arXiv subjects

Hai-Xiao Wang

Publications and source records attributed to Hai-Xiao Wang.

At least 19 recordsLinked to original sources

arXiv2026

Concentration of Regularized Sparse Random Matrices: Spectral Edge Bounds via Nonbacktracking Operators

In sparse random matrices, spectral outliers (eigenvalues and singular values located away from the bulk) emerge due to degree fluctuations: high degrees inflate the operator norm, while low column degrees reduce the least singular value. As proved by Feige and Ofek (2005) and Le, Levina, and Vershynin (2017), degree regularization enforces concentration at the expected norm scale. However, precise bounds incorporating the cutoffs remain unexplored and challenging since regularization introduces dependencies among entries. For the first time in the literature, we provide variance- and cutoff-dependent bounds for extreme singular values and eigenvalues of regularized inhomogeneous random matrices. In the absence of regularization, our lower bound for the least singular value matches the same leading constant obtained by Brailovskaya and van Handel (2024). Moreover, our error term vanishes under the milder condition $d/\log N\to\infty$, as opposed to their stronger requirement $d/(\log N)^4\to\infty$. A key ingredient is to extend spectral radius bounds for nonbacktracking matrices to the dependent setting. We build on approaches for independent cases established by Benaych-Georges, Bordenave, and Knowles (2020), as well as Dumitriu and Zhu (2024), and carefully handle edges traversed only once. Our proof framework separates deterministic spectral comparisons from probabilistic estimates: once Loewner inequalities and columnwise variance controls are established, the remaining probabilistic analysis boils down to verifying the graph moment conditions formulated in this paper. We hope this framework can be extended to handle general random matrices with more complex dependencies.

math.PR↗

arXiv2026

Majority Dynamics on Resampled Sparse Erdős--Rényi Graphs: Gaussian Winner Selection and Pace to Unanimity

We study the two-opinion majority dynamics process: at each time step, every vertex adopts the majority opinion among its neighbors, retaining its current opinion if there is a tie. Independently at each step, the interaction graph is resampled from the sparse Erdős--Rényi model $\mathbb G(N,p)$ with $p=b\log N/N$ and fixed $b>1$. Our results identify three regimes governed by the initial advantage $Δ_0=|B_0|-|R_0|$, where $|B_0|$ and $|R_0|$ denote the initial blue and red camps, respectively. First, an initial blue advantage above an explicit constant multiple of $N/\sqrt{\log N}$ leads to blue unanimity within two updates with high probability. Second, throughout the intermediate regime $\sqrt{N/\log N}\llΔ_0\lesssim N/\sqrt{\log N}$, we obtain explicit high-probability upper and lower bounds on the blue-unanimity time. Finally, uniformly in the critical window $Δ_0\sqrt p=O(1)$, the blue- and red-unanimity probabilities equal $Φ(\sqrt{2/π}\,Δ_0\sqrt p)+o(1)$ and $Φ(-\sqrt{2/π}\,Δ_0\sqrt p)+o(1)$, respectively, and unanimity is reached within $(1+o(1))\log N/\log\log N$ many updates with high probability. This resolves the resampled version of the \emph{optimal power-of-few} conjecture raised by Tran and Vu (2025).

math.PR↗

arXiv2026

Observation of biased random-flux-induced topological phase transition in gyromagnetic photonic crystals

The interplay between disorder and topological states has attracted growing interest. While previous studies have primarily addressed the effects of geometric or potential randomness, the exploration of topological phase transitions driven by random-flux remains experimentally elusive. Here, we report the first experimental realization of topological phase transitions driven by biased random-flux in gyromagnetic photonic crystals. By stochastically orienting the magnetization of constituent gyromagnetic rods, we implement a disordered Haldane model in which the sign of the next-nearest-neighbor hopping phases is randomly distributed. We demonstrate that the bulk band gap closes when the densities of positive and negative flux are balanced, i.e, restoring time-reversal symmetry in a statistical sense, and reopens when a net positive or negative flux is introduced. Microwave near-field measurements directly visualize the reversal of chiral edge states, confirming a transition between distinct topological phases. Our results establish a unique disorder-driven mechanism for realizing topological phase transitions and deepen our understanding of the interplay between disorder and topological phases in bosonic systems.

physics.optics↗

arXiv2026

Majority Dynamics on Assortative Sparse Stochastic Block Models

Majority dynamics is a two-opinion process in which each vertex repeatedly updates to the majority opinion among its neighbors. We study this process on a resampled sparse binary stochastic block model in the assortative regime. At each time step, a graph is sampled from the current opinion partition: vertices with the same opinion are joined with probability $α=a\log N/N$, while vertices with differing opinions are joined with probability $β=b\log N/N$, where $a>b>1$. Let $B_t$ and $R_t$ denote the blue and red camps at time $t$. We show that the weighted advantage $\widetildeΔ_t =b|B_t|-a|R_t|$, rather than the unweighted advantage $Δ_t=|B_t|-|R_t|$ alone, governs the pace to unanimity. Our results, which hold with high probability as $N\to\infty$, identify three regimes for blue unanimity under the initial blue advantage, i.e., $Δ_0>0$: constant time, subpolynomial time, and polynomial time. First, when $\widetildeΔ_0 \gtrsim -N/\sqrt{\log N}$, blue unanimity occurs within three updates. Second, when $\widetildeΔ_0 < 0$ and $|\widetildeΔ_0| = o(N)$, blue unanimity occurs within $N^{o(1)}$ updates. Furthermore, when $\widetildeΔ_0 < 0$, $|\widetildeΔ_0| = O(N)$, and $Δ_0\gg\sqrt{N/\log N}$, blue unanimity still occurs within $N^{I_0+o(1)}$ updates, where \[ I_0= \left(\mathbf{ReLU}\Big(\sqrt{a\frac{|R_0|}{N}}-\sqrt{b\frac{|B_0|}{N}}\Big)\right)^2, \] and $\mathbf{ReLU}(x)=\max\{x,0\}$. Conversely, away from the weighted threshold, when $|B_0|/|R_0|\le a/b-κ$ and $Δ_0>0$, $N^{I_0 - o(1)}$ updates are necessary for blue unanimity. Our analysis relies on detailed estimates for one-vertex flip probabilities in sparse binomial differences, which could be of independent interest.

math.PR↗

arXiv2026

Partial recovery and weak consistency in the non-uniform hypergraph Stochastic Block Model

We consider the community detection problem in sparse random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), a general model of random networks with community structure and higher-order interactions. When the random hypergraph has bounded expected degrees, we provide a spectral algorithm that outputs a partition with at least a $γ$ fraction of the vertices classified correctly, where $γ\in (0.5,1)$ depends on the signal-to-noise ratio (SNR) of the model. When the SNR grows slowly as the number of vertices goes to infinity, our algorithm achieves weak consistency, which improves the previous results in Ghoshdastidar and Dukkipati (2017) for non-uniform HSBMs. Our spectral algorithm consists of three major steps: (1) Hyperedge selection: select hyperedges of certain sizes to provide the maximal signal-to-noise ratio for the induced sub-hypergraph; (2) Spectral partition: construct a regularized adjacency matrix and obtain an approximate partition based on singular vectors; (3) Correction and merging: incorporate the hyperedge information from adjacency tensors to upgrade the error rate guarantee. The theoretical analysis of our algorithm relies on the concentration and regularization of the adjacency matrix for sparse non-uniform random hypergraphs, which can be of independent interest.

math.ST↗

arXiv2026

Optimal and exact recovery on the general nonuniform Hypergraph Stochastic Block Model

Consider the community detection problem in random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), where each hyperedge appears independently with some given probability depending only on the labels of its vertices. We establish, for the first time in the literature, a sharp threshold for exact recovery under this non-uniform case, subject to minor constraints; in particular, we consider the model with multiple communities. One crucial point here is that by aggregating information from all the uniform layers, we may obtain exact recovery even in cases when this may appear impossible if each layer were considered alone. Besides that, we prove a wide-ranging, information-theoretic lower bound on the number of misclassified vertices \emph{for any algorithm}, depending on a \emph{generalized Chernoff-Hellinger} divergence involving model parameters. We provide two efficient algorithms which successfully achieve exact recovery when above the threshold, and attain the lowest possible mismatch ratio when the exact recovery is impossible, proved to be optimal. The theoretical analysis of our algorithms relies on the concentration and regularization of the adjacency matrix for non-uniform random hypergraphs, which could be of independent interest. We also address some open problems regarding parameter knowledge and estimation.

math.ST↗

arXiv2026

Attention Mechanisms Through the Lens of Numerical Methods: Approximation Methods and Alternative Formulations

The attention mechanism is the computational core of modern Transformer architectures, but its quadratic complexity in the input sequence length is the bottleneck for large-scale inference. This has motivated a rapidly growing body of work aimed at accelerating attention through approximation and reformulation. In this survey, we revisit attention mechanisms through the lens of numerical analysis, with a particular emphasis on tools and perspectives from numerical linear algebra. Our goal is twofold: first, we aim to systematically review and classify fast approximation methods according to the numerical principles they exploit. These include sparsity and clustering approaches, low-rank and subspace projection techniques, randomized sketching methods, and tensor-based decompositions. We also discuss kernel-inspired reformulations of attention and recent architectural variants, such as Latent Attention, that modify the standard softmax formulation to improve efficiency. Second, by presenting these developments within a unified mathematical framework, we aim to bridge the gap between disciplines and highlight opportunities for further contributions from computational mathematics, particularly numerical linear algebra, to the design of scalable attention mechanisms.

math.NA↗

arXiv2025

Antichiral surface states and higher-order topological states based on a modified Haldane model

Antichiral surface states, characterized by unidirectional propagation on parallel surfaces, offer unique potential for controlling classical waves. However, their realization typically relies on complex implementations of the two-dimensional modified Haldane model, limiting practical applications. Here, we propose a simplified scheme to realize such states within the nodal-line semimetal phase of a single-layer honeycomb lattice, by emulating the essential physics of the modified Haldane model through an introduced layer degree of freedom. Furthermore, we demonstrate that unequal vertical interlayer couplings can generate valley higher-order topological partial bandgaps, hosting coexisting one-dimensional hinge states and gapped antichiral surface states. We numerically verify these multiple topological states in acoustic crystals, establishing a versatile platform for advanced wave manipulation.

physics.app-ph↗

arXiv2025

Information-Theoretic Limits and Strong Consistency on Binary Non-uniform Hypergraph Stochastic Block Models

We investigate the unsupervised node classification problem on random hypergraphs under the non-uniform Hypergraph Stochastic Block Model (HSBM) with two equal-sized communities. In this model, edges appear independently with probabilities depending only on the labels of their vertices. We identify the threshold for strong consistency, expressed in terms of the Generalized Hellinger distance. Below this threshold, strong consistency is impossible, and we derive the Information-Theoretic (IT) lower bound on the expected mismatch ratio. Above the threshold, the parameter space is typically divided into two disjoint regions. When only the aggregated adjacency matrices are accessible, while one-stage algorithms accomplish strong consistency with high probability in the region far from the threshold, they fail in the region closer to the threshold. We propose a new refinement algorithm which, in conjunction with the initial estimation, provably achieves strong consistency throughout the entire region above the threshold, and attains the IT lower bound when below the threshold, proving its optimality. This novel refinement algorithm applies the power iteration method to a weighted adjacency matrix, where the weights are determined by hyperedge sizes and the initial label estimate. Unlike the constant degree regime where a subset selection of uniform layers is necessary to enhance clustering accuracy, in the scenario with diverging degrees, each uniform layer contributes non-negatively to clustering accuracy. Therefore, aggregating information across all uniform layers yields better performance than using any single layer alone.

math.ST↗

arXiv2025

Manipulation of photonic topological edge and corner states via trivial claddings

Crystalline symmetry offers a powerful tool to realize photonic topological phases, in which additional trivial claddings are typically required to confine topological boundary states. However, the utility of the trivial cladding in manipulating topological waves is often overlooked. Here, we demonstrate two topologically distinct kagome photonic crystals (KPCs) based on different crystalline symmetries: \mathbit{C}_\mathbf{6}- symmetric KPCs exhibit a quantum spin Hall phase, while \mathbit{C}_\mathbf{3}-symmetric KPCs serve as trivial cladding. By tuning the geometric parameter of the trivial cladding, we observe that a pair of topological interface states featured with pseudospin-momentum locking undergoes a phase transition, accompanied by the appearance and disappearance of corner states in a finite hexagonal supercell. Such a geometry-induced band inversion is characterized by a sign change in the Dirac mass of the topological interface states and holds potential for applications such as rainbow trapping. Furthermore, we experimentally demonstrate the corner states, which is a hallmark of higher-order topology, also depend critically on the trivial cladding. Our work highlights the crucial role of trivial claddings on the formation of topological boundary states, and offers a novel approach for their manipulation.

physics.optics↗

arXiv2025

Singular values of sparse random rectangular matrices: Emergence of outliers at criticality

Consider the random bipartite Erdős-Rényi graph $\mathbb{G}(n, m, p)$, where each edge with one vertex in $V_{1}=[n]$ and the other vertex in $V_{2} =[m]$ is connected with probability $p$, and $n=\lfloor γm\rfloor$ for a constant aspect ratio $γ\geq 1$. It is well known that the empirical spectral measure of its centered and normalized adjacency matrix converges to the Marčenko-Pastur (MP) distribution. However, largest and smallest singular values may not converge to the right and left edges, respectively, especially when $p = o(1)$. Notably, it was proved by Dumitriu and Zhu (2024) that there are almost surely no singular value outside the compact support of the MP law when $np = ω(\log(n))$. In this paper, we consider the critical sparsity regime where $p = b\log(n)/\sqrt{mn}$ for some constant $b>0$. We quantitatively characterize the emergence of outlier singular values as follows. For explicit $b_{*}$ and $b^{*}$ as functions of $γ$, we prove that when $b > b_{*}$, there is no outlier outside the bulk; when $b^{*}< b < b_{*}$, outliers are present only outside the right edge of the MP law; and when $b < b^{*}$, outliers are present on both sides, all with high probability. Moreover, the locations of those outliers are precisely characterized by a function depending on the largest and smallest degree vertices of the random graph. We estimate the number of outliers as well. Our results follow the path forged by Alt, Ducatez and Knowles (2021), and can be extended to sparse random rectangular matrices with bounded entries.

math.PR↗

arXiv2024

Optimal Exact Recovery in Semi-Supervised Learning: A Study of Spectral Methods and Graph Convolutional Networks

We delve into the challenge of semi-supervised node classification on the Contextual Stochastic Block Model (CSBM) dataset. Here, nodes from the two-cluster Stochastic Block Model (SBM) are coupled with feature vectors, which are derived from a Gaussian Mixture Model (GMM) that corresponds to their respective node labels. With only a subset of the CSBM node labels accessible for training, our primary objective becomes the accurate classification of the remaining nodes. Venturing into the transductive learning landscape, we, for the first time, pinpoint the information-theoretical threshold for the exact recovery of all test nodes in CSBM. Concurrently, we design an optimal spectral estimator inspired by Principal Component Analysis (PCA) with the training labels and essential data from both the adjacency matrix and feature vectors. We also evaluate the efficacy of graph ridge regression and Graph Convolutional Networks (GCN) on this synthetic dataset. Our findings underscore that graph ridge regression and GCN possess the ability to achieve the information threshold of exact recovery in a manner akin to the optimal estimator when using the optimal weighted self-loops. This highlights the potential role of feature learning in augmenting the proficiency of GCN, especially in the realm of semi-supervised learning.

cs.LG↗

arXiv2024

Higher-order topological phases in bilayer phononic crystals and topological bound states in the continuum

Recent studies on the interplay between band topology and layer degree of freedom provide an effective way to realize exotic topological phases. Here we systematically study the $C_6$- and $C_3$-symmetric higher-order topological phases in bilayer spinless tight-binding lattice models. For concreteness, we consider bilayer phononic crystals as the realizations of these models. We find that for mirror-symmetric-stacking bilayer lattices, the interlayer couplings control the emergence and disappearance of the topological bound states in the continuum where we consider the corner states as possible bound states in the bulk continuum. For the bilayer phononic crystals formed by two different lattices with identical symmetry, the band topology is determined by both the band topology of each layer as well as their mutual couplings. The bilayer phononic crystals experience a phase transition from nontrivial to trivial band topology when the interlayer couplings are gradually increased. Our work unveils the rich physics and topological phases emerging in bilayer lattice systems that can be used to engineer interesting phenomena and induce emergent topological phases.

cond-mat.mes-hall↗

arXiv2023

Topological light guiding and trapping via shifted photonic crystal interfaces

Photonic crystals (PCs) are periodic dielectric structures that severed as an excellent platform to manipulate light. A conventional way to guide/trap light via PCs is to introduce a line or point defect by removing or modifying several unit cells. Here we show that the light can be effectively guided and trapped in the glided photonic crystal interfaces (GPCIs). The projected band gap of GPCIs, which depends on the glide parameter, is characterized by a Dirac mass. Interestingly, the GPCIs with zero Dirac mass is a glide-symmetric waveguide featured with excellent transmission performance even in the presence of sharp corners and disorders. Moreover, placing two GPCIs with opposite Dirac mass together results in a photonic bound state due to the Jackiw-Rebbi theory. Our work provides an alternative way towards the design of ultracompact photonic devices such as GPCIs-induced coupled cavity-waveguide system and waveguide splitter.

physics.optics↗

arXiv2023

Topological phononic metamaterials

The concept of topological energy bands and their manifestations have been demonstrated in condensed matter systems as a fantastic paradigm toward unprecedented physical phenomena and properties that are robust against disorders. Recent years, this paradigm was extended to phononic metamaterials (including mechanical and acoustic metamaterials), giving rise to the discovery of remarkable phenomena that were not observed elsewhere thanks to the extraordinary controllability and tunability of phononic metamaterials as well as versatile measuring techniques. These phenomena include, but not limited to, topological negative refraction, topological 'sasers' (i.e., the phonon analog of lasers), higher-order topological insulating states, non-Abelian topological phases, higher-order Weyl semimetal phases, Majorana-like modes in Dirac vortex structures and fragile topological phases with spectral flows. Here we review the developments in the field of topological phononic metamaterials from both theoretical and experimental perspectives with emphasis on the underlying physics principles. To give a broad view of topological phononics, we also discuss the synergy with non-Hermitian effects and cover topics including synthetic dimensions, artificial gauge fields, Floquet topological acoustics, bulk topological transport, topological pumping, and topological active matters as well as potential applications, materials fabrications and measurements of topological phononic metamaterials. Finally, we discuss the challenges, opportunities and future developments in this intriguing field and its potential impact on physics and materials science.

cond-mat.mes-hall↗

arXiv2022

Hybrid topological photonic crystals

Photonic topological phases offering unprecedented manipulation of electromagnetic waves have attracted much research interest which, however, have been mostly restricted to a single band gap. Here, we report on the experimental discovery of hybrid topological photonic crystals which host simultaneously quantum anomalous Hall and valley Hall phases in different photonic band gaps. The underlying hybrid topological phase manifests itself in the edge responses as the coexistence of the chiral edge states and valley Hall edge states in different frequency ranges. We experimentally verify such an emergent phenomenon and show that such a feature enables novel multiplexing of photon transport in the edge channels. Our study reveals a situation with coexisting topology of distinct nature in a single photonic system that may enable frequency-dependent filtering and manipulation of topological edge photons.

cond-mat.mes-hall↗

arXiv2021

Possible Realization of Optical Quadratic and Dirac Points in Woodpile Photonic Crystals

The simulation of fermionic relativistic physics, e.g., Dirac and Weyl physics, has led to the discovery of many unprecedented phenomena in photonics, of which the optical-frequency realization is, however, still challenging. Here, surprisingly, we discover that the woodpile photonic crystals commonly used for optical frequency applications host exotic fermion-like relativistic degeneracies: a Dirac nodal line and a fourfold quadratic point, as protected by the nonsymmorphic crystalline symmetry. Deforming the woodpile photonic crystal leads to the emergence of type-II Dirac points from the fourfold quadratic point. Such type-II Dirac points can be detected by its anomalous refraction property which is manifested as a giant birefringence in a slab setup. Our findings provide a promising route towards 3D optical Dirac physics in all-dielectric photonic crystals.

physics.optics↗

arXiv2021

Higher-order topological phases in tunable $C_3$-symmetric photonic crystals

We demonstrate that multiple higher-order topological transitions can be triggered via the continuous change of the geometry in kagome photonic crystals composed of three dielectric rods. By tuning a single geometry parameter, the photonic corner and edge states emerge or disappear with the higher-order topological transitions. Two distinct higher-order topological insulator phases and a normal insulator phase are revealed. Their topological indices are obtained from symmetry representations. A photonic analog of fractional corner charge is introduced to distinguish the two higher-order topological insulator phases. Our predictions can be readily realized and verified in configurable dielectric photonic crystals.

physics.optics↗