arXiv subjects

Ziwen Zhao

Publications and source records attributed to Ziwen Zhao.

8 recordsLinked to original sources

arXiv2026

Fixed Point Floer Cohomology of Dehn Twists I: Splitting Formulas

This paper is the first in a series following on our earlier work [arXiv:2205.14516, arXiv:2307.08180] studying the pair-of-pants product on fixed point Floer cohomology. In [arXiv:2205.14516, arXiv:2307.08180] we fully computed this product for Dehn twists on surfaces of genus greater or equal to 2, and used it to compute a version of the (small) "quantum cohomology" for nodal curves. In the present work, we develop tools for computing the fixed point Floer cohomology and the associated product in the case of Dehn twists in all higher dimensions: for iterated Dehn twists around a Lagrangian sphere in a Liouville domain, we show that the product and differential on the fixed point Floer cohomology split into local and Morse-theoretic contributions on the level of cochains, using some new confinement results for J-holomorphic curves. The local contributions are expected to recover a finite sector of the homology of (twisted) loop spaces of $S^n$ along with an associated Chas-Sullivan product, which we will examine in detail in future work. We also discuss some immediate applications and curiosities for future work.

math.SG↗

arXiv2026

Hierarchical Abstract Tree for Cross-Document Retrieval-Augmented Generation

Retrieval-augmented generation (RAG) enhances large language models with external knowledge, and tree-based RAG organizes documents into hierarchical indexes to support queries at multiple granularities. However, existing Tree-RAG methods designed for single-document retrieval face critical challenges in scaling to cross-document multi-hop questions: (1) poor distribution adaptability, where $k$-means clustering introduces noise due to rigid distribution assumptions; (2) structural isolation, as tree indexes lack explicit cross-document connections; and (3) coarse abstraction, which obscures fine-grained details. To address these limitations, we propose $\Psi$-RAG, a tree-RAG framework with two key components. First, a hierarchical abstract tree index built through an iterative "merging and collapse" process that adapts to data distributions without a priori assumption. Second, a multi-granular retrieval agent that intelligently interacts with the knowledge base with reorganized queries and an agent-powered hybrid retriever. $\Psi$-RAG supports diverse tasks from token-level question answering to document-level summarization. On cross-document multi-hop QA benchmarks, it outperforms RAPTOR by 25.9% and HippoRAG 2 by 7.4% in average F1 score. Code is available at https://github.com/Newiz430/Psi-RAG.

cs.LG↗

arXiv2025

A Survey on Self-Supervised Graph Foundation Models: Knowledge-Based Perspective

Graph self-supervised learning (SSL) is now a go-to method for pre-training graph foundation models (GFMs). There is a wide variety of knowledge patterns embedded in the graph data, such as node properties and clusters, which are crucial to learning generalized representations for GFMs. However, existing surveys of GFMs have several shortcomings: they lack comprehensiveness regarding the most recent progress, have unclear categorization of self-supervised methods, and take a limited architecture-based perspective that is restricted to only certain types of graph models. As the ultimate goal of GFMs is to learn generalized graph knowledge, we provide a comprehensive survey of self-supervised GFMs from a novel knowledge-based perspective. We propose a knowledge-based taxonomy, which categorizes self-supervised graph models by the specific graph knowledge utilized. Our taxonomy consists of microscopic (nodes, links, etc.), mesoscopic (context, clusters, etc.), and macroscopic knowledge (global structure, manifolds, etc.). It covers a total of 9 knowledge categories and more than 25 pretext tasks for pre-training GFMs, as well as various downstream task generalization strategies. Such a knowledge-based taxonomy allows us to re-examine graph models based on new architectures more clearly, such as graph language models, as well as provide more in-depth insights for constructing GFMs.

cs.LG↗

arXiv2024

Masked Graph Autoencoder with Non-discrete Bandwidths

Masked graph autoencoders have emerged as a powerful graph self-supervised learning method that has yet to be fully explored. In this paper, we unveil that the existing discrete edge masking and binary link reconstruction strategies are insufficient to learn topologically informative representations, from the perspective of message propagation on graph neural networks. These limitations include blocking message flows, vulnerability to over-smoothness, and suboptimal neighborhood discriminability. Inspired by these understandings, we explore non-discrete edge masks, which are sampled from a continuous and dispersive probability distribution instead of the discrete Bernoulli distribution. These masks restrict the amount of output messages for each edge, referred to as "bandwidths". We propose a novel, informative, and effective topological masked graph autoencoder using bandwidth masking and a layer-wise bandwidth prediction objective. We demonstrate its powerful graph topological learning ability both theoretically and empirically. Our proposed framework outperforms representative baselines in both self-supervised link prediction (improving the discrete edge reconstructors by at most 20%) and node classification on numerous datasets, solely with a structure-learning pretext. Our implementation is available at https://github.com/Newiz430/Bandana.

cs.LG↗

arXiv2023

Fixed point Floer cohomology and closed-string mirror symmetry for nodal curves

We show that for singular hypersurfaces, a version of their genus-zero Gromov-Witten theory may be described in terms of a direct limit of fixed point Floer cohomology groups, a construction which is more amenable to computation and easier to define than the technical foundations of the enumerative geometry of more general singular symplectic spaces. As an illustration, we give a direct proof of closed-string mirror symmetry for nodal curves of genus greater than or equal to 2, using calculations of (co)product structures on fixed point Floer homology of Dehn twists due to Yao-Zhao.

math.SG↗

arXiv2023

CSGCL: Community-Strength-Enhanced Graph Contrastive Learning

Graph Contrastive Learning (GCL) is an effective way to learn generalized graph representations in a self-supervised manner, and has grown rapidly in recent years. However, the underlying community semantics has not been well explored by most previous GCL methods. Research that attempts to leverage communities in GCL regards them as having the same influence on the graph, leading to extra representation errors. To tackle this issue, we define ''community strength'' to measure the difference of influence among communities. Under this premise, we propose a Community-Strength-enhanced Graph Contrastive Learning (CSGCL) framework to preserve community strength throughout the learning process. Firstly, we present two novel graph augmentation methods, Communal Attribute Voting (CAV) and Communal Edge Dropping (CED), where the perturbations of node attributes and edges are guided by community strength. Secondly, we propose a dynamic ''Team-up'' contrastive learning scheme, where community strength is used to progressively fine-tune the contrastive objective. We report extensive experiment results on three downstream tasks: node classification, node clustering, and link prediction. CSGCL achieves state-of-the-art performance compared with other GCL methods, validating that community strength brings effectiveness and generality to graph representations. Our code is available at https://github.com/HanChen-HUST/CSGCL.

cs.SI↗

arXiv2023

Legendrian embedded contact homology

We give a construction of embedded contact homology (ECH) for a contact $3$-manifold $Y$ with convex sutured boundary and a pair of Legendrians $Λ_+$ and $Λ_-$ contained in $\partial Y$ satisfying an exactness condition. The chain complex is generated by certain configurations of closed Reeb orbits of $Y$ and Reeb chords of $Λ_+$ to $Λ_-$. The main ingredients include: a general Legendrian adjunction formula for curves in $\mathbb{R} \times Y$ with boundary on $\mathbb{R} \times Λ$; a relative writhe bound for curves in contact $3$-manifolds asymptotic to Reeb chords; and a Legendrian ECH index with an accompanying ECH index inequality. The (action filtered) Legendrian ECH of any pair $(Y,Λ)$ of a closed contact $3$-manifold $Y$ and a Legendrian link $Λ$ can also be defined using this machinery after passing to a sutured link complement. This work builds on ideas present in Colin-Ghiggini-Honda's proof of the equivalence of Heegaard-Floer homology and ECH. The independence of our construction of choices of almost complex structure and contact form should require a new flavor of monopole Floer homology. It is beyond the scope of this paper.

math.SG↗

arXiv2022

Product and Coproduct on Fixed Point Floer Homology of Positive Dehn Twists

We compute the product and coproduct structures on the fixed point Floer homology of iterations on the single Dehn twist, subject to some mild topological restrictions. We show that the resulting product and coproduct structures are determined by the product and coproduct on Morse homology of the complement of the twist region, together with certain sectors of product and coproduct structures on the symplectic homology of $T^*S^1$. The computation is done via a direct enumeration of $J-$holomorphic sections: we use a local energy inequality to show that some of the putative holomorphic sections do not exist, and we use a gluing construction plus some Morse-Bott theory to construct the sections we could not rule out.

math.SG↗