Research Interests:

• Scientific computing, optimization and variational PDEs.

• Image processing and applications.

• Machine/Deep learning for manifold-structured data.

• Computational Mean Field Games/Controls problems and their connections to deep learning

Please see Google scholar for the most recent list:  

1. 1. H Liu, A Havrilla, R Lai, W Liao, Deep Nonparametric Estimation of Intrinsic Data Structures by Chart Autoencoders: Generalization Error and Robustness, arXiv preprint arXiv:2303.09863, 2023

   2. SC Schonsheck, S Mahan, T Klock, A Cloninger, R Lai, Semi-Supervised Manifold Learning with Complexity Decoupled Chart Autoencoders, arXiv preprint arXiv:2208.10570, 2022.

   3. H Huang, J Yu, J Chen, R Lai, Bridging Mean-Field Games and Normalizing Flows with Trajectory Regularization, arXiv preprint arXiv:2206.14990, 2022

   4. J Yu, R Lai, W Li, S Osher, Computational Mean-field Games on Manifolds, arXiv preprint arXiv:2206.01622, 2022.

   5. J Yu, R Lai, W Li, S Osher, A Fast Proximal Gradient Method and Convergence Analysis for Dynamic Mean Field Planning, arXiv preprint arXiv:2102.13260, 2021

   6. W. Lee, R. Lai, W. Li, S. Osher, Fast Algorithms for Generalized Unnormalized Optimal Transport, Journal of Computational Physics, 436, 110041, 2021

   7. A. Tasissa, R. Lai, “Low-rank Matrix Completion in a General Non-orthogonal Basis”, Linear Algebra and Applications, 625, 81-112, 2021. [PDF]

   8. S. Schonsheck, M. Bronstein, R. Lai, “Nonisometric Surface Registration via Conformal Laplace-Beltrami Basis Pursuit”, Journal of Scientific Computing, 86(3), 1-24, 2021. [PDF]

   9. R Xiang, R Lai, H Zhao, A Dual Iterative Refinement Method for Non-rigid Shape Matching, CVPR, 2021

   10. FL Fan, R Lai, G Wang, Quasi-equivalence of width and depth of neural networks, arXiv preprint arXiv:2002.02515

   11. NJ Tatro, SC Schonsheck, R Lai, Unsupervised Geometric Disentanglement for Surfaces via CFAN-VAE, arXiv preprint arXiv:2005.11622, 2020.

   12. NJ Tatro, PY Chen, P Das, I Melnyk, P Sattigeri, R Lai, Optimizing Mode Connectivity via Neuron Alignment, NeurIPS, 2020

   13. R Xiang, R Lai, H Zhao, Efficient and Robust Shape Correspondence via Sparsity-Enforced Quadratic Assignment, CVPR 2020. 

   14. H. Yuan, X. Gu, R. Lai and Z. Wen, “Global optimization with orthogonality constraints via stochastic diffusion on manifold”, Journal of Scientific Computing, 80(2), pp. 1139–1170. 2019.[PDF]

   15. O Azencot, R Lai, Shape analysis via functional map construction and bases pursuit, arXiv preprint arXiv:1909.13200, 2019.

   16. S Schonsheck, J Chen, R Lai, "Chart Auto-Encoders for Manifold Structured Data", arXiv preprint arXiv:1912.10094, 2019

   17. P. Jin, T.Lai, R Lai, B. Dong, NPTC-net: Narrow-Band Parallel Transport Convolutional Neural Network on Point Clouds, submitted, 2019 [PDF]

   18. R. Lai, H. Zhao, Solving PDEs on manifolds represented as point clouds and applications, to appear, Processing, Analyzing and Learning of Images, Shapes, and Forms: Part 2, Handbook of Numerical Analysis, 2019.

   19. Z. Liu, R. Lai, H. Zhang, C. Wu, “Triangulated Surface Denoising using Higher Order Regularization with Dynamic Weights” 41(1), B1–B26, SIAM Journal on Scientific Computing, 2019. [PDF]

   20. W. Cong, G. Wang, Q. Yang, J. Hsieh, J. Li, R. Lai, “CT Image Reconstruction in a Low Dimensional Manifold”, Inverse problem and Imaging, 13(3), pp 449–460, 2019. [PDF]

   21. S. Schonsheck, B. Dong, R. Lai, “Parallel Transport Convolution: A New Tool for Convolutional Neural Networks on Manifolds”, submitted. 2018. [PDF]

   22. Z Chen, F Chen, R Lai, X Zhang, CT Lu, “Rational Neural Networks for Approximating Jump Discontinuities of Graph Convolution Operator”, IEEE ICDM, 59-68. 2018. [PDF]

   23. A. Tasissa, R. Lai, “Exact Reconstruction of Euclidean Distance Geometry Problem Using Low-rank Matrix Completion”, 65(5), 3124-3144, IEEE. Transaction on Information Theory, 2018. [PDF] [Code]

   24. R. Lai, J. Li, “Manifold Based Low-rank Regularization for Image Restoration and Semi-supervised Learning”, Journal of Scientific Computing, 74(3), pp 1241–1263, 2018. [PDF]

   25. R. Lai, J. Lu, “Point Cloud Discretization of Fokker-Planck Operators for Committor Functions”, Multiscale Modeling & Simulation, 16(2), 710-726, 2018. [PDF]

   26. R. Lai, J. Li, “Solving Partial Differential Equations on Manifolds From Incomplete Inter-Point Distance”, SIAM Journal on Scientific Computing, 39(5), pp. 2231-2256, 2017. [PDF]

   27. R. Lai, H. Zhao, “Multi-scale Non-Rigid Point Cloud Registration Using Robust Sliced-Wasserstein Distance via Laplace-Beltrami Eigenmap”, SIAM Journal on Imaging Sciences, 10(2), pp. 449—483, 2017. [PDF]

   28. C. Kao, R. Lai, B. Osting, “Maximization of Laplace-Beltrami Eigenvalues on Closed Riemannian Surfaces”, 23(2), pp. 685-720, ESAIM: Control, Optimisation and Calculus of Variations, 2017. [PDF]

   29. R. Lai, J. Lu, “Localized Density Matrix Minimization and Linear Scaling Algorithms”, 315, pp. 194–210, Journal of Computational Physics, 2016. [PDF]

   30. Q. Yang, MK Kalra, A. Padole, J. Lai, E. Hilliard, R. Lai, G. Wang, "Big Data from CT Scanning", 2(1), 1003-1-1003-6, JSM Biomed. Imag, 2015.

   31. R. Lai, J. Lu and S. Osher, “Density Matrix Minimization with l_1 Regularization”, 13(8), pp. 2097–2117, Comm Math Sci. 2015. [PDF]

   32. Y. Shi, R. Lai, J.J. Wang, D. Pelletier, D. Mohr, N. Sicotte, A. W. Toga: “Metric Optimization for Surface Analysis in the Laplace-Beltrami Embedding Space”, 33(7), pp. 1447–1463, IEEE Trans. Medical Imaging, 2014. [PDF]

   33. R. Lai, S. Osher, “A splitting method for orthogonality constrained problems”, Journal of Scientific Computing, 58(2), pp. 431–449. 2014. [PDF]

   34. R. Lai, Z. Wen, W. Yin, X. Gu, and L. Lui, “Folding-Free Global Conformal Mapping for Genus-0 Surfaces by Harmonic Energy Minimization”, Journal of Scientific Computing, 58(3), pp. 705–725, 2014. [PDF]

   35. V. Ozolins, R. Lai, R. Caflisch, S. Osher, “Compressed Plane Waves yield Compactly Supported Multiresolution Basis for the Laplace Operator”, Proceedings of the National Academy of Sciences (PNAS), 111(5), pp. 1691–1696, 2014. [PDF]

   36. V. Ozolins, R. Lai, R. Caflisch, S. Osher， “Compressed Modes for Variational Problems in Mathematics and Physics”, Proceedings of the National Academy of Sciences (PNAS), 110 (46), pp. 18368–18373, 2013. [PDF] [Code]

   37. Y. Shi, R. Lai, A. Toga, “Cortical Surface Reconstruction via Unified Reeb Analysis of Geometric and Topological Outliers in Magnetic Resonance Images”, IEEE Trans. Medical Imaging, 32(3), pp.511–530, 2013. [PDF]

   38. R. Lai, J. Liang, H.K. Zhao, “A local Mesh Method for Solving PDEs on Point Clouds”, Inverse Problems and Imaging, 7(3), pp. 737-755, 2013. [PDF] [Code]

   39. Y. Shi, R. Lai, A. W. Toga, “Conformal Mapping via Metric Optimization with Application for Cortical Label Fusion”, Information Processing in Medical Imaging, pp. 244–255. Springer Berlin Heidelberg, 2013. [PDF]

   40. R. Lai, X. Tai, T. F. Chan, “A Ridge and Corner Preserving Model for Surface Restoration”, SIAM Journal on Scientific Computing, 35(2), pp. 675–695, 2013. [PDF]

   41. J. Zhang, R. Lai, C.-C. Jay Kuo, “Adaptive Directional Totoal-Variation Model for Latent Fingerprint Segmentation”, IEEE Transactions on Information Forensics and Security, 8(8), pp. 1261–1273, 2013. [PDF]

   42. V. Estellers, D. Zosso, R. Lai, J-P. Thiran, S. Osher and X. Bresson, “An Efficient Algorithm for Level Set Method Preserving Distance Function”, IEEE Transactions on Image Processing, 21(12), pp. 4722–4734, 2012. [PDF]

   43. J. Zhang, R. Lai, C.-C. Jay Kuo, “Latent Fingerprint Detection and Segmentation with A Directional Total Variation Model”, IEEE International Conference on Image Processing (ICIP), pp. 1145–1148, 2012. [PDF]

   44. J. Zhang, R. Lai, C.-C. Kuo, “Latent Fingerprint Segmentation with Adaptive Total Variation Model”, IEEE International Conference on Biometrics, pp. 189–195, 2012. [PDF]

   45. J. Liang, R. Lai, T.W. Wong, H.K. Zhao, “Geometric Understanding of Point Clouds Using Laplace-Beltrami Operator”, Computer Vision and Pattern Recognition (CVPR), IEEE Conference on, pp. 214–221, 2012. [PDF]

   46. Y. Shi, R. Lai, A. Toga, “Unified Geometry and Topology Correction for Cortical Surface Reconstruction with Intrinsic Reeb Analysis”, Medical Image Computing and Computer-Assisted Intervention (MICCAI), pp. 601–608, Springer Berlin Heidelberg, 2012. [PDF]

   47. R. Lai, Y. Shi, N. Sicotte, A. Toga, “Automated Corpus Callosum Extraction via Laplace-Beltrami Nodal Parcellation and Intrinsic Geodesic Curvature Flows on Surfaces”, International Conference on Computer Vision (ICCV), pp. 2034-2040, 2011. [PDF]

   48. Y. Shi, R. Lai, R. Gill, N. Sicotte, D. Pelletier, D. Mohr, A. W. Toga, “Conformal Metric Optimization on Surface (CMOS) for Deformation and Mapping in Laplace-Beltrami Embedding Space”, Medical Image Computing and Computer Assisted Intervention (MICCAI), pp. 327-334. Springer Berlin Heidelberg, 2011. [PDF]

   49. R. Lai, T. F. Chan, “A Framework for Intrinsic Image Processing on Surfaces”, Computer Vision and Image Understanding, 115, pp. 1647–1661, 2011. [PDF]

   50. Y. Shi, R. Lai, A. W. Toga, “CoRPORATE: Cortical Reconstruction by Pruning Outliers with Reeb Analysis and Topology-preserving Evolution”, Information Processing on Medical Imaging (IPMI), pp. 233-244. Springer Berlin Heidelberg, 2011. [PDF]

   51. Y. Shi, R. Lai, Jonathan H. Morra, Ivo Dinov, Paul M. Thompson, A. W. Toga, “Robust Surface Reconstruction via Laplace-Beltrami Eigen-Projection and Boundary Deformation”, IEEE Trans. Medical Imaging, 29(12), pp. 2009–2022, 2010. [PDF]

   52. W. Gao, R. Lai, Y. Shi, I. Dinov, A. W. Toga, “A Narrowband Approach for Approximating the Laplace-Beltrami Spectrum of 3D shapes”, International Conference of Numerical Analysis and Applied Mathematics (ICNAAM), vol. 1281, pp. 1010–1013, 2010. [PDF]

   53. Y. Shi, B. Sun, R. Lai, I. Dinov, A. W. Toga, “Automated Sulci Identification via Intrinsic Modeling of Cortical Anatomy”, Medical Image Computing and Computer Assisted Intervention (MICCAI), pp. 49–56. Springer Berlin Heidelberg, 2010. [PDF]

   54. R. Lai, Y.Shi, K. Scheibel, S. Fears, R. Woods, A. W. Toga, T. F. Chan, “Metric-Induced Optimal Embedding for Intrinsic 3D Shape Analysis”, Computer Vision and Pattern Recognition (CVPR), IEEE Conference on, pp. 2871–2878, 2010. [PDF]

   55. R. Lai, Y.Shi, I. Dinov, T.F.Chan, A.W.Toga, “Laplace-Beltrami Nodal Counts: a new signature for 3D shape analysis”, IEEE International Symposium on Biomedical Imaging (ISBI), pp. 694-697, 2009. [PDF]

   56. Y. Shi, R. Lai, S. Krishna, N. Sicotte, I. Dinov, A. W. Toga, “Anisotropic Laplace-Beltrami Eigenmaps: Bridging Reeb Graphs and Skeletons”, IEEE, Mathematical Methods in Biomedical Image Analysis, pp. 1–7, 2008. [PDF]

   57. Y. Shi, R. Lai, K. Kern, N. Sicotte, I. Dinov, A. W. Toga, “Harmonic Surface Mapping With Laplace-Beltrami Eigenmaps”, The Medical Image Computing and Computer Assisted Intervention (MICCAI), pp. 147–154. Springer Berlin Heidelberg, 2008. [PDF]