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Xing Liang

Publications and source records attributed to Xing Liang.

At least 19 recordsLinked to original sources
arXiv2026

Variational Principles and Rearrangement Inequalities for asymmetric Operators on Periodic Lattices

In this paper, we establish variational formulas and a rearrangement inequality for principal eigenvalues of asymmetric second-order difference operators on periodic lattices. For a general irreducible nearest-neighbor operator, positive right and left eigenvectors give an explicit saddle point and hence equal minimum--maximum and maximum--minimum formulas over positive profiles and periodic logarithmic correctors. The corrector is unique and satisfies a nonlinear discrete flux-conservation law. For exponentially tilted symmetric diffusion, the formula separates the discrete Dirichlet and potential terms from an exact nonlinear periodic correction. In one dimension the flux is constant, and an elementary scalar-flux representation yields a bell-shaped cyclic rearrangement that maximizes the tilted principal eigenvalue for every tilt.

math.AP↗
arXiv2026

On advective nonlocal operators: multiplicity of principal eigenpairs

We study the existence and multiplicity of principal eigenvalues and eigenfunctions for a periodically heterogeneous nonlocal dispersal model with advection. The operator we consider is resolvent-positive but not resolvent-compact; therefore, the classical Krein-Rutman theory cannot be applied directly. When the advection coefficient has a constant sign, we prove the existence and uniqueness of the principal eigenvalue and the corresponding normalized eigenfunction. In sharp contrast, when the advection does not have a constant sign, the problem is more involved and leads to surprising results. Depending on the coefficients of the equation, the principal eigenproblem can either have a unique normalized solution or a continuum of solutions, at the boundary of which there exists a principal eigenvector with a singular measure component. In the latter situation, all the constructed eigenvalues are embedded in the continuous spectrum of our operator. We completely characterize the eigenvalues associated with positive eigenvectors, even when the eigenvector is a Radon measure. Finally, we discuss an application to a nonlinear KPP-type equation with nonlocal dispersal, which possesses a continuum of nontrivial stationary solutions, a different behavior from the classical KPP equation with local diffusion.

math.AP↗
arXiv2026

Hybrid quantum-classical neural network for sentiment analysis

Quantum machine learning has recently emerged as a promising paradigm that leverages the expressive power of quantum circuits to address complex learning tasks. In this work, we investigate the applicability of hybrid quantum-classical neural networks to sentiment analysis, a central problem in natural language processing. We focus on a dataset of tweets related to COVID-19, where the textual content is vectorized using TF-IDF and fed into both classical feedforward networks and hybrid architectures incorporating parameterized quantum circuits. Our results show that hybrid models can achieve accuracy comparable to the classical baseline, while exhibiting distinct learning dynamics, especially in terms of validation loss and accuracy, that suggest a richer representational capacity. Moreover, when applying transfer learning to an SMS spam classification task, the hybrid models consistently outperform the classical counterpart, achieving an accuracy increase of 15 percentage points (from 66% to 81%) on the spam class, demonstrating enhanced generalization. These findings highlight the feasibility of employing QML for natural language processing and point toward the potential advantages of hybrid models as quantum hardware continues to advance.

cs.LG↗
arXiv2026

Spreading speeds for Fisher-KPP equations with slowly decaying initial data in an almost periodic setting

This paper investigates the long-times behavior of the Fisher-KPP equation with slowly decaying initial data in an almost periodic medium. We mainly focus on two classes of initial data: exponentially decaying initial data and inital data that decay more slowly than any exponential function. Employing the Hamilton-Jacobi approach, we provide a unified framwork for analyzing the Cauchy problem with initial data in both cases. We demonstrate that the level sets of the solution can be estimated by the generalized principal eigenvalue of the linearized operator and the decay rate of the initial data.

math.AP↗
arXiv2026

Spectral Geometry and Bosonic-Bloch Probes: Explorations in Quantum Learning

This paper studies how spectral geometry emerges in quantum learning models and how it can be diagnosed with physically grounded probes. In graph-regularized quantum networks, training reorganizes the output similarity graph, increases the effective spectral dimension Delta S = +0.23, and reshapes the Laplacian spectrum. Edge-resolved two-boson interference directly probes this restructuring: the bosonic enhancement Delta P_uv correlates with the Fiedler edge split |Delta v_2| (r = -0.50), linking learned spectral partitions to interference signatures. A phase diagram shows a nonmonotonic dependence of performance on coupling strength gamma and noise delta, with graph regularization improving fidelity only in a restricted regime; hardware experiments confirm the predicted interference behavior within shot-noise uncertainty. We also analyze a hybrid quantum autoencoder and introduce Bloch-space drift as a geometric diagnostic of its latent representation. With an unsupervised benign-data threshold, the model achieves high ranking performance (ROC-AUC about 0.99) and negligible false-negative rates. Absolute Bloch drift strongly discriminates anomalies (ROC-AUC at least about 0.9), while consecutive drift is near random (ROC-AUC about 0.5), showing that detection arises from persistent state-space displacement rather than local fluctuations. Through the geometry of reduced single-qubit states and associated quantum Fisher information, these results show that learning-induced spectral organization appears as measurable quantum-state structure, establishing a unified spectral-geometric framework for diagnosing quantum learning systems with bosonic and Bloch probes.

quant-ph↗
arXiv2026

Compression-Driven Anomaly Detection in Brain MRI Using an Interpretable Quantum Autoencoder

We study a quantum autoencoder (QAE) for compression-driven anomaly detection in brain MRI data. The approach leverages angle encoding to map image patches into quantum states, followed by a variational encoder-decoder architecture trained to discard information via auxiliary trash qubits. Anomaly scores reflect the degree to which inputs resist compression relative to normal data, with higher scores corresponding to deviations from the learned normal manifold. Evaluated on publicly available brain MRI DICOM datasets, the method achieves a slice-level ROC-AUC of approximately 0.95 and a patch-level ROC-AUC of approximately 0.813, outperforming classical autoencoder and PCA baselines. Analysis of the learned parameters reveals a pronounced encoder-decoder asymmetry, where effective anomaly detection arises from structured information compression within the encoder rather than increased parameter magnitude or decoder expressivity. This results in a controlled compression-reconstruction trade-off with a clear operating regime that supports principled threshold selection. Qualitative evaluation further shows that the QAE produces spatially localized anomaly heatmaps aligned with tumorous regions. The results, supported by promising baseline performances, demonstrate that quantum autoencoders provide an interpretable and controllable mechanism for anomaly detection based on incompressibility with respect to a learned latent representation. This work highlights the potential of quantum autoencoders as a principled tool for studying compression dynamics in quantum machine learning, with promising implications for decision support in medical imaging workflows.

quant-ph↗
arXiv2026

Quantum Generative Diffusion Model for Real-World Time Series

Generative models have achieved remarkable success in data synthesis, though recent advances driven by increasing model scale have introduced challenges in computational cost and efficiency. Quantum machine learning offers a promising alternative, representing complex data distributions using compact, highly expressive models. Here, we propose QDiffusion-TS, the first quantum generative diffusion model for time series synthesis, and validate it on the IQM quantum processor. The framework extends a classical diffusion architecture by replacing feed-forward components within the denoising transformer with quantum neural networks, yielding a hybrid quantum transformer that reduces the number of trainable parameters in each replaced component by nearly three orders of magnitude. Evaluated on financial time series from Apple and Amazon, the model generates synthetic data that more accurately reproduces the real distributions, reducing Wasserstein distance by approximately 44% relative to its classical counterpart across both datasets. In a downstream forecasting task, augmentation with the generated data improves predictive performance by up to 71% in RMSE over a baseline trained solely on real data. These results show that quantum enhanced architectures can consistently match and frequently surpass classical performance with substantially fewer parameters, establishing a practical framework towards more efficient and scalable data-driven generative modelling.

cs.LG↗
arXiv2025

Solving larger Travelling Salesman Problem networks with a penalty-free Variational Quantum Algorithm

The Travelling Salesman Problem (TSP) is a well-known NP-Hard combinatorial optimisation problem, with industrial use cases such as last-mile delivery. Although TSP has been studied extensively on quantum computers, it is rare to find quantum solutions of TSP network with more than a dozen locations. In this paper, we present high quality solutions in noise-free Qiskit simulations of networks with up to twelve locations using a hybrid penalty-free, circuit-model, Variational Quantum Algorithm (VQA). Noisy qubits are also simulated. To our knowledge, this is the first successful VQA simulation of a twelve-location TSP on circuit-model devices. Multiple encoding strategies, including factorial, non-factorial, and Gray encoding are evaluated. Our formulation scales as $\mathcal{O}(nlog_2(n))$ qubits, requiring only 29 qubits for twelve locations, compared with over 100 qubits for conventional approaches scaling as $\mathcal{O}(n^2)$. Computational time is further reduced by almost two orders of magnitude through the use of Simultaneous Perturbation Stochastic Approximation (SPSA) gradient estimation and cost-function caching. We also introduce a novel machine-learning model, and benchmark both quantum and classical approaches against a Monte Carlo baseline. The VQA outperforms the classical machine-learning approach, and performs similarly to Monte Carlo for the small networks simulated. Additionally, the results indicate a trend toward improved performance with problem size, outlining a pathway to solving larger TSP instances on quantum devices.

quant-ph↗
arXiv2025

Generalized principal eigenvalues of elliptic operators and spreading speeds of Fisher-KPP equations in two-scale almost periodic media

This paper is concerned with the asymptotic behavior of the generalized principal eigenvalues of elliptic operators and spreading speeds of Fisher-KPP equations in two-scale almost periodic media where one scale is fixed and another one approaches zero or infinity. We transform the problem into the homogenization of certain effective Hamiltonian and then establish the asymptotic limits and the convergence rates. Based on the analysis of the asymptotic behavior of effective Hamiltonians, we investigate how the heterogeneity of the advection and growth rates affect on the propagation in the case where the media has very rapid or slow spatial oscillation: We show a normal scale perturbation of the growth rate with mean zero can accelerate the propagation in the media with rapid or slow oscillation; and an advection with slow oscillation and mean zero can decelerate the propagation in 1-D case.

math.AP↗
arXiv2024

The influence of advection on the propagation phenomena of reaction-diffusion equations with KPP-bistable nonlinearity

This paper is devoted to propagation phenomena for a reaction-diffusion-advection equation in a one-dimensional heterogeneous environment, where heterogeneity is reflected by the nonlinearity term -- being KPP type on $(-\infty, -L]$ and being bistable type on $[L,+\infty)$ for some $L>0$. A comprehensive analysis is presented on the influence of advection and heterogeneous reactions, based on various values of the advection rate $c$. Denote by $c_m$ and $c_b$ the spreading speeds of KPP and bistable reactions, respectively. When $c>-c_m$, it is shown that propagation can always occur with leftward spreading speed $c_m+c$ and rightward spreading speed $\min\big(\max(c_b-c,0),c_m-c\big)$. Moreover, a logarithmic delay of the level sets in the left direction is discovered. When $c \le -c_m$, propagation phenomena are determined by the initial data and by the sign of $c_b$. In particular, when $c_b>0$, the leftward propagation speed is $c_b+c$ if the initial population is "large enough"; whereas extinction occurs if the initial value is located in the bistable region and is "relatively small". In addition, the attractiveness of the bistable traveling wave is obtained when the leftward spreading speed is $c_b+c$ and/or when the rightward spreading speed is $c_b-c$.

math.AP↗
arXiv2022

Bistable pulsating fronts in slowly oscillating environments *

We consider reaction-diffusion fronts in spatially periodic bistable media with large periods. Whereas the homogenization regime associated with small periods had been well studied for bistable or Fisher-KPP reactions and, in the latter case, a formula for the limit minimal speeds of fronts in media with large periods had also been obtained thanks to the linear formulation of these minimal speeds and their monotonicity with respect to the period, the main remaining open question is concerned with fronts in bistable environments with large periods. In bistable media the unique front speeds are not linearly determined and are not monotone with respect to the spatial period in general, making the analysis of the limit of large periods more intricate. We show in this paper the existence of and an explicit formula for the limit of bistable front speeds as the spatial period goes to infinity. We also prove that the front profiles converge to a family of front profiles associated with spatially homogeneous equations. The main results are based on uniform estimates on the spatial width of the fronts, which themselves use zero number properties and intersection arguments.

math.AP↗
arXiv2022

Almost-periodic ground state of the non-self-adjoint Jacobi operator and its applications

We study the ground states of the one-dimensional non-self-adjoint Jacobi operators in the almost periodic media by using the method of dynamical systems. We show the existence of the ground state. Particularly, in the quasi-periodic media, we show that the lower regularity of coefficients can guarantee the existence of ground states. Besides that, we give two applications: the first application is to show the existence and uniqueness of the positive steady state of the discrete Fisher-KPP type equation; the second application is to investigate the asymptotic behavior of the discrete stationary parabolic equation with large lower order terms.

math.DS↗
arXiv2021

Traveling fronts for Fisher-KPP lattice equations in almost periodic media

This paper investigates the existence of almost periodic traveling fronts for Fisher-KPP lattice equations in one-dimensional almost periodic media. By the Lyapunov exponent of the linearized operator near the unstable steady state, we give sufficient condition of the existence of minimal speed of traveling fronts. Furthermore, it is showed that almost periodic traveling fronts share the same recurrence property as the structure of the media. As applications, we give some typical examples which have minimal speed, and the proof of this depends on dynamical system approach to almost periodic Schrodinger operator.

math.AP↗
arXiv2020

A Multi-modal Machine Learning Approach and Toolkit to Automate Recognition of Early Stages of Dementia among British Sign Language Users

The ageing population trend is correlated with an increased prevalence of acquired cognitive impairments such as dementia. Although there is no cure for dementia, a timely diagnosis helps in obtaining necessary support and appropriate medication. Researchers are working urgently to develop effective technological tools that can help doctors undertake early identification of cognitive disorder. In particular, screening for dementia in ageing Deaf signers of British Sign Language (BSL) poses additional challenges as the diagnostic process is bound up with conditions such as quality and availability of interpreters, as well as appropriate questionnaires and cognitive tests. On the other hand, deep learning based approaches for image and video analysis and understanding are promising, particularly the adoption of Convolutional Neural Network (CNN), which require large amounts of training data. In this paper, however, we demonstrate novelty in the following way: a) a multi-modal machine learning based automatic recognition toolkit for early stages of dementia among BSL users in that features from several parts of the body contributing to the sign envelope, e.g., hand-arm movements and facial expressions, are combined, b) universality in that it is possible to apply our technique to users of any sign language, since it is language independent, c) given the trade-off between complexity and accuracy of machine learning (ML) prediction models as well as the limited amount of training and testing data being available, we show that our approach is not over-fitted and has the potential to scale up.

cs.CV↗
arXiv2018

Transition semi-wave solutions of reaction diffusion equations with free boundaries

In this paper, we define the transition semi-wave solution of the following reaction diffusion equation with free boundaries \begin{equation}\label{0.1} \left\{ \begin{aligned} u_{t}=u_{xx}+f(t,x,u),\ \ &t\in\Real, x 1, $$ we prove that the semi-wave connecting $1$ and $0$ is unique provided it exists. Furthermore, we prove that any bounded transition semi-wave connecting $1$ and 0 is exactly the semi-wave. In the cases where $f$ is KPP-Fisher type and almost periodic in time (space), i.e., $f(t,x,u)=u(c(t)-u)$ (resp. $u(a(x)-u)$) with $c(t)$ (resp. $a(x)$) being almost periodic, applying totally different method, we also prove any bounded transition semi-wave connecting the unique almost periodic positive solution of $u_{t}=u(c(t)-u)$ (resp. $u_{xx}+u(a(x)-u)=0$) and $0$ is exactly the unique almost periodic semi-wave. Finally, we provide an example of the heterogeneous equation to show the existence of the transition semi-wave without any global mean speeds.

math.AP↗
arXiv2018

Spreading speeds of KPP-type lattice systems in heterogeneous media

In this paper, we investigate spreading properties of the solutions of the Kolmogorov-Petrovsky-Piskunov-type, (to be simple,KPP-type) lattice system \begin{equation}\label{firstequation}\overset{.}u_{i}(t) =d^{\prime}_{i}(u_{i+1}(t)-u_{i}(t))+d_{i}(u_{i-1}(t)-u_{i}(t))+f(i,u_{i}).\end{equation} we develop some new discrete Harnack-type estimates and homogenization techniques for this lattice system to construct two speeds $\overlineω\leq \underline ω$ such that $\displaystyle{\lim_{t\rightarrow+\infty}}\sup \limits_{i\geqωt}|u_i(t)|=0$ for any $ω>\overlineω$, and $\displaystyle{\lim_{t\rightarrow+\infty}}\sup \limits_{0\leq i\leqωt}|u_i(t)-1|=0$ for any $ω<\underlineω$. These speeds are characterized by two generalized principal eigenvalues of the linearized systems. In particular, we derive the exact spreading speed when the coefficients are random stationary ergodic or almost periodic (where $\underlineω= \overlineω$). Finally, in the case where $f_{s}^{\prime}(i,0)$ is almost periodic in $i$ and the diffusion rate $d_i'=d_i$ is independent of $i$, we show that the spreading speeds in the positive and negative directions are identical even if $ f(i,u_{i})$ is not invariant with respect to the reflection.

math.AP↗
arXiv2018

Spreading speeds of nonlocal KPP equations in heterogeneous media

In this paper, we prove the existence of the spreading speed of nonlocal KPP equations in two cases: 1. The media is almost periodic and the kernel of diffusion is continuous; 2. The media is periodic and the diffusion is not continuous but weak irreducible. To do this, we develop the theory of generalized principal eigenvalues, the method homogenization and special Harnack's inequalities of nonlocal diffusion equations.

math.AP↗