arXiv subjects

Allen Zhang

Publications and source records attributed to Allen Zhang.

11 recordsLinked to original sources

arXiv2026

$R$-Bunce-Deddens Algebras

We construct an $R$-algebraic analog of the Bunce-Deddens algebra using the theory of Leavitt labelled path algebras. In particular, we will construct a labelled space whose associated partial action on tight filters is exactly the odometer action on the Cantor set that induces the classical Bunce-Deddens algebra. After defining $R$-Bunce-Deddens algebras, we will prove $R$-algebraic analogs of various results for classical Bunce-Deddens algebras. As our primary application of these results, we will show that, for any field $K$, a $K$-Bunce-Deddens algebra is not Morita equivalent to any Leavitt path algebra.

math.RA↗

arXiv2025

On-Demand Inverse Design for Narrowband Nanophotonic Structures Based on Generative Model and Tandem Network

Inverse design in nanophotonics remains challenging due to its ill-posed nature and sensitivity to input inaccuracies. We present a novel framework that combines a Conditional Variational Autoencoder (CVAE) with a tandem network, enabling robust and efficient on-demand inverse design of photonic structures. Unlike prior approaches that use CVAEs or tandem networks in isolation, our method integrates spectral adjustment and structural prediction in a unified architecture. Specifically, the CVAE adjusts the idealized target spectra, such as Lorentzian-shaped notches, making them more physically realizable and consistent with the training data distribution. This adjusted spectrum is then passed to the tandem network, which predicts the corresponding structural parameters. The framework effectively handles both narrowband (<50 nm) and highly complex spectra, while addressing the one-to-many mapping challenge inherent in inverse design. The model achieves high accuracy, and the designed spectra closely match full-wave simulation results, validating its practicality for advanced nanophotonic applications.

physics.optics↗

arXiv2025

Relative Generalized Boolean Dynamical System Algebras

We study an algebraic analog of a C*-algebra associated to a generalized Boolean dynamical system which parallels the relation between graph C*-algebras and Leavitt path algebras. We prove that such algebras are Cuntz-Pimsner algebras and partial skew group rings and use these facts to prove a graded uniqueness theorem. We then describe the notion of a relative generalized Boolean dynamical system and generalize the graded uniqueness theorem to relative generalized Boolean dynamical system algebras. We use the graded uniqueness theorem to characterize the graded ideals of a relative generalized Boolean dynamical system algebra in terms of the underlying dynamical system. We prove that every generalized Boolean dynamical system algebra is Morita equivalent to an algebra associated to a generalized Boolean dynamical system with no singularities. Finally, we give an alternate characterization of the class of generalized Boolean dynamical system algebras that uses an underlying graph structure derived from Stone duality.

math.RA↗

arXiv2025

Morita Equivalence of Subrings with Applications to Inverse Semigroup Algebras

We develop a technique to show the Morita equivalence of certain subrings of a ring with local units. We then apply this technique to develop conditions that are sufficient to show the Morita equivalence of subalgebras induced by partial subactions on generalized Boolean algebras and, subsequently, strongly $E^{\ast}$-unitary inverse subsemigroups. As an application, we prove that the Leavitt path algebra of a graph is Morita equivalent to the Leavitt path algebra of certain subgraphs and use this to calculate the Morita equivalence class of some Leavitt path algebras. Finally, as the main application, we prove a desingularization result for labelled Leavitt path algebras.

math.RA↗

arXiv2025

Partial Actions on Generalized Boolean Algebras with Applications to Inverse Semigroups and Combinatorial $R$-Algebras

We define the notion of a partial action on a generalized Boolean algebra and associate to every such system and commutative unital ring $R$ an $R$-algebra. We prove that every strongly $E^{\ast}$-unitary inverse semigroup has an associated partial action on the generalized Boolean algebra of compact open sets of tight filters in the meet semilattice of idempotents. Using these correspondences, we associate to every strongly $E^{\ast}$-unitary inverse semigroup and commutative unital ring $R$ an $R$-algebra, and show that it is isomorphic to the Steinberg algebra of the tight groupoid. As an application, we show that there is a natural unitization operation on an inverse semigroup that corresponds to a unitization of the corresponding $R$-algebra. Finally, we show that Leavitt path algebras and labelled Leavitt path algebras can be realized as the $R$-algebra associated to a strongly $E^{\ast}$-unitary inverse semigroup.

math.RA↗

arXiv2025

CG-RAG: Research Question Answering by Citation Graph Retrieval-Augmented LLMs

Research question answering requires accurate retrieval and contextual understanding of scientific literature. However, current Retrieval-Augmented Generation (RAG) methods often struggle to balance complex document relationships with precise information retrieval. In this paper, we introduce Contextualized Graph Retrieval-Augmented Generation (CG-RAG), a novel framework that integrates sparse and dense retrieval signals within graph structures to enhance retrieval efficiency and subsequently improve generation quality for research question answering. First, we propose a contextual graph representation for citation graphs, effectively capturing both explicit and implicit connections within and across documents. Next, we introduce Lexical-Semantic Graph Retrieval (LeSeGR), which seamlessly integrates sparse and dense retrieval signals with graph encoding. It bridges the gap between lexical precision and semantic understanding in citation graph retrieval, demonstrating generalizability to existing graph retrieval and hybrid retrieval methods. Finally, we present a context-aware generation strategy that utilizes the retrieved graph-structured information to generate precise and contextually enriched responses using large language models (LLMs). Extensive experiments on research question answering benchmarks across multiple domains demonstrate that our CG-RAG framework significantly outperforms RAG methods combined with various state-of-the-art retrieval approaches, delivering superior retrieval accuracy and generation quality.

cs.IR↗

arXiv2025

Eliza: A Web3 friendly AI Agent Operating System

AI Agent, powered by large language models (LLMs) as its cognitive core, is an intelligent agentic system capable of autonomously controlling and determining the execution paths under user's instructions. With the burst of capabilities of LLMs and various plugins, such as RAG, text-to-image/video/3D, etc., the potential of AI Agents has been vastly expanded, with their capabilities growing stronger by the day. However, at the intersection between AI and web3, there is currently no ideal agentic framework that can seamlessly integrate web3 applications into AI agent functionalities. In this paper, we propose Eliza, the first open-source web3-friendly Agentic framework that makes the deployment of web3 applications effortless. We emphasize that every aspect of Eliza is a regular Typescript program under the full control of its user, and it seamlessly integrates with web3 (i.e., reading and writing blockchain data, interacting with smart contracts, etc.). Furthermore, we show how stable performance is achieved through the pragmatic implementation of the key components of Eliza's runtime. Our code is publicly available at https://github.com/ai16z/eliza.

cs.AI↗

arXiv2024

Representation Learning of Geometric Trees

Geometric trees are characterized by their tree-structured layout and spatially constrained nodes and edges, which significantly impacts their topological attributes. This inherent hierarchical structure plays a crucial role in domains such as neuron morphology and river geomorphology, but traditional graph representation methods often overlook these specific characteristics of tree structures. To address this, we introduce a new representation learning framework tailored for geometric trees. It first features a unique message passing neural network, which is both provably geometrical structure-recoverable and rotation-translation invariant. To address the data label scarcity issue, our approach also includes two innovative training targets that reflect the hierarchical ordering and geometric structure of these geometric trees. This enables fully self-supervised learning without explicit labels. We validate our method's effectiveness on eight real-world datasets, demonstrating its capability to represent geometric trees.

cs.LG↗

arXiv2024

GRASP: GRAph-Structured Pyramidal Whole Slide Image Representation

Cancer subtyping is one of the most challenging tasks in digital pathology, where Multiple Instance Learning (MIL) by processing gigapixel whole slide images (WSIs) has been in the spotlight of recent research. However, MIL approaches do not take advantage of inter- and intra-magnification information contained in WSIs. In this work, we present GRASP, a novel lightweight graph-structured multi-magnification framework for processing WSIs in digital pathology. Our approach is designed to dynamically emulate the pathologist's behavior in handling WSIs and benefits from the hierarchical structure of WSIs. GRASP, which introduces a convergence-based node aggregation mechanism replacing traditional pooling mechanisms, outperforms state-of-the-art methods by a high margin in terms of balanced accuracy, while being significantly smaller than the closest-performing state-of-the-art models in terms of the number of parameters. Our results show that GRASP is dynamic in finding and consulting with different magnifications for subtyping cancers, is reliable and stable across different hyperparameters, and can generalize when using features from different backbones. The model's behavior has been evaluated by two expert pathologists confirming the interpretability of the model's dynamic. We also provide a theoretical foundation, along with empirical evidence, for our work, explaining how GRASP interacts with different magnifications and nodes in the graph to make predictions. We believe that the strong characteristics yet simple structure of GRASP will encourage the development of interpretable, structure-based designs for WSI representation in digital pathology. Data and code can be found in https://github.com/AIMLab-UBC/GRASP

cs.CV↗

arXiv2023

Non-Euclidean Spatial Graph Neural Network

Spatial networks are networks whose graph topology is constrained by their embedded spatial space. Understanding the coupled spatial-graph properties is crucial for extracting powerful representations from spatial networks. Therefore, merely combining individual spatial and network representations cannot reveal the underlying interaction mechanism of spatial networks. Besides, existing spatial network representation learning methods can only consider networks embedded in Euclidean space, and can not well exploit the rich geometric information carried by irregular and non-uniform non-Euclidean space. In order to address this issue, in this paper we propose a novel generic framework to learn the representation of spatial networks that are embedded in non-Euclidean manifold space. Specifically, a novel message-passing-based neural network is proposed to combine graph topology and spatial geometry, where spatial geometry is extracted as messages on the edges. We theoretically guarantee that the learned representations are provably invariant to important symmetries such as rotation or translation, and simultaneously maintain sufficient ability in distinguishing different geometric structures. The strength of our proposed method is demonstrated through extensive experiments on both synthetic and real-world datasets.

cs.LG↗

arXiv2023

VOLTA: an Environment-Aware Contrastive Cell Representation Learning for Histopathology

In clinical practice, many diagnosis tasks rely on the identification of cells in histopathology images. While supervised machine learning techniques require labels, providing manual cell annotations is time-consuming due to the large number of cells. In this paper, we propose a self-supervised framework (VOLTA) for cell representation learning in histopathology images using a novel technique that accounts for the cell's mutual relationship with its environment for improved cell representations. We subjected our model to extensive experiments on the data collected from multiple institutions around the world comprising of over 700,000 cells, four cancer types, and cell types ranging from three to six categories for each dataset. The results show that our model outperforms the state-of-the-art models in cell representation learning. To showcase the potential power of our proposed framework, we applied VOLTA to ovarian and endometrial cancers with very small sample sizes (10-20 samples) and demonstrated that our cell representations can be utilized to identify the known histotypes of ovarian cancer and provide novel insights that link histopathology and molecular subtypes of endometrial cancer. Unlike supervised deep learning models that require large sample sizes for training, we provide a framework that can empower new discoveries without any annotation data in situations where sample sizes are limited.

eess.IV↗