Supervisor: Dr. Chun Che Li

Duration: Jan - June 2021 (1st year Bachelor)

This project was part of the 2020-21 Undergraduate Research Opportunity Program (UROP) organized by the Mathematics Department at CUHK. In this project, I read and presented textbooks and research literature regarding the study of prime number theory. I also had the opportunity to present one of the most well-known results in prime number theory - the Bertrand's Postulate, which states that there is always a prime number strictly between n and 2n, where n is a natural number. 

Although our work was below the research level most of the time, this project gave me a valuable taste of studying challenging materials and presenting in front of a small cohort. I would like to express my deepest gratitude to Dr. Li for providing me with such opportunity.

Supervisor: Prof. Chandra Nair

Duration: Apr 2022 - Aug 2022 (2nd year Bachelor)

This project was part of the 2022 Summer Research Internship organized by the Faculty of Engineering at CUHK. In this project, I worked on the problem of Marton inner bound under the guidance of Prof. Nair. 

The Marton inner bound, proposed in 1979. is an inner bound regarding the capacity region of broadcast channel. However, it's optimality remains unknown to this today. At the beginning of the project, I quickly reviewed some fundamental topics in probability theory, information theory, and network information theory at the beginning. I then reproduced some recent results regarding Marton inner bound and developed simulations for the purpose of observe and identify potential new properties that might help with a mathematical proof for its optimality. At the end, we proposed (and proved) several new properties of the optimizers. This line of research is currently continued by one of Prof. Nair's Ph.D. student.

A summary report can be found here (a bit badly written).

Supervisor: Prof. Amin Gohari

Duration: Apr 2023 - Aug 2023 (3rd year Bachelor)

This project was part of the 2022 Summer Research Internship organized by the Faculty of Engineering at CUHK. In this project, I worked on the problem of zero-error Shannon capacity of graphs under the guidance of Prof. Gohari.

The problem of maximum independent set from graph theory seems to be unrelated to the problem of the zero-capacity of a discrete memoryless channel from information theory. In 1979, Lovász's paper 'On the Shannon capacity of a graph' established a close connection between them. In this project, we are interested in the supremum of the (normalized) size of the largest independent set of a given graph G with respect to all its self-power G^n, where the product here is the strong product for graphs. This quantity is hard to analysis and remain unsolved even for many simple and structured graphs such as a cycle of odd number of (at least 7) vertices. In light of this, we formulate this quantity as a triple optimization (min-max-min) problem on some suitably defined information functionals, and try to analyze it with known results for these information measures and try to apply the technique of single-letterization from information theory to relate the products of graph to the original graph. At the end, we proved several properties of this optimization problem.

A summary report can be found here. This project won the Best Project Award.

Supervisor: Prof. Pascal Vontobel

Duration: Sep 2023 - May 2024 (final year Bachelor)

This project was my graduation thesis for my Bachelor degree in Mathematics and Information Engineering. In this project, I worked on the Bethe estimation on the permanent of pattern maximum likelihood (PML) matrices.

The permanent of a matrix with (non-negative entries) is an important quantity appears in various fields in mathematics such as combinatorics. It is defined in a similar way as a determinant of a matrix. However, while the determinant can be computed in polynomial time, permanent has been shown to be as hard as an NP problem. Consequently, various approximation methods for estimating and bounding permanent of matrices are proposed in the past decades, among which the Bethe approximation is one of the most well-known schemes. 

Previous studies have shown that the Bethe approximation may perform poorly in the worst case for general matrices. However, when restricted to a smaller class of matrices, the Bethe approximation provides surprisingly accurate estimation with probabilistic guarantee. In this project, we focused on the class of 'PML matrices', which appears in the study of information theory and statistical inference frequently. At the end, we established several theoretical guarantee on the Bethe approximation for PML matrix of specific dimensions. This line of research is currently continued by one of Prof. Vontobel's Ph.D. student.

A short unpolished version of my graduation thesis can be found here.

Supervisor: Prof. Lele Wang

Duration: Sep 2024 - Now 

This project marks the beginning of my Ph.D. study. In this project, we study the 'Subgraph Alignment Problem' as an extension and integration of the 'graph alignment problem' and 'subgraph isomorphism problem. This project is also a collaboration with Prof. Hei Victor Cheng from Aarhus University, Denmark.

Subgraph searching is a common task that appears in many fields of application such as computer vision, bioinformatics, network analysis. The problem of subgraph isomorphism problem captures the exact idea of searching a subgraph with a specific structure (isomorphic copy) from a larger base graph. However, the subgraph isomorphism problem is known to be NP complete. 

On the other hand, the graph isomorphism problem, which aim to recover the vertex correspondence between two isomorphic graphs with anonymized label is also 'hard' (the complexity of graph isomorphism problem is actually unknown and there is a specific complexity class GI for it). However, if you allow a vanishing error probability in the recovery criterion and instead analyze the average case behavior, there are actually polynomial time algorithms achieving almost-sure achievability, this is known as the graph alignment problem. In light of this, we formulate the subgraph alignment problem in a similar sense, and established information-theoretic limits for both achievability and converse results for exact recovery.

The formulation and some preliminary results on this project can be found here. We're currently wrapping up this project. A full version paper of this problem is coming soon!

Collaborator: Dr. Chih Wei Ling

Duration: Jun 2025 - Now

This project is my first collaboration with my friend (actually, my previous TA in the course 'Discrete Math and Probability'). We worked on analyzing high-dimensional lattice quantization scheme and provide theoretical guarantee on the performance in various application scenarios.

In the development of model trainings, the number of parameters and size of dataset growth exponentially. Federated learning (FL) is a common solution to deal with these computational demanding tasks by distributing the tasks to a class of clients. However, FL poses two key challenges, (1) communication efficiency between the clients (and the server if it's centralized), and (2) privacy adaptability to avoid data leakage between different clients. Unlike some previous studies focus on addressing each problem independently by designing some suitable algorithms, we proposed a joint scheme that achieves both communication-efficiency and privacy adaptability simultaneously. The 'magic' behind the scheme leverages the power of a novel universal quantization scheme (introduced in 2024) to construct quantizer whose resulting distortion can be used as our privacy mechanism.

In this project, we proposed the joint scheme as an algorithm with the construction of the quantizer and established theoretical privacy guarantee. Also, we demonstrated that our proposed scheme outperforms some other commonly used baselines.

Our recent work can be found here. We are currently working on some extensions of this problem.

Collaborator: Dr. Yanxiu Liu, Prof. Deniz Gündüz, Prof. Lele Wang

Duration: Nov 2025 - Now

This project is my second collaboration with my friend (my previous TA in the course 'Information Theory'). We worked on analyzing the generalization error bound on specific class of algorithms -- the class of differentially private algorithms. Though the mutual information upper bound is celebrated, it could be hard to compute in general. In our work, we proposed a tighter and easy-to-compute upper bound for these special classes of algorithms via a typicality analysis.

Our recent work can be found here. We are currently working on some extensions of this problem.

Supervisor: Prof. Lele Wang

Duration: Aug 2025 - Now

I don't have a specific research problem that I'm currently working on. However, I'm actively reading and preseting literatures in quantum information and exploring potential research problems. In particular, I'm currently delving into the study of quantum error correction codes and local Hamiltonian problem, and I may choose one of them for my doctoral dissertation.

Host: Prof. Pascal Vontobel

Duration: Sep 2023 - Apr 2024

Main Reading Material: The book 'Quantum Computation and Quantum Information', by Michael A. Nielsen and Isaac L. Chuang

This book is an extended exploration of quantum information after taking the graduate course `Quantum Information Processing'. Two other Ph.D. students and I take turns to present contents from the book (from Ch.1 to Ch. 12 - the entire book). We also went through some interesting paper related to the CHSH inequality.

Host: Prof. Lele Wang

Duration: May 2024 - Aug 2024

Main Reading Material: Course note 'Algorithmic Counting and Sampling' from MIT by Prof. Kuikui Liu 

In this reading group, we went through most of the above lecture notes. I gave several presentations, including the Heilmann-Lieb theorem for matching counting algorithm, Lee-Yang circle theorem for the ferromagnetic Ising model, and establishing spectral independence using correlation decay and zero-freeness respectively.

Here is the hand-written presentation notes for Heilmann-Lieb theorem. But I lost the rest of the notes:(

Host: Prof. Lele Wang

Duration: May 2025 - Aug 2025

Main Reading Material: Course note 'Diffusion Models, Sampling and Stochastic Localization' from ETH Zurich by Prof. Yuansi Chen

In this reading group, we went through most of the above lecture notes and some related papers. I gave two presentations, which focused on viewing stochastic localization from an information-theoretic perspective.

• Here is the beamer (without annotation) on the paper 'Information-Theoretic Proofs for Diffusion Sampling'.

• Here is the beamer (without annotation) on the paper 'An Information-Theoretic View of Stochastic Localization'.

Host: Me

Duration: Sep 2025 - Dec 2025

Main Reading Material: The book 'A First Course in Harmonic Analysis,' by Anton Deitmar

This is my first time hosting a reading group with my friends and colleagues. In this reading group, we brought together students from University of British Columbia, Chinese University of Hong Kong, City University of Hong Kong, UC Santa Barbara, and Oxford University. We planned to go through the entire book and possibly go through some interesting topics in harmonic analysis if time permits. I gave presentations on the first two chapters of the book, and I'm going to give a presentation on Ch.9 soon.

• Here is the beamer (without annotation) for Ch.1 of the book.

• Here is the beamer (without annotation) for Ch.2 of the book.

Host: Me

Duration: Jan 2026 - Now

Main Reading Material: Classical textbooks in Analysis

In this reading group, we brought together undergraduate, master, doctoral students from University of British Columbia, University of Melbourne, and Oxford University. Our audience has different background, including mathematics, physics, computer science, etc. I am the main presenter and our main goal is to equip people from different background with a solid foundation in analysis, which would be helpful in understanding research-level literature across multiple fields.

Host: Me

Duration: Mar 2026 - Now

Main Reading Material: The book 'Quantum Computation and Quantum Information', by Michael A. Nielsen and Isaac L. Chuang, and the book 'Quantum Information Theory' by Mark M. Wilde

In this reading group, we brought together doctoral students, postdocs and professors from University of British Columbia, University of Toronto, and University of Aarhus. I am the main presenter in the reading group and my goal is to equip audience with a solid background in quantum information processing, and ultimately understanding literature in quantum information theory.