arXiv subjects

Zhongyang Li

Publications and source records attributed to Zhongyang Li.

At least 73 records · Page 4Linked to original sources

arXiv2020

Exact Recovery of Community Detection in k-partite Graph Models

We study the vertex classification problem on a graph whose vertices are in $k\ (k\geq 2)$ different communities, edges are only allowed between distinct communities, and the number of vertices in different communities are not necessarily equal. The observation is a weighted adjacency matrix, perturbed by a scalar multiple of the Gaussian Orthogonal Ensemble (GOE), or Gaussian Unitary Ensemble (GUE) matrix. For the exact recovery of the maximum likelihood estimation (MLE) with various weighted adjacency matrices, we prove sharp thresholds of the intensity $σ$ of the Gaussian perturbation. These weighted adjacency matrices may be considered as natural models for the electric network. Surprisingly, these thresholds of $σ$ do not depend on whether the sample space for MLE is restricted to such classifications that the number of vertices in each group is equal to the true value. In contrast to the $\ZZ_2$-synchronization, a new complex version of the semi-definite programming (SDP) is designed to efficiently implement the community detection problem when the number of communities $k$ is greater than 2, and a common region (independent of $k$) for $σ$ such that SDP exactly recovers the true classification is obtained.

math.PR↗

arXiv2020

Conformal invariance of dimer heights on isoradial double graphs

An isoradial graph is a planar graph in which each face is inscribable into a circle of common radius. We study the 2-dimensional perfect matchings on a bipartite isoradial graph, obtained from the union of an isoradial graph and its interior dual graph. Using the isoradial graph to approximate a simply-connected domain bounded by a simple closed curve, by letting the mesh size go to zero, we prove that in the scaling limit, the distribution of height is conformally invariant and converges to a Gaussian free field.

math.PR↗

arXiv2020

Constrained percolation, Ising model and XOR Ising model on planar lattices

We study constrained percolation models on planar lattices including the $[m,4,n,4]$ lattice and the square tilings of the hyperbolic plane, satisfying certain local constraints on faces of degree 4, and investigate the existence of infinite clusters. The constrained percolation models on these lattices are closely related to Ising models and XOR Ising models on regular tilings of the Euclidean plane or the hyperbolic plane. In particular, we obtain a complete picture of the number of infinite "$+$" and "$-$" clusters of the ferromagnetic Ising model with the free boundary condition on a vertex-transitive triangular tiling of the hyperbolic plane with all the possible values of coupling constants. Our results show that for the Ising model on a vertex-transitive triangular tiling of the hyperbolic plane, it is possible that its random cluster representation has no infinite open clusters, while the Ising model has infinitely many infinite "$+$"-clusters and infinitely many infinite "$-$"-clusters. We also study different behaviors the infinite "$+$" and "$-$" clusters of XOR Ising models on regular tilings of the Euclidean plane and the hyperbolic plane for different coupling constants. A by-product we prove is the result that the critical random cluster model with $q\geq 1$ and the wired boundary condition on a quasi-transitive, non-amenable, unimodular graph almost surely has no infinite open clusters.

math.PR↗

arXiv2019

Modeling Event Background for If-Then Commonsense Reasoning Using Context-aware Variational Autoencoder

Understanding event and event-centered commonsense reasoning are crucial for natural language processing (NLP). Given an observed event, it is trivial for human to infer its intents and effects, while this type of If-Then reasoning still remains challenging for NLP systems. To facilitate this, a If-Then commonsense reasoning dataset Atomic is proposed, together with an RNN-based Seq2Seq model to conduct such reasoning. However, two fundamental problems still need to be addressed: first, the intents of an event may be multiple, while the generations of RNN-based Seq2Seq models are always semantically close; second, external knowledge of the event background may be necessary for understanding events and conducting the If-Then reasoning. To address these issues, we propose a novel context-aware variational autoencoder effectively learning event background information to guide the If-Then reasoning. Experimental results show that our approach improves the accuracy and diversity of inferences compared with state-of-the-art baseline methods.

cs.CL↗

arXiv2019

Cubic graphs and the golden mean

The connective constant $μ(G)$ of a graph $G$ is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. We investigate the validity of the inequality $μ\ge ϕ$ for infinite, transitive, simple, cubic graphs, where $ϕ:= \frac12(1+\sqrt 5)$ is the golden mean. The inequality is proved for several families of graphs including (i) Cayley graphs of infinite groups with three generators and strictly positive first Betti number, (ii) infinite, transitive, topologically locally finite (TLF) planar, cubic graphs, and (iii) cubic Cayley graphs with two ends. Bounds for $μ$ are presented for transitive cubic graphs with girth either $3$ or $4$, and for certain quasi-transitive cubic graphs.

math.CO↗

arXiv2019

ELG: An Event Logic Graph

The evolution and development of events have their own basic principles, which make events happen sequentially. Therefore, the discovery of such evolutionary patterns among events are of great value for event prediction, decision-making and scenario design of dialog systems. However, conventional knowledge graph mainly focuses on the entities and their relations, which neglects the real world events. In this paper, we present a novel type of knowledge base - Event Logic Graph (ELG), which can reveal evolutionary patterns and development logics of real world events. Specifically, ELG is a directed cyclic graph, whose nodes are events, and edges stand for the sequential, causal, conditional or hypernym-hyponym (is-a) relations between events. We constructed two domain ELG: financial domain ELG, which consists of more than 1.5 million of event nodes and more than 1.8 million of directed edges, and travel domain ELG, which consists of about 30 thousand of event nodes and more than 234 thousand of directed edges. Experimental results show that ELG is effective for the task of script event prediction.

cs.AI↗

arXiv2019

Self-avoiding walks and connective constants

The connective constant $μ(G)$ of a quasi-transitive graph $G$ is the asymptotic growth rate of the number of self-avoiding walks (SAWs) on $G$ from a given starting vertex. We survey several aspects of the relationship between the connective constant and the underlying graph $G$. $\bullet$ We present upper and lower bounds for $μ$ in terms of the vertex-degree and girth of a transitive graph. $\bullet$ We discuss the question of whether $μ\geϕ$ for transitive cubic graphs (where $ϕ$ denotes the golden mean), and we introduce the Fisher transformation for SAWs (that is, the replacement of vertices by triangles). $\bullet$ We present strict inequalities for the connective constants $μ(G)$ of transitive graphs $G$, as $G$ varies. $\bullet$ As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a non-trivial group element is declared to be a further generator. $\bullet$ We describe so-called graph height functions within an account of "bridges" for quasi-transitive graphs, and indicate that the bridge constant equals the connective constant when the graph has a unimodular graph height function. $\bullet$ A partial answer is given to the question of the locality of connective constants, based around the existence of unimodular graph height functions. $\bullet$ Examples are presented of Cayley graphs of finitely presented groups that possess graph height functions (that are, in addition, harmonic and unimodular), and that do not. $\bullet$ The review closes with a brief account of the "speed" of SAW.

math.CO↗

arXiv2019

Weighted self-avoiding walks

We study the connective constants of weighted self-avoiding walks (SAWs) on infinite graphs and groups. The main focus is upon weighted SAWs on finitely generated, virtually indicable groups. Such groups possess so-called 'height functions', and this permits the study of SAWs with the special property of being bridges. The group structure is relevant in the interaction between the height function and the weight function. The main difficulties arise when the support of the weight function is unbounded, since the corresponding graph is no longer locally finite. There are two principal results, of which the first is a condition under which the weighted connective constant and the weighted bridge constant are equal. When the weight function has unbounded support, we work with a generalized notion of the 'length' of a walk, which is subject to a certain condition. In the second main result, the above equality is used to prove a continuity theorem for connective constants on the space of weight functions endowed with a suitable distance function.

math.CO↗

arXiv2019

Learning to Rank for Plausible Plausibility

Researchers illustrate improvements in contextual encoding strategies via resultant performance on a battery of shared Natural Language Understanding (NLU) tasks. Many of these tasks are of a categorical prediction variety: given a conditioning context (e.g., an NLI premise), provide a label based on an associated prompt (e.g., an NLI hypothesis). The categorical nature of these tasks has led to common use of a cross entropy log-loss objective during training. We suggest this loss is intuitively wrong when applied to plausibility tasks, where the prompt by design is neither categorically entailed nor contradictory given the context. Log-loss naturally drives models to assign scores near 0.0 or 1.0, in contrast to our proposed use of a margin-based loss. Following a discussion of our intuition, we describe a confirmation study based on an extreme, synthetically curated task derived from MultiNLI. We find that a margin-based loss leads to a more plausible model of plausibility. Finally, we illustrate improvements on the Choice Of Plausible Alternative (COPA) task through this change in loss.

cs.CL↗

arXiv2019

Story Ending Prediction by Transferable BERT

Recent advances, such as GPT and BERT, have shown success in incorporating a pre-trained transformer language model and fine-tuning operation to improve downstream NLP systems. However, this framework still has some fundamental problems in effectively incorporating supervised knowledge from other related tasks. In this study, we investigate a transferable BERT (TransBERT) training framework, which can transfer not only general language knowledge from large-scale unlabeled data but also specific kinds of knowledge from various semantically related supervised tasks, for a target task. Particularly, we propose utilizing three kinds of transfer tasks, including natural language inference, sentiment classification, and next action prediction, to further train BERT based on a pre-trained model. This enables the model to get a better initialization for the target task. We take story ending prediction as the target task to conduct experiments. The final result, an accuracy of 91.8%, dramatically outperforms previous state-of-the-art baseline methods. Several comparative experiments give some helpful suggestions on how to select transfer tasks. Error analysis shows what are the strength and weakness of BERT-based models for story ending prediction.

cs.CL↗

arXiv2019

Simultaneous generation of two THz waves with bulk LiNbO3 and four THz waves with PPLN by coupled optical parametric generation

We present a theoretical research concerning simultaneous generation of two terahertz (THz) waves with bulk LiNbO3 and four THz waves with periodically poled LiNbO3 (PPLN) by coupled optical parametric generation (COPG). First, we investigate collinear phase matching of COPG generating two orthogonally polarized THz waves with two types of phase matching of o = e + e and o = e + o with bulk LiNbO3. The two orthogonally polarized THz waves are generated from stimulated polariton scattering (SPS) with A1 and E symmetric transverse optical (TO) modes in bulk LiNbO3, respectively. Then, we find that perturbations of phase mismatch for o = e + e and o = e + o can be compensated by a same grating vector of PPLN. As a result, four THz waves are simultaneously generated with a PPLN crystal and a pump laser. We calculate third-order nonlinear optical coefficients of o = e + o generating THz waves from E symmetric TO modes. The intensities of four THz waves are calculated by solving coupled wave equations. The calculation results demonstrate that the COPG generating four THz waves have high photon conversion efficiencies.

physics.optics↗

arXiv2018

Mixing time of Markov chains for the 1-2 model

A 1-2 model configuration is a subset of edges of a hexagonal lattice satisfying the constraint that each vertex is incident to 1 or 2 edges. We introduce Markov chains to sample the 1-2 model configurations on 2D hexagonal lattice and prove that the mixing time of these chains is polynomial in the sizes of the graphs for a large class of probability measures.

math.PR↗

arXiv2018

Positive speed self-avoiding walks on graphs with more than one end

A self-avoiding walk (SAW) is a path on a graph that visits each vertex at most once. The mean square displacement of an $n$-step SAW is the expected value of the square of the distance between the ending point and the starting point of an $n$-step SAW, where the expectation is taken with respect to the uniform measure on $n$-step SAWs starting from a fixed vertex. It is conjectured that the mean square displacement of an $n$-step SAW is asymptotically $n^{2ν}$, where $ν$ is a constant. Computing the exact values of the exponent $ν$ on various graphs has been a challenging problem in mathematical and scientific research for long. In this paper we show that on any locally finite Cayley graph of an infinite, finitely-generated group with more than two ends, the number of SAWs whose end-to-end distances are linear in lengths has the same exponential growth rate as the number of all the SAWs. We also prove that for any infinite, finitely-generated group with more than one end, there exists a locally finite Cayley graph on which SAWs have positive speed - this implies that the mean square displacement exponent $ν=1$ on such graphs. These results are obtained by proving more general theorems for SAWs on quasi-transitive graphs with more than one end, which make use of a variation of Kesten's pattern theorem in a surprising way, as well as the Stalling's splitting theorem. Applications include proving that SAWs have positive speed on the square grid in an infinite cylinder, and on the infinite free product graph of two connected, quasi-transitive graphs.

math-ph↗

arXiv2018

Fluctuations of dimer heights on contracting square-hexagon lattices

We study perfect matchings on the square-hexagon lattice with $1\times n$ periodic edge weights such that the boundary condition is given by either (1) each remaining vertex on the bottom boundary is followed by $(m-1)$ removed vertices; (2) the bottom boundary can be divided into finitely many alternating line segments where all the vertices along each line segment are either removed or remained. In Case (1), we show that under certain homeomorphism from the liquid region to the upper half plane, the height fluctuations converge to the Gaussian free field in the upper half plane. In Case (2), when the edge weights $x_1,\ldots,x_n$ in one period satisfy the condition that $x_{i+1}=O\left(\frac{x_i}{N^α}\right)$, where $α>0$ is a constant independent of $N$, we show that the height fluctuations converge to an independent Gaussian free field in each connected component of the liquid region.

math.PR↗

arXiv2018

Locality of connective constants

The connective constant $μ(G)$ of a quasi-transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. We prove a locality theorem for connective constants, namely, that the connective constants of two graphs are close in value whenever the graphs agree on a large ball around the origin (and a further condition is satisfied). The proof exploits a generalized bridge decomposition of self-avoiding walks, which is valid subject to the assumption that the underlying graph is quasi-transitive and possesses a so-called unimodular graph height function.

math.CO↗

arXiv2018

Constructing Narrative Event Evolutionary Graph for Script Event Prediction

Script event prediction requires a model to predict the subsequent event given an existing event context. Previous models based on event pairs or event chains cannot make full use of dense event connections, which may limit their capability of event prediction. To remedy this, we propose constructing an event graph to better utilize the event network information for script event prediction. In particular, we first extract narrative event chains from large quantities of news corpus, and then construct a narrative event evolutionary graph (NEEG) based on the extracted chains. NEEG can be seen as a knowledge base that describes event evolutionary principles and patterns. To solve the inference problem on NEEG, we present a scaled graph neural network (SGNN) to model event interactions and learn better event representations. Instead of computing the representations on the whole graph, SGNN processes only the concerned nodes each time, which makes our model feasible to large-scale graphs. By comparing the similarity between input context event representations and candidate event representations, we can choose the most reasonable subsequent event. Experimental results on widely used New York Times corpus demonstrate that our model significantly outperforms state-of-the-art baseline methods, by using standard multiple choice narrative cloze evaluation.

cs.AI↗

arXiv2017

Critical surface of the 1-2 model

The 1-2 model on the hexagonal lattice is a model of statistical mechanics in which each vertex is constrained to have degree either $1$ or $2$. There are three types of edge, and three corresponding parameters $a$, $b$, $c$. It is proved that, when $a \ge b \ge c > 0$, the surface given by $\sqrt a = \sqrt b + \sqrt c$ is critical. The proof hinges upon a representation of the partition function in terms of that of a certain dimer model. This dimer model may be studied via the Pfaffian representation of Fisher, Kasteleyn, and Temperley. It is proved, in addition, that the two-edge correlation function converges exponentially fast with distance when $\sqrt a \ne \sqrt b + \sqrt c$. Many of the results may be extended to periodic models.

math.PR↗

arXiv2016

Constrained percolation in two dimensions

We prove absence of infinite clusters and contours in a class of critical constrained percolation models on the square lattice. The percolation configuration is assumed to satisfy certain hard local constraints, but only weak symmetry and ergodicity conditions are imposed on its law. The proofs use new combinatorial techniques exploiting planar duality. Applications include absence of infinite clusters of diagonal edges for critical dimer models on the square-octagon lattice, as well as absence of infinite contours and infinite clusters for critical XOR Ising models on the square grid. We also prove that there exists at most one infinite contour for high-temperature XOR Ising models, and no infinite contour for low-temperature XOR Ising model.

math.PR↗