arXiv subjects

Shizhe Liang

Publications and source records attributed to Shizhe Liang.

7 recordsLinked to original sources

arXiv2025

A census of graph-drawing algorithms based on generalized transversal structures

We present two graph drawing algorithms based on the recently defined "grand-Schnyder woods", which are a far-reaching generalization of the classical Schnyder woods. The first is a straight-line drawing algorithm for plane graphs with faces of degree 3 and 4 with no separating 3-cycle, while the second is a rectangular drawing algorithm for the dual of such plane graphs. In our algorithms, the coordinates of the vertices are defined in a global manner, based on the underlying grand-Schnyder woods. The grand-Schnyder woods and drawings are computed in linear time. When specializing our algorithms to special classes of plane graphs, we recover the following known algorithms: (1) He's algorithm for rectangular drawing of 3-valent plane graphs, based on transversal structures, (2) Fusy's algorithm for the straight-line drawing of triangulations of the square, based on transversal structures, (3) Bernardi and Fusy's algorithm for the orthogonal drawing of 4-valent plane graphs, based on 2-orientations, (4) Barriere and Huemer's algorithm for the straight-line drawing of quadrangulations, based on separating decompositions. Our contributions therefore provide a unifying perspective on a large family of graph drawing algorithms that were originally defined on different classes of plane graphs and were based on seemingly different combinatorial structures.

math.CO↗

arXiv2025

Large Language Models for Bioinformatics

With the rapid advancements in large language model (LLM) technology and the emergence of bioinformatics-specific language models (BioLMs), there is a growing need for a comprehensive analysis of the current landscape, computational characteristics, and diverse applications. This survey aims to address this need by providing a thorough review of BioLMs, focusing on their evolution, classification, and distinguishing features, alongside a detailed examination of training methodologies, datasets, and evaluation frameworks. We explore the wide-ranging applications of BioLMs in critical areas such as disease diagnosis, drug discovery, and vaccine development, highlighting their impact and transformative potential in bioinformatics. We identify key challenges and limitations inherent in BioLMs, including data privacy and security concerns, interpretability issues, biases in training data and model outputs, and domain adaptation complexities. Finally, we highlight emerging trends and future directions, offering valuable insights to guide researchers and clinicians toward advancing BioLMs for increasingly sophisticated biological and clinical applications.

q-bio.QM↗

arXiv2024

Mathematics and Machine Creativity: A Survey on Bridging Mathematics with AI

This paper presents a comprehensive overview on the applications of artificial intelligence (AI) in mathematical research, highlighting the transformative role AI has begun to play in this domain. Traditionally, AI advancements have heavily relied on theoretical foundations provided by mathematics and statistics. However, recent developments in AI, particularly in reinforcement learning (RL) and large language models (LLMs), have demonstrated the potential for AI to contribute back to mathematics by offering flexible algorithmic frameworks and powerful inductive reasoning capabilities that support various aspects of mathematical research. This survey aims to establish a bridge between AI and mathematics, providing insights into the mutual benefits and fostering deeper interdisciplinary understanding. In particular, we argue that while current AI and LLMs may struggle with complex deductive reasoning, their "inherent creativity", the ability to generate outputs at high throughput based on recognition of shallow patterns, holds significant potential to support and inspire mathematical research. This creative capability, often overlooked, could be the key to unlocking new perspectives and methodologies in mathematics. Furthermore, we address the lack of cross-disciplinary communication: mathematicians may not fully comprehend the latest advances in AI, while AI researchers frequently prioritize benchmark performance over real-world applications in frontier mathematical research. This paper seeks to close that gap, offering a detailed exploration of AI fundamentals, its strengths, and its emerging applications in the mathematical sciences.

cs.AI↗

arXiv2024

HELENE: Hessian Layer-wise Clipping and Gradient Annealing for Accelerating Fine-tuning LLM with Zeroth-order Optimization

Fine-tuning large language models (LLMs) poses significant memory challenges, as the back-propagation process demands extensive resources, especially with growing model sizes. Recent work, MeZO, addresses this issue using a zeroth-order (ZO) optimization method, which reduces memory consumption by matching the usage to the inference phase. However, MeZO experiences slow convergence due to varying curvatures across model parameters. To overcome this limitation, we introduce HELENE, a novel scalable and memory-efficient optimizer that integrates annealed A-GNB gradients with a diagonal Hessian estimation and layer-wise clipping, serving as a second-order pre-conditioner. This combination allows for faster and more stable convergence. Our theoretical analysis demonstrates that HELENE improves convergence rates, particularly for models with heterogeneous layer dimensions, by reducing the dependency on the total parameter space dimension. Instead, the method scales with the largest layer dimension, making it highly suitable for modern LLM architectures. Experimental results on RoBERTa-large and OPT-1.3B across multiple tasks show that HELENE achieves up to a 20x speedup compared to MeZO, with average accuracy improvements of 1.5%. Furthermore, HELENE remains compatible with both full parameter tuning and parameter-efficient fine-tuning (PEFT), outperforming several state-of-the-art optimizers. The codes will be released after reviewing.

cs.LG↗

arXiv2024

Grand Schnyder Woods

We define a far-reaching generalization of Schnyder woods which encompasses many classical combinatorial structures on planar graphs. Schnyder woods are defined for planar triangulations as certain triples of spanning trees covering the triangulation and crossing each other in an orderly fashion. They are of theoretical and practical importance, as they are central to the proof that the order dimension of any planar graph is at most 3, and they are also underlying an elegant drawing algorithm. In this article we extend the concept of Schnyder wood well beyond its original setting: for any integer d>2 we define a ``grand-Schnyder'' structure for (embedded) planar graphs which have faces of degree at most d and non-facial cycles of length at least d. We prove the existence of grand-Schnyder structures, provide a linear construction algorithm, describe 4 different incarnations (in terms of tuples of trees, corner labelings, weighted orientations, and marked orientations), and define a lattice for the set of grand Schnyder structures of a given planar graph. We show that the grand-Schnyder framework unifies and extends several classical constructions: Schnyder woods and Schnyder decompositions, regular edge-labelings (a.k.a. transversal structures), and Felsner woods.

math.CO↗

arXiv2024

Evaluation of OpenAI o1: Opportunities and Challenges of AGI

This comprehensive study evaluates the performance of OpenAI's o1-preview large language model across a diverse array of complex reasoning tasks, spanning multiple domains, including computer science, mathematics, natural sciences, medicine, linguistics, and social sciences. Through rigorous testing, o1-preview demonstrated remarkable capabilities, often achieving human-level or superior performance in areas ranging from coding challenges to scientific reasoning and from language processing to creative problem-solving. Key findings include: -83.3% success rate in solving complex competitive programming problems, surpassing many human experts. -Superior ability in generating coherent and accurate radiology reports, outperforming other evaluated models. -100% accuracy in high school-level mathematical reasoning tasks, providing detailed step-by-step solutions. -Advanced natural language inference capabilities across general and specialized domains like medicine. -Impressive performance in chip design tasks, outperforming specialized models in areas such as EDA script generation and bug analysis. -Remarkable proficiency in anthropology and geology, demonstrating deep understanding and reasoning in these specialized fields. -Strong capabilities in quantitative investing. O1 has comprehensive financial knowledge and statistical modeling skills. -Effective performance in social media analysis, including sentiment analysis and emotion recognition. The model excelled particularly in tasks requiring intricate reasoning and knowledge integration across various fields. While some limitations were observed, including occasional errors on simpler problems and challenges with certain highly specialized concepts, the overall results indicate significant progress towards artificial general intelligence.

cs.CL↗

arXiv2023

A Schnyder-type drawing algorithm for 5-connected triangulations

We define some Schnyder-type combinatorial structures on a class of planar triangulations of the pentagon which are closely related to 5-connected triangulations. The combinatorial structures have three incarnations defined in terms of orientations, corner-labelings, and woods respectively. The wood incarnation consists in 5 spanning trees crossing each other in an orderly fashion. Similarly as for Schnyder woods on triangulations, it induces, for each vertex, a partition of the inner triangles into face-connected regions (5~regions here). We show that the induced barycentric vertex-placement, where each vertex is at the barycenter of the 5 outer vertices with weights given by the number of faces in each region, yields a planar straight-line drawing.

math.CO↗