Wenjing Liao  Associate Professor School of Mathematics Georgia Institute of Technology Office: Skiles 115 Email: wliao60@gatech.edu

My research interests are imaging, signal processing,  high dimensional data analysis and machine learning. I finished my Ph.D. in Mathematics under the supervision of Prof. Albert Fannjiang at University of California, Davis. I was a visiting assistant professor at Duke University and an assistant research scientist at Johns Hopkins University before joining Georgia Tech. I worked with Prof. Mauro Maggioni at Duke and JHU. My research is supported by NSF and DOE,  including NSF CAREER and DOE CAREER awards. Here is my CV.

I am looking for motivated students who are interested in deep learning theory, manifold learning, PDE identification and super-resolution in imaging.  Please send me an email if you are interested.

Teaching at Georgia Tech: 

• Math 3215 and 4210 in Fall 2023
  
• Math4803 Data Science in Fall 2019: This course becomes Math 4210.
  

Research Workshop:

• 2025 Georgia Scientific Computing Symposium (GSCS), 2025
  
• IPAM on Sampling, Inference, and Data-Driven Physical Modeling in Scientific Machine Learning, July 14 -18, 2025
  

Student Workshop:

• Southeast Florida-Georgia Applied Computational Math Student Workshop: 2023 at FSU, 2024 at Georgia Tech, 2025 at Auburn Univ.

• Understanding In-Context Learning on Structured Manifolds: Bridging Attention to Kernel Methods, arXiv, 2025.
  Zhaiming Shen, Alex Hsu, Rongjie Lai, Wenjing Liao
• IDENT Review: Recent Advances in Identification of Differential Equations from Noisy Data, arXiv, 2025.
  Roy He, Hao Liu, Wenjing Liao, Sung Ha Kang
  
• Identification of Differential Equations by Dynamics-Guided Weighted Weak From with Voting, arXiv, 2025.
  Jiahui Cheng, Sung Ha Kang, Haomin Zhou, Wenjing Liao
• Single-Shot Prediction of Parametric  Partial Differential Equations, arXiv, 2025.
  Khalid Rafiq, Wenjing Liao, Aditya G. Nair
  
• Transformers for Learning on Noisy and Task-Level Manifold: Approximation and Generalization Insights, arXiv, 2025.
  Zhaiming Shen, Alex Havrilla, Rongjie Lai, Alex Cloninger, Wenjing Liao
• Optimality of Gradient-MUSIC for Spectral Estimation, arXiv, 2025.
  Albert Fannjiang, Weilin Li, Wenjing Liao
• Coefficient-to-Basis Network: A Fine-Tunable Operator Learning Framework for Inverse Problems with Adaptive Discretizations and Theoretical Guarantees, Philosophical Transactions A, 2025.
  Zecheng Zhang, Hao Liu, Wenjing Liao, Guang Lin

• Neural Scaling Laws of Deep ReLU and Deep Operator Network: A Theoretical Study, arXiv, 2024.
  Hao Liu, Zecheng Zhang, Wenjing Liao, Hayden Schaeffer
  
• Understanding Scaling Laws with Statistical and Approximation Theory for Transformer Neural Networks on Intrinsically Low-Dimensional Data, NeurIPS, 2024.
  Alex Havrilla, Wenjing Liao
• Deep Neural Networks Are Adaptive to Function Regularity and Data Distribution in Approximation and Estimation, submitted to Journal of Machine Learning Research, 2024.
  Hao Liu, Jiahui Cheng, Wenjing Liao
• Generalization Error Guaranteed Auto-Encoder-Based Nonlinear Model Reduction for Operator Learning, Applied and Computational Harmonic Analysis, 2024.
  Hao Liu, Biraj Dahal, Rongjie Lai, Wenjing Liao
• Learning Functions Varying along a Central Subspace, SIAM Journal on Mathematics of Data Science, 2024.
  Hao Liu, Wenjing Liao

• DFU: Scale-Robust Diffusion Model for Zero-Shot Super-Resolution Image Generation, NeurIPS Workshop on Diffusion Models, 2023.
  Alex Havrilla, Kevin Rojas, Wenjing Liao, Molei Tao
• Fourier Features for Identifying Differential Equations (FourierIdent), arXiv, 2023.
  Mengyi Tang, Hao Liu, Wenjing Liao, Sung Ha Kang
• Effective Minkowski Dimension of Deep Nonparametric Regression, ICML, 2023.
  Zixuan Zhang, Minshuo Chen, Mengdi Wang, Wenjing Liao, Tuo Zhao
  
• Group Projected Subspace Pursuit for Identification of Variable Coefficient Differential Equations (GP-IDENT), Journal of Computational Physics, 2023.
  Yuchen He, Sung Ha Kang, Wenjing Liao, Hao Liu, Yingjie Liu
  
• Deep Nonparametric Estimation of Intrinsic Data Structures by Chart Autoencoders: Generalization Error and Robustness, Applied and Computational Harmonic Analysis, 2023.
  Hao Liu, Alex Havrilla, Rongjie Lai, Wenjing Liao

• On Deep Generative Models for Approximation and Estimation of Distributions on Manifolds, NeurIPS 2022.
  Biraj Dahal*, Alex Havrilla*, Minshuo Chen, Tuo Zhao, Wenjing Liao
• A Manifold Two-Sample Test Study: Integral Probability Metric with Neural Networks, IMA Information and Inference, 2023.
  Jie Wang, Minshuo Chen, Tuo Zhao, Wenjing Liao, Yao Xie
• Benefits of Overparametrized Convolutional Residual Networks: Function Approximation under Smoothness Constraint, ICML 2022.
  Hao Liu, Minshuo Chen, Siawpeng Er, Wenjing Liao, Tong Zhang, Tuo Zhao
• Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces, Journal of Machine Learning Research, 2024.
  Hao Liu, Haizhao Yang, Minshuo Chen, Tuo Zhao, Wenjing Liao
• WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimming, Journal of Computational Physics, 2023
  Mengyi Tang, Wenjing Liao, Rachel Kuske, Sung Ha Kang
• High Dimensional Binary Classification under Label Shift: Phase Transition and Regularization, Sampling Theory, Signal Processing, and Data Analysis, 2023.
  Jiahui Cheng*, Minshuo Chen*, Hao Liu, Tuo Zhao, Wenjing Liao
  

• Nonparametric Regression on Low-Dimensional Manifolds Using Deep ReLU Networks, IMA Information and Inference, 2022.
  Minshuo Chen, Haoming Jiang, Wenjing Liao, Tuo Zhao
• Doubly Robust Off-Policy Learning on Low-Dimensional Manifolds by Deep Neural Networks, arXiv, 2021
  Minshuo Chen*, Hao Liu*, Wenjing Liao, Tuo Zhao
  
• Besov Function Approximation and Binary Classification on Low-Dimensional Manifolds Using Convolutional Residual Networks, ICML, 2021.
  Hao Liu, Minshuo Chen, Tuo Zhao, Wenjing Liao
• Statistical Guarantees of Generative Adversarial Networks for Distribution Estimation, arXiv, 2020.
  Minshuo Chen, Wenjing Liao, Hongyuan Zha, Tuo Zhao
• Efficient Approximation of Deep ReLU Networks for Functions on Low-Dimensional Manifolds, NeurIPS 2019.
  Minshuo Chen, Haoming Jiang, Wenjing Liao, Tuo Zhao
  

• Multiscale Regression on Unknown Manifolds, Mathematics in Engineering, 2022.
  Wenjing Liao, Mauro Maggioni, Stefano Vigogna
• Adaptive Geometric Multiscale Approximations for Intrinsically Low-Dimensional Data, Journal of Machine Learning Research, 2020.
  Wenjing Liao, Mauro Maggioni
• Learning Adaptive Multiscale Approximations to Data and Functions near Low-Dimensional Sets, Proceedings of the IEEE Information Theory Workshop 2016
  Wenjing Liao, Mauro Maggioni, Stefano Vigogna

• Robust Identification of Differential Equations by Numerical Techniques from A Single Set of Noisy Observation, SIAM Journal on Scientific Computing, 2022.
  Yuchen He, Sung Ha Kang, Wenjing Liao, Hao Liu, Yingjie Liu
• Identifying Differential Equations with Numerical Time Evolution, Journal of Scientific Computing, 2021.
  Sung Ha Kang, Wenjing Liao, Yingjie Liu
• Numerical Identification of Nonlocal Potential Aggregation, Communication in Computational Physics, 2022.
  Yuchen He, Sung Ha Kang, Wenjing Liao, Hao Liu, Yingjie Liu

• Stability and Super-Resolution of MUSIC and ESPRIT for Multi-Snapshot Spectral Estimation, IEEE Transactions on Signal Processing, 2022.
  Weilin Li, Zengying Zhu, Weiguo Gao, Wenjing Liao
  
• Super-Resolution Limit of the ESPRIT Algorithm, IEEE Transactions on Information Theory 2020.
  Weilin Li, Wenjing Liao, Albert Fannjiang
  
• Conditioning of Restricted Fourier Matrices and Super-Resolution of MUSIC, Proceeding of the Sampling Theory and Applications, 2019.
  Weilin Li, Wenjing Liao
• Stable Super-Resolution Limit and Smallest Singular Value of Restricted Fourier Matrices, Applied and Computational Harmonic Analysis (ACHA), 2020. [ACHA 2021 Charles Chui Young Researcher Best Paper Award]
  Weilin Li, Wenjing Liao
• Sensor Calibration for Off-the-Grid Spectral Estimation, Applied and Computational Harmonic Analysis, 2018.
  Yonina Eldar, Wenjing Liao, Sui Tang
  
• MUSIC for Single-Snapshot Spectral Estimation: Stability and Super-Resolution, Applied and Computational Harmonic Analysis (ACHA), 2016.
  Wenjing Liao, Albert Fannjiang
  
• MUSIC for Multidimensional Spectral Estimation: Stability and Super-Resolution, IEEE Transactions on Signal Processing, 2015
  Weilin Li, Wenjing Liao
• MUSIC for Joint Frequency Estimation: Stability with Compressive Measurements, Proceedings of the IEEE Global Conference on Signal and Information Processing, 2014.
  Wenjing Liao
  
• Coherence-Pattern Guided Compressive Sensing with Unresolved Grids, SIAM Journal on Imaging Sciences, 2012.
  Albert Fannjiang, Wenjing Liao
  
• Super-Resolution by Compressive Sensing Algorithms, 2012 Asilomar Conference on Signals, Systems and Computers.
  Albert Fannjiang, Wenjing Liao
• Mismatch and Resolution in Compressive Imaging, Wavelets and Sparsity XIV, Proceedings of SPIE, 2011.
  Albert Fannjiang, Wenjing Liao

• Fourier Phasing with Phase-Uncertain Mask, Inverse Problems, 2013. [Matlab code]
  Albert Fannjiang, Wenjing Liao
  
• Phase Retrieval with Random Phase Illumination, Journal of the Optical Society of America, 2012. Top 5 of the most downloaded articles on image processing from JOSA A and Optical Express over 2012 and 2013. [Matlab code]
  Albert Fannjiang, Wenjing Liao
• Compressed Sensing Phase Retrieval, 2011 Asilomar Conference on Signals, Systems and Computers.
  Albert Fannjiang, Wenjing Liao

• Biraj Dahal, 2021 - now
  

Postdoc

• Zhaiming Shen, 2024 - now
  

Alumni

• Alex Havrilla, ML PhD student, Georgia Tech in 2021 - 2025, now Research Scientist at Google DeepMind
  
• Jiahui Cheng, CSE PhD student, Georgia Tech in 2020 - 2025, now Machine Learning Engineer at Meta
  
• Minshuo Chen, ML PhD student, Georgia Tech in 2018 - 2023, co-advised with Prof. Tuo Zhao in ISyE, now Assistant Professor at Northwestern University
  
• Hao Liu, postdoc at Georgia Tech in 2018 - 2021, now Assistant Professor at Hong Kong Baptist University