christopher R. Genovese, Jiashun Jin and Larry Wasserman). (2012). Journal of Machine Learning Research 13 2107-2143 [PDF]

 

 

[GRANTS]

Our work has been supported by Singapore Ministry of Education (MOE) Academic Research Fund grants --

·       MOE Tier 1 (Startup), Modern Statistical Methods for Complex Data, 2014-2017,

·       MOE Tier 2, Magnetoencephalography (MEG) Inverse Problem, 2017-2020,

·       MOE Tier 2, Learning in MEG/EEG imaging-- from localization to behaviour prediction, 2021-2024,

·       MOE Tier 2, Cryo-Electron Microscopy: Reconstruction and Uncertainty Quantification via Manifold Learning, 2023-2027,

·       MOE Tier 2, Manifold Landscape in Data Science: Learning beyond Euclidean Space (with Shing-Tung Yau), 2024-2028 (pending),

and by various grants from the National University of Singapore.

 

[RECRUITING]

Ø  PhD Candidates: Interested students are encouraged to contact me directly for possible funded positions. Preferences will be given to those with solid mathematical/statistical background and who are willing to work hard on challenging problems. Any exceptional student without TOEFL/GRE scores will also be considered.

For NUS students, I normally expect at least one semester of positive research interaction before discussing joining the group.

Ø  Postdocs: Candidates interested in these open positions are encouraged to get in touch directly.

 

[RECENT/UPCOMING SEMINARS]

     

 

 

1.     SIAM Conference on Applied Linear Algebra, Paris (May 2024)

2.     Plenary Talk, Current Developments in Mathematics and Physics, Beijing (April 2024)

3.     YMSC, Tsinghua (Feb 2024)

4.     Invited Lecture (45-min), International Congress of Chinese Mathematicians (ICCM), Shanghai (Jan 2024)

5.     Kent, LSE and Cambridge, under London Mathematical Society and Royal Statistical Society (Nov 2023)

6.     First Symposium of Geometry and Statistics in China, Beijing (July 2023)

7.     Pujiang Innovation Forum, hosted by Ministry of Science and Technology of China, Shanghai Municipal Government (July 2023)

8.     The Inaugural International Congress of Basic Science, Guest of Honor, People's Hall of China, Beijing (July 2023)

9.     YMSC, Tsinghua (July 2023)

10.  YMSC, Tsinghua (June 2023)

11.  Statistics Centre, Peking U (May 2023)

12.  Applied/Computational Mathematics/Statistics, Notre Dame (April 2023)

13.  Mathematics, Duke (March 2023)

14.  Statistics and Geometry Conference, Harvard (Feb 2023)

15.  Probability Seminar, Harvard (Feb 2023)

16.  Mathematics, Berkeley (Feb 2023)

17.  Biostatistics, Berkeley (Feb 2023)

18.  Statistics/Biostatistics, Stanford (Jan 2023)

19.  Simons Center, Stony Brook (Nov 2022)

20.  Statistics, George Mason (Nov 2022)

21.  Statistics, Columbia (Oct 2022)

22.  Statistics, UW-Madison (Oct 2022)

23.  CMSA, Harvard (Oct 2022)

24.  Statistics, Carnegie Mellon (Oct 2022)

25.  Statistics, Pitt (Sep 2022)

26.  Mathematical Sciences, WPI (Sep 2022)

27.  Statistics, Harvard (Sep 2022)

                                                                            

 

[CONFERENCES & WORKSHOPS]

     

 

 

Ø  Meeting the Statistical Challenges in High Dimensional Data and Complex Networks, Singapore, February 5-16, 2018 (Co-organizer) [IMPRINTS]

Ø  Interactions of Statistics and Geometry (ISAG I), February 14-18, 2022 (Co-organizer)

Ø  Harvard Conference on Geometry and Statistics, Center of Mathematical Sciences and Applications, Harvard University, Feb 27-March 1, 2023 (Organizer) [Poster]

Ø  The First Symposium of Geometry and Statistics in China, Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Yau Center at Tsinghua University, Beijing, July 29-31, 2023 (Organizer)

Ø  Interactions of Statistics and Geometry (ISAG II), Singapore, October 14-25, 2024 (Co-organizer)

Ø  The Second Symposium of Geometry and Statistics in China, The Tsinghua Sanya International Mathematics Forum (TSIMF), December 11-15 (tentatively), 2024 (Organizer)

                                                                            

[Professional Society]

     

 

Elected Member, International Statistical Institute (ISI), 2018-present

 

[University Service]

     

 

Member,  University Research Committee (URC) Expert Panel, 2024-present

Member,  University Research Committee (URC) Expert Panel (Informatics and Mathematics), 2022-2023

 

 

>>Back to top

 

 

 

 

counter customizable