arXiv subjects

Zuoqiang Shi

Publications and source records attributed to Zuoqiang Shi.

At least 19 recordsLinked to original sources

arXiv2026

A New Formulation of the Elastica Model with Application to Image Inpainting

Curves and interfaces play a central role in computer vision and image processing, and Euler's elastica energy is one of the most successful tools to model and manipulate them. The elastica energy is usually computed through a level-set or a phase field representation of the curve, in which the curvature term makes the energy nonlinear, non-convex and, in the level-set case, non-differentiable, so that its minimization is notoriously difficult. In this paper, we propose a new formulation of the elastica model based on a binary representation of curves. The key ingredient is a very simple formula for the curvature: if $ϕ$ is the $\pm1$ binary function associated with a curve $Γ$, then along $Γ$ the curvature is given by $-\frac{1}{C_0ε}\left(R_ε*ϕ\right)+O(ε)$, where $R_ε$ is a radial kernel supported in a ball of radius $ε$ and $C_0$ is an explicit constant. No derivative of $ϕ$ is involved and the convolution is evaluated by FFT. The resulting energy is a weighted total variation, so that formally the new formulation is a generalization of the well-known ROF model; a phase field type model is obtained from the same curvature formula. Combining FFT with the split Bregman iteration, which for the present problem coincides with an augmented Lagrangian method (ALM), we obtain efficient algorithms whose subproblems are either linear systems solved by conjugate gradient or explicit pointwise formulas. Numerical experiments on the inpainting of binary images are presented to demonstrate the effectiveness of the proposed algorithms.

cs.CV↗

arXiv2026

Reconstruction and Range Characterization for a Directional Confocal Non-Line-of-Sight Imaging Model

We study the reconstruction of a directional albedo field from confocal non-line-of-sight measurements. For mirror-symmetric vector fields in Sobolev spaces, radial preprocessing reduces the data to spherical means of the divergence with centers on the relay wall. Full data determine this divergence uniquely, with divergence-free fields forming the entire ambiguity. An explicit Fourier--sine formula recovers the irrotational Helmholtz component and reproduces the data. The Fourier--sine transforms of the model data form a weighted Hilbert space that we characterize exactly, with norm equal to the Sobolev norm of the reconstructed field. This weighted range remains well defined even when the preprocessed data fail to belong to standard Sobolev spaces. For Schwartz fields in the model class, such Sobolev regularity holds exactly when the divergence has zero depth integral. Measurements on an open subset of the relay wall, for all radii $0<r<R_{\max}$, uniquely determine the divergence in the union of the corresponding balls. An FFT-based algorithm with Stolt interpolation implements the reconstruction in $O(N^3\log N)$ operations on an $N^3$ grid. Tests on explicitly defined fields assess reconstruction accuracy and invariance under divergence-free perturbations. We present imaging examples of potential and vector-field reconstruction from model-generated, rendered, and measured transients.

math.NA↗

arXiv2026

Weighted Laplacian Flow: A Deterministic Particle Flow with Exponential Convergence

We introduce weighted Laplacian flow (WLF), a deterministic particle-flow framework for sampling from a target distribution known only up to normalization. The proposed method evolves the logarithmic density ratio through a transport equation and determines the particle velocity from a target-weighted Poisson problem, resulting in a nonlocal and kernel-free mechanism for redistributing mass. For bounded smooth domains, we establish global well-posedness of the flow for Lipschitz initial data and prove that the flow contracts the discrepancy between the evolving and target densities at an exact exponential rate in terms of the oscillation of the log-density ratio, as well as in the associated projective metric. The relative entropy also satisfies a precise dissipation relation involving the two directional Kullback--Leibler divergences. In addition, we identify a variational interpretation of WLF. The method can be viewed as a gradient flow of the reverse Kullback--Leibler divergence under a target-anchored metric. Consequently, the method achieves an explicit relaxation scale without requiring log-concavity or spectral-gap assumptions on the target distribution. Numerical experiments on multimodal, heavy-tailed, and high-dimensional targets demonstrate the effectiveness of the proposed approach in capturing long-range mass transport.

math.NA↗

arXiv2026

Accelerating Langevin Dynamics with Weighted Laplacian Flow: Global Well-Posedness and Exponential Convergence

We study an accelerated Langevin equation in which the classical overdamped Langevin drift is supplemented by a nonlocal gradient velocity determined through a target-weighted Poisson equation. The target $p$ is only assumed to be strictly positive and need not be normalized. Writing the dynamics in terms of $ω=\log(p/q)$ yields a coupled parabolic-elliptic system whose parabolic component contains the usual Langevin diffusion, whereas the elliptic component generates a global transport. The system is invariant under multiplication of $p$ by a positive constant. On a bounded smooth domain with homogeneous Neumann boundary conditions, we establish global existence and uniqueness of classical solutions for strictly positive $C^{2+α}$ target weights and compatible $C^{2+α}$ initial data. For long-time behavior, the oscillation of $ω$ decays at the explicit rate $e^{-t}$ and the solution density exponentially converges to target density. Moreover, an exact entropy-dissipation identity holds relative to the target density, in which the relative Fisher information and the Jeffreys divergence give the two dissipation mechanisms associated with the Langevin and weighted-Poisson components. In particular, KL divergence also converges to zero with rate $e^{-t}$ without a convexity assumption on the target.

math.PR↗

arXiv2026

High-Order Quadrature for Implicitly Defined Surfaces and Regions

We propose a high-order quadrature method for integrating over curves, surfaces, and enclosed regions defined implicitly as zero level sets of smooth functions. The method combines a mesh adjustment procedure with local change-of-variables parametrizations on cut elements. The mesh adjustment moves background mesh vertices away from the interface, ensuring a simple and nondegenerate intersection pattern between the level set and the simplicial mesh. On each cut element, the interface is parametrized by solving one-dimensional nonlinear equations along prescribed rays, reducing surface and region integration to standard Gauss--Legendre quadrature on fixed reference domains. The resulting curve and surface quadrature rules are local, nonrecursive, and have strictly positive weights. We prove high-order accuracy for curve, surface, and region integrals in two and three dimensions under natural smoothness and mesh consistency assumptions. Numerical experiments on representative implicit geometries confirm the predicted convergence rates.

math.NA↗

arXiv2026

A High-Order Surface Finite Element Method Based on Intersections with Background Tetrahedral Meshes

This paper develops a high-order surface finite element method for elliptic equations posed on a smooth closed surface implicitly defined as the zero level set of a function in three dimensions. In contrast to classical surface finite element methods that start from a prescribed triangulation of the surface, the proposed method constructs the discrete surface space from the intersections between the exact surface and an ambient tetrahedral mesh. More precisely, an active tetrahedral shell is generated around the implicit surface, and each cut tetrahedron contributes either a triangular or a quadrilateral surface patch according to its intersection pattern with the zero level set. The finite element space is first defined on the resulting piecewise planar cut surface and is then lifted to the exact surface by local level-set-based parameterizations. The resulting method combines features of surface FEM and unfitted/trace methods. Like surface FEM, it produces a conforming finite element space on the exact surface after lifting; however, the geometry and finite element space are induced by intersections with background tetrahedra like unfitted/trace methods. We prove that the local lifting maps agree pointwise across common faces and assemble into a global homeomorphism. We also derive explicit formulas for tangent vectors, Gram matrices, mass and stiffness integrals, and prove stability and high-order derivative estimates for the lifting maps. The latter estimates are formulated in broken Sobolev norms and lead to a Cea-type energy-norm error bound for the proposed high-order lifted surface finite element method.

math.NA↗

arXiv2026

Ultrasound Tomography of Musculoskeletal Tissues with Generative Neural Physics

Ultrasound Tomography (UT) is a radiation-free, high-resolution modality, but remains limited for musculoskeletal imaging due to the high computational cost and instability of full-waveform inversion in strongly scattering media. We propose a generative neural physics framework that couples generative networks with physics-informed neural simulation for fast, high-fidelity 3D UT. By learning a compact surrogate of ultrasonic wave propagation from a limited set of cross-modality images, our method merges the accuracy of wave modeling with the efficiency and stability of deep learning. This enables accurate quantitative imaging of in vivo musculoskeletal tissues, producing spatial maps of acoustic properties beyond reflection-mode images. On synthetic and in vivo data of breasts, arms, and legs, we reconstruct 3D maps of tissue parameters in under ten minutes, with sensitivity to acoustic variations in musculoskeletal tissues and resolution comparable to MRI. By overcoming computational bottlenecks in strongly scattering regimes, this approach demonstrates the feasibility of quantitative UT for musculoskeletal imaging and advances its development toward future routine clinical use.

cs.CV↗

arXiv2026

Neural Born Series Operator for Biomedical Ultrasound Computed Tomography

Ultrasound Computed Tomography (USCT) provides a radiation-free option for high-resolution clinical imaging. Despite its potential, the computationally intensive Full Waveform Inversion (FWI) required for tissue property reconstruction limits its clinical utility. This paper introduces the Neural Born Series Operator (NBSO), a novel technique designed to speed up wave simulations, thereby facilitating a more efficient USCT image reconstruction process through an NBSO-based FWI pipeline. Thoroughly validated on comprehensive brain and breast datasets, simulated under experimental USCT conditions, the NBSO proves to be accurate and efficient in both forward simulation and image reconstruction. This advancement demonstrates the potential of neural operators in facilitating near real-time USCT reconstruction, making the clinical application of USCT increasingly viable and promising.

eess.IV↗

arXiv2026

UNIFUSION: Adapting Autoregressive Language Models into Discrete Diffusion under a Unified Reverse-Rate Objective

Existing methods mainly adapt pretrained autoregressive (AR) language models to masked diffusion, whereas we directly adapt them to uniform-noise diffusion, where every token remains editable during sampling. However, adapting AR checkpoints across corruption kernels remains challenging because existing DLMs use different objectives and prediction parameterizations. We establish connections among SEDD, MDLM/GIDD, M2S, and Neural CTMC by expressing their conditional losses as a single generalized Kullback--Leibler objective over model reverse rates. We further derive conversions from clean-token predictions to concrete-score, posterior-mean, and exit-rate/jump parameterizations, yielding a shared $x_0$ interface that supports switching between mask and uniform kernels. Building on these connections, we propose \ours{}, a simple continual pre-training approach for directly adapting pretrained GPT2 checkpoints to uniform-noise diffusion. Through systematic evaluation of 124M- and 355M-parameter models, we show that \ours{} steadily improves the trade-off between generative perplexity (GenPPL) and unigram entropy as the sampling budget increases from 16 to 256 steps. At 256 steps, \ours{}-S and \ours{}-M achieve GenPPL/entropy pairs of $97.783/5.2626$ and $71.516/5.6669$, respectively; no evaluated model at the same scale simultaneously outperforms \ours{} on both metrics. At both scales, \ours{} also achieves the highest WinoGrande, SIQA, and BBH accuracy among the compared diffusion models.

cs.LG↗

arXiv2026

Mean-to-Score Discrete Diffusion: Posterior-Mean Denoisers for Score Entropy

Score Entropy Discrete Diffusion (SEDD) parameterizes discrete reverse processes with unconstrained positive score ratios. While positivity guarantees nonnegative reverse jump rates, it does not ensure Bayes realizability: ratios at a noisy state need not be jointly induced by any clean-token posterior under the forward kernel. The score-entropy loss has the correct population optimum but does not enforce this constraint away from it. In a trained pure-uniform SEDD checkpoint, roughly one quarter of complete score vectors violate the coordinate box, while more than half lie inside it yet remain materially incompatible with any valid posterior. Such violations can produce negative pre-normalization weights in finite-step sampling. Projecting raw scores onto the bridge polytope removes all observed negative weights and improves external generative PPL from $203.6$ to $175.1$ without changing the sampler. We introduce \emph{mean-to-score} (M2S), which predicts a clean-token posterior mean and converts it to the score through an exact kernel-dependent linear map. The construction applies to any known coordinate-wise continuous-time Markov chain (CTMC) satisfying a mild support condition. For uniform corruption, it maps the probability simplex onto the bridge polytope; for absorbing-mask corruption, the resulting objective recovers MD4 exactly. In a controlled 28.4M-parameter CIFAR-10 comparison, M2S lowers test BPD from $3.173$ to $3.129$ and FID-50k from $\CifarSEDDFID$ to $\CifarMtwoSFID$. A 170M-parameter M2S model trained on about 262B OpenWebText token slots outperforms the evaluated pure-uniform SEDD, GIDD, and Neural CTMC checkpoints at every tested sampling budget, reaching generative PPL $143.3$ at 128 steps versus $183.6$ for the strongest pure-uniform baseline.

cs.LG↗

arXiv2026

OLEDLM: A Unified Language Model for OLED Molecular Design

The development of organic light-emitting diode (OLED) materials faces the compounded challenges of an astronomically large chemical space, stringent quantum-chemical constraints, and a scarcity of labeled data. Although the question of OLED generation is important, few models have been trained effectively for this specific domain. We propose an inverse molecular design framework based on causal language models: given target optoelectronic properties (e.g., excitation energy, oscillator strength), our model directly generates OLED SMILES sequences satisfying the specified constraints. We employ a multi-stage strategy: first, we establish a foundational chemical language model using a LLaMA-style transformer architecture. To the best of our knowledge, this represents the first successful adaptation of LLMs specifically for the OLED domain, bridging the gap between generic molecular generation and the stringent structural requirements of optoelectronic materials. Second, we fine-tune property predictors based on a BERT model pre-trained on our large-scale OLED dataset. Then, we perform Reinforcement Learning on our fine-tuned model, leveraging our property predictor, for better SMILES generation. Finally, through DFT verification, we demonstrate that our framework can efficiently navigate the OLED chemical space, generating novel candidates with high structural validity and optimized optoelectronic properties.

cs.LG↗

arXiv2026

Non-line-of-sight imaging with arbitrary relay surface geometries via 3D Gaussian Transient Rendering

Imaging objects hidden outside the direct line of sight expands the effective field of view and is critical for applications such as autonomous driving and robotic perception. Despite impressive progress in time-of-flight (ToF)-based non-line-of-sight (NLOS) imaging, real-world deployment remains challenging because practical measurements are often collected over spatially limited, arbitrarily shaped relay regions-conditions that violate the planar-wall and dense-sampling assumptions made by most existing methods. To address these limitations, we propose a LOS-guided NLOS imaging pipeline that imposes no geometric assumptions on the relay surface and naturally supports both confocal and non-confocal configurations. Our method represents the hidden scene using 3D Gaussian primitives and couples them with an efficient, differentiable transient rendering model, enabling end-to-end optimization directly from measured transients. We validate our approach on real-world measurements from both a public dataset and a custom-built capture system. Across settings, our method achieves state-of-the-art reconstruction fidelity under spatially limited, sparsely sampled conditions, and significantly outperforms existing methods on complex, arbitrary relay surface geometries.

physics.optics↗

arXiv2026

Generalization Error Curves for Analytic Spectral Algorithms under Power-law Decay

The generalization error curve of certain kernel regression method aims at determining the exact order of generalization error with various source condition, noise level and choice of the regularization parameter rather than the minimax rate. In this work, under mild assumptions, we rigorously provide a full characterization of the generalization error curves of the kernel gradient descent method (and a large class of analytic spectral algorithms) in kernel regression. Consequently, we could sharpen the near inconsistency of kernel interpolation and clarify the saturation effects of kernel regression algorithms with higher qualification, etc. Thanks to the neural tangent kernel theory, these results greatly improve our understanding of the generalization behavior of training the wide neural networks. A novel technical contribution, the analytic functional argument, might be of independent interest.

cs.LG↗

arXiv2026

ND-TNN: Tensor-Neural-Network Approximation for High-Dimensional Nonlocal Diffusion Models

We study a numerical method, built on the tensor neural network (TNN) architecture introduced in \cite{wang2022tensor}, for solving nonlocal diffusion models in high-dimensional spaces. The tensor-product structure of the TNN ansatz, combined with the separability of the Gaussian kernel, reduces the high-dimensional integrals in the nonlocal energy to products of low-dimensional integrals, which are evaluated by Gauss--Legendre quadrature; nonseparable source and boundary data are handled by a TNN-based preconditioning step. For the Dirichlet boundary condition, we establish the asymptotically compatible $L^2$ error estimate \[ \|u_{\mathrm{loc}}-u_{δ,p}\|_{L^2(Ω)} \le C\!\left(\frac{\varepsilon_f}{\sqrtδ} +\frac{\varepsilon_g}δ +\frac{\varepsilon_u}{\sqrtδ} +η_{\mathrm{opt}}\right) +C\sqrtδ, \] where $\varepsilon_f$, $\varepsilon_g$ and $\varepsilon_u$ are the data and trial-class approximation errors and $η_{\mathrm{opt}}$ is the optimization residual. For the Neumann boundary condition, the $L^2$ estimate is improved to $O(\varepsilon_f+\varepsilon_g/\sqrtδ+\varepsilon_u +η_{\mathrm{opt}}+δ)$, and an $H^1$ gradient estimate is further obtained through a smoothing post-processing step. Numerical experiments on tensor-product domains up to $d=20$ support the theoretical results, and additional tests on two- and three-dimensional $L$-shaped domains demonstrate the practical robustness of the method beyond the smooth-domain setting covered by the analysis.

math.NA↗

arXiv2026

A Nonlocal $p$-Laplacian Interface Model with Sharp Interface

We propose an energy-based nonlocal $p$-Laplacian interface problem. Neumann interface conditions are naturally formulated via the energy, while Dirichlet conditions are enforced through a penalty term. A key feature is that the model retains a sharp interface, which facilitates extension to other interface problems; we illustrate this by developing a nonlocal approximation for the $p$-Laplacian interface problem with membrane conditions. By establishing $Γ$-convergence and compactness, we prove that as the nonlocal horizon vanishes, minimizers of the nonlocal functionals converge to those of the local counterparts. Numerical experiments using an efficient finite element method confirm the convergence.

math.AP↗

arXiv2026

Constant-Target Energy Matching: A Unified Framework for Continuous and Discrete Density Estimation

Density estimation is a central primitive in probabilistic modeling, yet continuous, discrete, and mixed-variable domains are often treated by separate objectives, limiting the ability to exploit a common statistical structure across data types. Continuous score-based methods rely on log-density gradients, while discrete extensions typically use concrete score whose unbounded targets become unstable near low-probability states. We introduce Constant-Target Energy Matching (CTEM), a unified energy-based framework for density estimation on general state spaces. CTEM replaces ordinary density-ratio regression with a bounded energy-difference transform and derives from it a sample-only training objective with the constant target 1. The learned scalar potential recovers log p without partition-function estimation or explicit unbounded ratio regression. Across continuous, discrete, and mixed-variable benchmarks, CTEM substantially improves density estimation over competitive baselines and yields higher-quality samples under standard sampling procedures.

cs.AI↗

arXiv2026

Neural Continuous-Time Markov Chain: Discrete Diffusion via Decoupled Jump Timing and Direction

Discrete diffusion models based on continuous-time Markov chains (CTMCs) have shown strong performance on language and discrete data generation, yet existing approaches typically parameterize the reverse rate matrix monolithically -- through proxies such as concrete scores (SEDD) or clean-data predictions (MDLM, GIDD) -- rather than aligning the parameterization with the intrinsic CTMC decomposition into jump timing and jump direction. We propose \textbf{Neural CTMC}, which exploits the underlying Poisson structure of CTMC dynamics by separately parameterizing the reverse process through an \emph{exit rate} (when to jump) and a \emph{jump distribution} (where to jump) via two dedicated network heads. We show that the evidence lower bound (ELBO) reduces to a path-space KL divergence between the true and learned reverse processes that factorizes into a Poisson KL for timing and a categorical KL for direction, and admits a tractable, gradient-equivalent and consistent loss. Experimentally, scored by Gemma2-9B, our pure-uniform Neural CTMC achieves $16.36$ generative perplexity on TinyStories (vs.\ GIDD $37.60$ and MDLM $42.66$). On OpenWebText, it attains the best perplexity at the same training-token budget across 16--128 sampling steps among the methods we compare (e.g., at 128 steps: Neural CTMC $183.6$ vs.\ MDLM $210.5$ and GIDD $249.8$). To facilitate reproducibility, we release our pretrained weights at https://huggingface.co/Jiangxy1117/Neural-CTMC.

cs.LG↗

arXiv2026

A Second-Order Nonlocal Approximation to Manifold Poisson Models with Neumann Boundary

In this paper, we propose a class of nonlocal models to approximate the Poisson model on manifolds with homogeneous Neumann boundary condition, where the manifolds are assumed to be embedded in high dimensional Euclid spaces. In comparison to the existing nonlocal approximation of Poisson models with Neumann boundary, we optimize the truncation error of model by adding an augmented function involving the second order normal derivative along the $2δ$ layer of boundary, with $2δ$ be the nonlocal interaction horizon. The 2nd normal derivative is expressed as the difference between the interior Laplacian and the boundary Laplacian. The concentration of our paper is on the construction of nonlocal model, the well-posedness of model, and its second-order convergence rate to its local counterpart. The localization rate of our nonlocal model is currently optimal among all related works even for the case of high dimensional Euclid spaces.

math.NA↗