arXiv subjects

Runze Wang

Publications and source records attributed to Runze Wang.

49 records · Page 3Linked to original sources

arXiv2024

Graph identification index

We introduce the \emph{ID-index} of a finite simple connected graph. For a graph $G=(V,\ E)$ with diameter $d$, we let $f:V\longrightarrow \mathbb{R}$ assign \emph{ranks} to the vertices, then under $f$, each vertex $v$ gets a \emph{string}, which is a $d$-vector with the $i$-th coordinate being the sum of the ranks of the vertices that are of distance $i$ from $v$. The \emph{ID-index} of $G$, denoted by $IDI(G)$, is defined to be the minimum number $k$ for which there is an $f$ with $|f(V)|=k$, such that each vertex gets a distinct string under $f$. We present some relations between ID-graphs, which were defined by Chartrand, Kono, and Zhang, and their ID-indices; give a lower bound on the ID-index of a graph; and determine the ID-indices of paths, grids, cycles, prisms, complete graphs, some complete multipartite graphs, and some caterpillars.

math.CO↗

arXiv2024

On the size of sets avoiding a general structure

Given a finite abelian group $G$ and a subset $S\subseteq G$, we let $N_{G,\ S}$ be the smallest integer $N$ such that for any subset $A\subseteq G$ with $N$ elements, we have $g+S\subseteq A$ for some $g\in G$. Using the probabilistic method, we prove that \begin{align*} \frac{|H_G(S)|-1}{|H_G(S)|}|G|+\Biggl\lceil\biggl(\frac{|G|}{|H_G(S)|}\biggr)^{1-|H_G(S)|/|S|}\Biggr\rceil\le N_{G,\ S}\le \biggl\lfloor\frac{|S|-1}{|S|}|G|\biggr\rfloor+1, \end{align*} where $H_G(S)$ is the stabilizer of $S$.

math.CO↗

arXiv2024

A note on the threshold numbers of cycles

A graph $G=(V,E)$ is said to be a \textit{$k$-threshold graph} with \textit{thresholds} $θ_1<θ_2<...<θ_k$ if there is a map $r: V \longrightarrow \mathbb{R}$ such that $uv\in E$ if and only if $θ_i\le r(u)+r(v)$ holds for an odd number of $i\in [k]$. The \textit{threshold number} of $G$, denoted by $Θ(G)$, is the smallest positive integer $k$ such that $G$ is a $k$-threshold graph. In this paper, we determine the exact threshold numbers of cycles by proving \[ Θ(C_n)=\begin{cases} 1 & if\ n=3, 2 & if\ n=4, 4 & if\ n\ge 5, \end{cases} \] where $C_n$ is the cycle with $n$ vertices.

math.CO↗

arXiv2024

How Can Large Language Models Understand Spatial-Temporal Data?

While Large Language Models (LLMs) dominate tasks like natural language processing and computer vision, harnessing their power for spatial-temporal forecasting remains challenging. The disparity between sequential text and complex spatial-temporal data hinders this application. To address this issue, this paper introduces STG-LLM, an innovative approach empowering LLMs for spatial-temporal forecasting. We tackle the data mismatch by proposing: 1) STG-Tokenizer: This spatial-temporal graph tokenizer transforms intricate graph data into concise tokens capturing both spatial and temporal relationships; 2) STG-Adapter: This minimalistic adapter, consisting of linear encoding and decoding layers, bridges the gap between tokenized data and LLM comprehension. By fine-tuning only a small set of parameters, it can effectively grasp the semantics of tokens generated by STG-Tokenizer, while preserving the original natural language understanding capabilities of LLMs. Extensive experiments on diverse spatial-temporal benchmark datasets show that STG-LLM successfully unlocks LLM potential for spatial-temporal forecasting. Remarkably, our approach achieves competitive performance on par with dedicated SOTA methods.

cs.LG↗

arXiv2024

Hecke algebras for tame genuine principal series and local Shimura correspondence

In this paper, we extend the local Shimura correspondence building upon the groundwork laid by Gordan Savin. As preparations, we review part of the type theory of Bushnell and Kutzko which equally applies to covering groups. By adapting the method of linear algebraic groups, we describe the structure of Hecke algebras associated with genuine principal series components and construct the types. In particular, we show each of them shares the same affine Hecke algebra part as one corresponding Hecke algebra of the principal endoscopy group. However, we give a counter example to show they are not isomorphic in general.

math.RT↗

arXiv2024

CO3: Low-resource Contrastive Co-training for Generative Conversational Query Rewrite

Generative query rewrite generates reconstructed query rewrites using the conversation history while rely heavily on gold rewrite pairs that are expensive to obtain. Recently, few-shot learning is gaining increasing popularity for this task, whereas these methods are sensitive to the inherent noise due to limited data size. Besides, both attempts face performance degradation when there exists language style shift between training and testing cases. To this end, we study low-resource generative conversational query rewrite that is robust to both noise and language style shift. The core idea is to utilize massive unlabeled data to make further improvements via a contrastive co-training paradigm. Specifically, we co-train two dual models (namely Rewriter and Simplifier) such that each of them provides extra guidance through pseudo-labeling for enhancing the other in an iterative manner. We also leverage contrastive learning with data augmentation, which enables our model pay more attention on the truly valuable information than the noise. Extensive experiments demonstrate the superiority of our model under both few-shot and zero-shot scenarios. We also verify the better generalization ability of our model when encountering language style shift.

cs.CL↗

arXiv2023

Formal degrees of genuine Iwahori-spherical representations

For square-integrable genuine Iwahori-spherical representations of central covers, we verify the Hiraga--Ichino--Ikeda formula for their formal degrees. We also compute the Whittaker dimensions of these representations, when their associated modules over the genuine Iwahori--Hecke algebra are one-dimensional.

math.RT↗

arXiv2023

Extremal problems of double stars

In a generalized Turán problem, two graphs $H$ and $F$ are given and the question is the maximum number of copies of $H$ in an $F$-free graph of order $n$. In this paper, we study the number of double stars $S_{k,l}$ in triangle-free graphs. We also study an opposite version of this question: what is the maximum number edges/triangles in graphs with double star type restrictions, which leads us to study two questions related to the extremal number of triangles or edges in graphs with degree-sum constraints over adjacent or non-adjacent vertices.

math.CO↗

arXiv2022

Hardness of Graph-Structured Algebraic and Symbolic Problems

In this paper, we study the hardness of solving graph-structured linear systems with coefficients over a finite field $\mathbb{Z}_p$ and over a polynomial ring $\mathbb{F}[x_1,\ldots,x_t]$. We reduce solving general linear systems in $\mathbb{Z}_p$ to solving unit-weight low-degree graph Laplacians over $\mathbb{Z}_p$ with a polylogarithmic overhead on the number of non-zeros. Given the hardness of solving general linear systems in $\mathbb{Z}_p$ [Casacuberta-Kyng 2022], this result shows that it is unlikely that we can generalize Laplacian solvers over $\mathbb{R}$, or finite-element based methods over $\mathbb{R}$ in general, to a finite-field setting. We also reduce solving general linear systems over $\mathbb{Z}_p$ to solving linear systems whose coefficient matrices are walk matrices (matrices with all ones on the diagonal) and normalized Laplacians (Laplacians that are also walk matrices) over $\mathbb{Z}_p$. We often need to apply linear system solvers to random linear systems, in which case the worst case analysis above might be less relevant. For example, we often need to substitute variables in a symbolic matrix with random values. Here, a symbolic matrix is simply a matrix whose entries are in a polynomial ring $\mathbb{F}[x_1, \ldots, x_t]$. We formally define the reducibility between symbolic matrix classes, which are classified in terms of the degrees of the entries and the number of occurrences of the variables. We show that the determinant identity testing problem for symbolic matrices with polynomial degree $1$ and variable multiplicity at most $3$ is at least as hard as the same problem for general matrices over $\mathbb{R}$.

cs.DS↗

arXiv2022

McQueen: a Benchmark for Multimodal Conversational Query Rewrite

The task of query rewrite aims to convert an in-context query to its fully-specified version where ellipsis and coreference are completed and referred-back according to the history context. Although much progress has been made, less efforts have been paid to real scenario conversations that involve drawing information from more than one modalities. In this paper, we propose the task of multimodal conversational query rewrite (McQR), which performs query rewrite under the multimodal visual conversation setting. We collect a large-scale dataset named McQueen based on manual annotation, which contains 15k visual conversations and over 80k queries where each one is associated with a fully-specified rewrite version. In addition, for entities appearing in the rewrite, we provide the corresponding image box annotation. We then use the McQueen dataset to benchmark a state-of-the-art method for effectively tackling the McQR task, which is based on a multimodal pre-trained model with pointer generator. Extensive experiments are performed to demonstrate the effectiveness of our model on this task\footnote{The dataset and code of this paper are both available in \url{https://github.com/yfyuan01/MQR}

cs.CL↗

arXiv2021

Domains of Convergence for Polyhedral Packings

Polyhedral circle packings are generalizations of the Apollonian packing. We develop the theory of the Apollonian group, Descartes quadratic form, and related objects for all polyhedral packings. We use these tools to determine the domain of absolute convergence of a generating function that can be associated to any polyhedral packing. This domain of convergence is the Tits cone for an infinite root system.

math.NT↗

arXiv2020

Can You be More Social? Injecting Politeness and Positivity into Task-Oriented Conversational Agents

Goal-oriented conversational agents are becoming prevalent in our daily lives. For these systems to engage users and achieve their goals, they need to exhibit appropriate social behavior as well as provide informative replies that guide users through tasks. The first component of the research in this paper applies statistical modeling techniques to understand conversations between users and human agents for customer service. Analyses show that social language used by human agents is associated with greater users' responsiveness and task completion. The second component of the research is the construction of a conversational agent model capable of injecting social language into an agent's responses while still preserving content. The model uses a sequence-to-sequence deep learning architecture, extended with a social language understanding element. Evaluation in terms of content preservation and social language level using both human judgment and automatic linguistic measures shows that the model can generate responses that enable agents to address users' issues in a more socially appropriate way.

cs.CL↗

arXiv2020

Joint Contextual Modeling for ASR Correction and Language Understanding

The quality of automatic speech recognition (ASR) is critical to Dialogue Systems as ASR errors propagate to and directly impact downstream tasks such as language understanding (LU). In this paper, we propose multi-task neural approaches to perform contextual language correction on ASR outputs jointly with LU to improve the performance of both tasks simultaneously. To measure the effectiveness of this approach we used a public benchmark, the 2nd Dialogue State Tracking (DSTC2) corpus. As a baseline approach, we trained task-specific Statistical Language Models (SLM) and fine-tuned state-of-the-art Generalized Pre-training (GPT) Language Model to re-rank the n-best ASR hypotheses, followed by a model to identify the dialog act and slots. i) We further trained ranker models using GPT and Hierarchical CNN-RNN models with discriminatory losses to detect the best output given n-best hypotheses. We extended these ranker models to first select the best ASR output and then identify the dialogue act and slots in an end to end fashion. ii) We also proposed a novel joint ASR error correction and LU model, a word confusion pointer network (WCN-Ptr) with multi-head self-attention on top, which consumes the word confusions populated from the n-best. We show that the error rates of off the shelf ASR and following LU systems can be reduced significantly by 14% relative with joint models trained using small amounts of in-domain data.

cs.CL↗