🤓☝️ About Me

The photo was taken near 📍 浮間橋, Tokyo,
which once appeared as a landmark in the
anime TIGER×DRAGON (とらドラ！).

I am a highly motivated undergraduate student majoring in Computer Science who is especially interested in Theory. My previous research experiences lie in areas such as Information Theory, Statistical Learning Theory, Reinforcement Learning Theory, Computer Graphics and Computational Fluid Dynamics.

Another thing about me is that I have a strong passion for math, and have had a decent amont of exposure to physics. I enjoy exploring new mathematical fields and concepts, and have encorporated them into my research. Those tools often provide me with new perspectives and insights, as well as some refreshing beauty, for instance in the study of large sample analysis, leveraging tools from Harmonic Analysis, martingale, and truncation techniques is stunningly gorgeous.
Additionally, rigorous math provided me with guarantees and insights into previous commonly used methods as a "physics-person". For example, the famous Calculus of Variations can only be rigorized using the fact that the Schwartz Function Class is dense in L^p space, fascinating connections between generalized function space and the duality of L^p and L^q spaces. I've also formulate a proof for the minimum surface equation using mechanical equilibrium as a small project for my Differential Geometry course, which is also the first course I achieved full marks in my university life. There are way more examples than I can list here.

Expand: About autimatic theorem prover

I am also interested in the possibility of AI agents such as LLMs combined with interactive theorem provers for autimatic theorem proving and discovering. As a theory person, I am always eager to make sure that proofs are correct and rigorous, which is a subtle thing given that mathematicians uses natural language for derivation, and can sometimes be hand-waving.

Click here to see my Philosophy of Mathematics and Its Relation to Other Subjects.

📚 Research Interests Research Statement

The Theory Part:

As someone interested in theory, my research interests always follows from the math that I'm interested in. My purpose for research is to apply rigorous math to interesting topics in computer science, namely, machine learning theory. My current research interests (And also the topics I've been doing) includes Information Theory with Asymptotic Analysis in probability and its intersection with Statistical Learning Theory, such topics may include Information-theoretic hardness (Le Cam's two points method etc.), Algorithm Stability and Differential Privacy. I am also interested in Functional Analysis and its application in learning theory, such as Kernels, NTKs, RKHSs, RKBSs, topics relating to Entrophy Numbers, which is naturally related to the compactness of a complete metric space; Gelfand Widths, Rademacher Complexity and Duality between estimation and approximation.

There are also many interesting math that I'm not yet so familar with, and could consequently lead to potential research topics and ideas. Some of them include:

• Stochastic Analysis
• High Dimentional Statistics
• Random Matrix Theory
• Optimal Transport
• Geometry and Topology in Data Science

Despite my interest in general, I do have some preferences in my taste for the theory I typically enjoy working on. For example, I prefer the view of function spaces, general normed vector spaces to simply finite dimensional vector spaces; I prefer measure-theoretic probability spaces to discrete probability spaces; I prefer dealing with the spectrum of operators to simply the eigenvalues of matrices, as I mentioned in my philosophy of math, the true power of math as a pure "artificial" language comes from the concept of infinity and further more, contunity. But of course, I would always choose the right tool for the right job, and would agree that in certain circumstances, the finite perspective maintains the essence of the ideas and is more efficient to work on (e.g. this is exactly the case for RL theory).

The Empirical Part:

Despite my general interest in theory, I do have very strong coding skills and have done some pretty complicated empirical work. Project wise I have done some work in Real-time Ray Tracing, which is a super complex coding experience with exposure to very basic interaction between CPUs and GPUs, and I have also built up LBM code from scratch for fluid simulation, which is also a very complicated task in scientific computing. I've also conducted experiments on information-theoretic properties of the Gibbs algorithm, which provided me interesting results and insights.

I am experienced in and willing for conducting empirical and applied tasks on relatively interesting topics as a complement to theoretical research.

📖 Education

CV & transcripts-Chinese & transcripts-English

Huazhong University of Science and Technology (HUST) Sep. 2021 - now

Shenzhen Middle School (High School) Sep. 2018 - Jun. 2021

Shenzhen Middle School Sep. 2015 - Jun. 2018

🧱 Internship

Research Intern, Department of Computer Science, University of Illinois at Urbana-Champaign (UIUC). Jul. 2024 - Now

Supervised by Prof. Nan Jiang. I also audited Harmonic Analysis and Topology during the semester.

Research Intern (remote), ECE Department, University of Florida. Jul. 2023 - May 2024

Supervised by Prof. Yuheng Bu.

🏆 Selected Honors and Awards

Scholarship of Weichai Power, Weichai Power Corporation, in 2023

Scholarship for Academic Excellence, Huazhong University of Science and Technology, in 2023

Merit Student, Huazhong University of Science and Technology, in 2022

Awarded to top 5% students

Scholarship for Rising Stars in Optic Valley, Committee of Wuhan East Lake High-tech Development Zone, in 2022

Bronze Medal 🥉 at 37^th Chinese Physics Olympiad (CPhO), Final, the Chinese Physical Society, in 2020

First Price at 37^th Chinese Physics Olympiad (CPhO), Semi-final, the Chinese Physical Society, in 2020

Rank 5 in Guangdong Province

First Price at 36^th Chinese Physics Olympiad (CPhO), Semi-final, the Chinese Physical Society, in 2019

Rank 13 in Guangdong Province

📝 Publications

• Information-theoretic Analysis of the Gibbs Algorithm: An Individual Sample Approach [arXiv] [pdf]
  Youheng Zhu, Yuheng Bu.
  to appear in, IEEE Information Theory Workshop (ITW), Shenzhen, China, Nov. 2024.

  Abstract: Recent progress has shown that the generalization error of the Gibbs algorithm can be exactly characterized using the symmetrized KL information between the learned hypothesis and the entire training dataset. However, evaluating such a characterization is cumbersome, as it involves a high-dimensional information measure. In this paper, we address this issue by considering individual sample information measures within the Gibbs algorithm. Our main contribution lies in establishing the asymptotic equivalence between the sum of symmetrized KL information between the output hypothesis and individual samples and that between the hypothesis and the entire dataset. We prove this by providing explicit expressions for the gap between these measures in the non-asymptotic regime. Additionally, we characterize the asymptotic behavior of various information measures in the context of the Gibbs algorithm, leading to tighter generalization error bounds. An illustrative example is provided to verify our theoretical results, demonstrating our analysis holds in broader settings.

📘 Notes

• Differential Geometry: An Equilibrium Proof to Minimum Surface Equation [See Projects]
  Youheng Zhu. July 2024
  

• A Note on Belief Space Binning for POMDPs: Guarantees under Smoothness Condition [In process]
  Youheng Zhu. October 2024
  

📜 Interesting Projects and Side Projects (Selected)

• Fluid Simulation: Simple LBM on Taichi

  Lid-driven cavity flow is a classical benchmark problem for CFD. In this scenario, the fluid is placed in a 2-Dimensional box, and the lid on the top of the box is moving at a fixed velocity. The fluid inside the box is then driven by the movement of the lid before it stabilizes. I also showcased a visualization of a simple model for blood vessel tumor, and the corresponding simulation.

  The videos below show the evolution of the velocity field with regards to the iteration count using the Lattice-Boltzmann Method (LBM). The task was implemented using taichi. The source code can be found at simple-LBM-based-on-taichi.

• Differential Geometry: An Equilibrium Proof to Minimum Surface Equation

  In physics, the minimization of energy or action represents equilibrium states and real motion. This concept transitions from classical mechanics to analytical mechanics. In analytical mechanics, energy extremization replaces force balance, and the Lagrange equations replace Newton's second law, meaning that every extremization problem carries an equivalent "classical mechanics perspective." From this viewpoint, the goal of this passage is to derive the minimal surface equation starting from force equilibrium.

  Show/Hide PDF

👥 Erdős Number

My Erdős number is 4:

📧 Contact

I am happy to discuss any problems related to math, physics, and computer science.
And I am happy to discuss any opportunities for collaboration in research areas we are both interested in.
I am also actively looking for a Ph.D. position, and I am open to any suggestions and advice.

📧 Email 1: youhengzhu@hust.edu.cn
📧 Email 2: youheng@illinois.edu
📧 Email 3: yhengwilliams@gmail.com

🐱 Github: Zhu Youheng