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Zongben Xu

Publications and source records attributed to Zongben Xu.

At least 19 recordsLinked to original sources
arXiv2026

The solution to Kadison's problem on orthonormal bases of unitaries for type $\mathrm{II}_1$ factors

In 1967, Kadison asked whether every type $\mathrm{II}_1$ factor admits an orthonormal basis, with respect to its trace, consisting of unitaries. We resolve this problem in full generality and, more broadly, characterize the diffuse finite von Neumann algebras admitting such bases consisting of symmetries. Let $M$ be a diffuse finite von Neumann algebra with a faithful normal tracial state $τ$, let $κ$ be the density character of $L^2(M,τ)$, and identify $M$ with its canonical image in $L^2(M,τ)$. We prove that there exists a family $B\subset {s\in M:s=s^*=s^{-1},\ τ(s)=0}$ such that ${1}\cup B$ is an orthonormal basis of $L^2(M,τ)$ if and only if the density character of $L^2(zM,τ(z)^{-1}τ|_{zM})$ equals $κ$ for every nonzero central projection $z\in Z(M)$. In particular, every type $\mathrm{II}_1$ factor admits an orthonormal basis consisting of unitaries, thereby answering Kadison's question affirmatively. We also provide a Lean 4 formalization of the main results.

math.OA↗
arXiv2026

The modified Cartan conjecture

In 1928 H. Cartan stated a conjecture about holomorphic curves in $\mathbb{P}^n$ parametrized by the unit disc, omitting $n+2$ hyperplanes in general position. He proved it for $n=2$. In 1996 the first-named author constructed counterexamples for all $n\geq 3$, and proposed a modified form of Cartan's conjecture which he proved for $n=3$. In this paper we prove this modified conjecture in all dimensions. As a byproduct we prove a quantitative version of the classical theorem that analytic functions are linearly dependent if and only if their Wronskian determinant is zero. We interpret our results in terms of the Kobayashi--Royden pseudometric on the complement of $n+2$ hyperplanes, and on smooth algebraic varieties in the algebraic torus.

math.CV↗
arXiv2026

Coupled Optimal Transport with Landmark Constraints

Existing optimal transport (OT) models primarily seek an OT map or plan between distributions by minimizing a prescribed transport cost or distortion. However, minimizing transport cost or distortion alone may fail to identify a geometrically meaningful transformation between the two distributions. To address this limitation, this paper proposes a novel coupled OT framework that leverages a small number of annotated landmarks to guide the recovery of an underlying deformation governing the distribution transformation. The coupled OT framework integrates the optimization of the transport plan and the deformation field into a unified model, where the landmark-guided deformation field and the cost-driven transport plan are coupled through a mutual-consistency constraint. As a result, the deformation is jointly determined by the annotated landmarks and cost-driven distribution matching. The proposed framework provides a principled connection between landmark-based registration and transport-based distribution matching, enabling the recovery of transport maps from sparse geometric supervision. We establish the well-definedness of the proposed model in a general variational setting and develop a finite-element-based numerical algorithm for computation whose convergence properties are systematically analyzed. The practical effectiveness of the proposed approach is verified in shape matching.

cs.CV↗
arXiv2026

Harnessing the Potential of Optimizing Data Mixtures via Bayesian Domain Reweighting

The performance of Large Language Models (LLMs) is fundamentally influenced by the distributional composition of multi-domain pre-training data. While manual heuristics were prevalent in early models, they increasingly fail to capture the intricate synergies between domains as data complexity grows. To overcome the issue, a dominant approach seeks to fit a proxy function mapping between domain weights and their corresponding validation losses, and then find the optimal domain weights to minimize validation losses. These methods rely on strong structural assumptions, such as rank invariance or scaling laws, which are often violated, resulting in non-negligible estimation bias. A promising approach is to directly optimize the weighting scheme from data. However, it suffers from unstable optimization trajectory and prohibitive computational overhead, limiting its potential to search better domain weights configurations. This paper presents a Bayesian domain weighting method to infer the weights from a Dirichlet distribution via introducing Gamma prior information learned from observations. Experimental results demonstrate that proposed method could achieve stable and efficient domain weights learning, and identifies optimal mixtures while consuming substantially less data than search-based function-fitting methods, revitalizing optimization-based domain weighting for large-scale applications.

cs.LG↗
arXiv2026

Keypoint-Guided Optimal Transport: Models, Algorithms, and Applications

Existing Optimal Transport (OT) methods mainly derive the optimal transport plan/matching under the criterion of transport cost/distance minimization, which may cause incorrect matching in some cases. In real applications, annotating a few matched keypoints across domains is reasonable or even efortless in annotation burden. It is valuable to investigate how to leverage the annotated keypoints to guide the correct matching in OT. In this paper, we propose a novel KeyPoint-Guided model by ReLation preservation (KPG-RL)that searches for the optimal matching (i.e., transport plan) guided by the keypoints in OT.KPG-RL exploits a mask-based constraint of the transport plan to preserve the matching of keypoint pairs in transport, and guides the matching by relation of each data point to the keypoints. The KPG-RL is developed in both balanced and unbalanced/partial transport settings in Kantorovich and Gromov-Wasserstein formulations. Moreover, we deduce the dual formulation of $χ^2$-regularized KPG-RL model, from which we learn the transport based on deep learning techniques, scaling better to larger numbers of data. With the learned transport plan, two novel neural transport strategies, named manifold barycentric projection and manifold sampling, are developed to transport source data to the target domain data manifold. As applications, we apply the proposed approach to the heterogeneous domain adaptation, multi-omic single-cell alignment, and image-to-image translation. Experiments verifed the efectiveness of our approach.

cs.CV↗
arXiv2026

Sharp tangent inequalities in noncommutative $L_p$-spaces

Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful normal semifinite weight, and let $L_p(\mathcal{M})$ denote the associated Haagerup noncommutative $L_p$-space. For $1<p<\infty$, let $q=p/(p-1)$. For $z\in L_q(\mathcal{M})$ and $y\in L_p(\mathcal{M})$, denote by $\langle z,y\rangle=\operatorname{Re}\operatorname{Tr}(z^*y)$ the canonical real duality pairing between $L_q(\mathcal{M})$ and $L_p(\mathcal{M})$. Define the normalized duality map by $J_p(x)=\|x\|_p^{\,2-p}u|x|^{p-1}$ for $x=u|x|\neq0$, and set $J_p(0)=0$. We establish the sharp tangent inequality $$\|x+y\|_p^2\ge\|x\|_p^2+2\langle J_p(x),y\rangle+(p-1)\|y\|_p^2,\qquad x,y\in L_p(\mathcal{M}),\quad 1<p\le2.$$ The reverse inequality holds for $2\le p<\infty$, and the constant $p-1$ is optimal. Our inequality extends Xu's tangent inequality \cite{Xu89} from classical $L_p$-spaces to general noncommutative $L_p$-spaces and provides the sharp tangent formulation of the Ricard--Xu convexity inequality \cite{RX16}. When specialized to Schatten $p$-classes, it also yields the corresponding tangent formulation of the Ball--Carlen--Lieb convexity inequality \cite{BCL94}. Under a suitable positivity assumption, we further establish a sharp complement to Xu's tangent inequality for Schatten $p$-classes in the range $0<p<1$.

math.OA↗
arXiv2026

Flash EQ-Linear: Accelerating Equivariant Linear Layers via Group-wise Discrete Fourier Transform

Equivariant networks embed geometric symmetries as structural priors through weight sharing, achieving remarkable parameter efficiency across vision tasks. However, this parameter efficiency does not translate into compute efficiency: existing implementations unroll the structured weights into dense matrices and dispatch them to generic dense kernels, so the FLOPs of an equivariant layer are no smaller than those of a non-equivariant counterpart. In this paper, we observe that the equivariant linear (EQ-Linear) layer---the most fundamental and frequently used module in modern equivariant architectures---is essentially a circular convolution along the group dimension composed with a linear transform along the channel dimension. Building on this observation, we propose Flash EQ-Linear, an exact acceleration algorithm that reduces the complexity from $\mathcal{O}(NDC)$ to $\mathcal{O}(NDC/T)$ by combining the Fourier convolution theorem along the group dimension with the conjugate symmetry of the real DFT. We further provide dedicated CUDA kernels for Flash EQ-Linear, covering both forward and backward passes and both FP32 and FP16 precision. At the operator level, Flash EQ-Linear achieves up to ${2\times}$ forward speedup over PyTorch's F.linear; at the network level, Flash EQ-ViT and Flash EQ-Swin achieve up to ${1.7\times}$ end-to-end speedup over both equivariant and non-equivariant baselines. To our knowledge, this is the first time equivariant networks strictly dominate their non-equivariant counterparts along all three axes simultaneously: accuracy, parameter efficiency, and inference speed.Code is available at https://github.com/zhongchenzhao/FlashEQLinear.

cs.CV↗
arXiv2026

BaRA: Bayesian Adaptive Rank Allocation for Parameter-Efficient Fine-Tuning

While Low-rank adaptation (LoRA) enables highly efficient fine-tuning by constraining task-specific updates to fixed low-rank subspaces, this rigid design limits representational flexibility and often results in overconfident predictions and miscalibrated uncertainty, especially in low-data regimes. Recent Bayesian LoRA variants improve uncertainty estimation by modeling posterior distributions over adaptation parameters. However, these approaches typically rely on fixed or heuristically determined ranks, overlooking the inherently context-dependent nature of adaptation capacity. In this paper, we propose BaRA, a Bayesian Adaptive Rank Allocation framework for parameter-efficient fine-tuning. Drawing inspiration from probabilistic topic models, BaRA dynamically allocates adaptation capacity by activating a sparse, context-dependent subset of disentangled latent factors, enabling instance-wise variation in effective rank. This Bayesian formulation provides principled, data-driven capacity control, mitigating over-parameterization while preserving expressiveness. Beyond the modeling contribution, we provide a complexity-theoretic generalization analysis showing that the generalization gap of BaRA depends on the learned joint effective rank $\bar{s}_{Φ,θ}$ induced by the global-local gate, rather than the maximum rank $r$. This result explains why sparse adaptive rank allocation can reduce the effective hypothesis complexity while preserving input-dependent expressiveness. Extensive experiments on diverse natural language benchmarks demonstrate that BaRA consistently improves predictive performance, robustness, and uncertainty calibration compared to standard LoRA and existing Bayesian LoRA variants.

cs.LG↗
arXiv2026

A Limit Theory of Foundation Models: A Mathematical Approach to Understanding Emergent Intelligence and Scaling Laws

Emergent intelligence have played a major role in the modern AI development. While existing studies primarily rely on empirical observations to characterize this phenomenon, a rigorous theoretical framework remains underexplored. This study attempts to develop a mathematical approach to formalize emergent intelligence from the perspective of limit theory. Specifically, we introduce a performance function E(N, P, K), dependent on data size N, model size P and training steps K, to quantify intelligence behavior. We posit that intelligence emerges as a transition from finite to effectively infinite knowledge, and thus recast emergent intelligence as existence of the limit $\lim_{N,P,K \to \infty} \mathcal{E}(N,P,K)$, with emergent abilities corresponding to the limiting behavior. This limit theory helps reveal that emergent intelligence originates from the existence of a parameter-limit architecture (referred to as the limit architecture), and that emergent intelligence rationally corresponds to the learning behavior of this limit system. By introducing tools from nonlinear Lipschitz operator theory, we prove that the necessary and sufficient conditions for existence of the limit architecture. Furthermore, we derive the scaling law of foundation models by leveraging tools of Lipschitz operator and covering number. Theoretical results show that: 1) emergent intelligence is governed by three key factors-training steps, data size and the model architecture, where the properties of basic blocks play a crucial role in constructing foundation models; 2) the critical condition Lip(T)=1 for emergent intelligence provides theoretical support for existing findings. 3) emergent intelligence is determined by an infinite-dimensional system, yet can be effectively realized in practice through a finite-dimensional architecture. Our empirical results corroborate these theoretical findings.

cs.LG↗
arXiv2026

Aligning Network Equivariance with Data Symmetry: A Theoretical Framework and Adaptive Approach for Image Restoration

Image restoration is an inherently ill posed inverse problem. Equivariant networks that embed geometric symmetry priors can mitigate this ill posedness and improve performance. However, current understanding of the relationship between network equivariance and data symmetry remains largely heuristic. Particularly for real world data with imperfect symmetry, existing research lacks a systematic theoretical framework to quantify symmetry, select transformation groups, or evaluate model data alignment. To bridge this gap, we conduct an analysis from an optimization perspective and formalize the intrinsic relationship among data symmetry priors, model equivariance, and generalization capability. Specifically, we propose for the first time a quantifiable definition of non strict symmetry at the dataset level (rather than sample level) and use it as a constraint to formulate the restoration inverse problem. We then show that the equivariance for restoration models can be naturally derived from this inverse problems incorporated the proposed symmetry constraints, and that the equivariance error of the optimal restoration operator is strictly bounded by the data symmetry error and the discretization mesh size. Furthermore, by analyzing the network's empirical risk, we demonstrate that aligning equivariance with data symmetry optimizes the bias variance trade off, minimizing the total expected risk. Guided by these insights, we propose a Sample Adaptive Equivariant Network that uses a hypernetwork and transformation learnable equivariant convolutions to dynamically align with each sample's inherent symmetry. Extensive experiments on super resolution, denoising, and deraining validate our theoretical findings and show significant superiority over standard baselines and traditional equivariant models. Our code and supplementary material are available at https://github.com/tanfy929/SA-Conv.

cs.CV↗
arXiv2026

Rotation Equivariant Mamba for Vision Tasks

Rotation equivariance constitutes one of the most general and crucial structural priors for visual data, yet it remains notably absent from current Mamba-based vision architectures. Despite the success of Mamba in natural language processing and its growing adoption in computer vision, existing visual Mamba models fail to account for rotational symmetry in their design. This omission renders them inherently sensitive to image rotations, thereby constraining their robustness and cross-task generalization. To address this limitation, we incorporate rotation symmetry, a universal and fundamental geometric prior in images, into Mamba-based architectures. Specifically, we introduce EQ-VMamba, the first rotation equivariant visual Mamba architecture for vision tasks. The core components of EQ-VMamba include a carefully designed rotation equivariant cross-scan strategy and group Mamba blocks. Moreover, we provide a rigorous theoretical analysis of the intrinsic equivariance error, demonstrating that the proposed architecture enforces end-to-end rotation equivariance throughout the network. Extensive experiments across multiple benchmarks -- including high-level image classification, mid-level semantic segmentation, and low-level image super-resolution -- demonstrate that EQ-VMamba consistently improves rotation robustness and achieves superior or competitive performance compared to non-equivariant baselines, while requiring approximately 50\% fewer parameters. These results indicate that embedding rotation equivariance not only effectively bolsters the robustness of visual Mamba models against rotation transformations, but also enhances overall performance with significantly improved parameter efficiency. Code is available at https://github.com/zhongchenzhao/EQ-VMamba.

cs.CV↗
arXiv2026

Unsupervised Posterior Sampling for Seismic Data Recovery via Score-Based Generative Priors

Seismic data restoration is a fundamental task in seismic exploration, yet remains challenging under complex and unknown degradations. Traditional model-driven or task-specific learning methods often require retraining for each degradation type and fail to generalize effectively to unseen field data. In this work, we introduce an unsupervised Posterior Sampling Framework (PSF) built upon Score-based Generative Models (SGMs) for unified seismic data restoration. PSF leverages the pre-trained unconditional SGMs as a seismic-aware generative prior and derives a generalized conditional score function associated with the forward operator of each inverse problem. This enables posterior sampling across different seismic restoration tasks without retraining or supervision. Additionally, an adaptive noise-level estimation mechanism is incorporated to dynamically regulate the noise suppression strength during sampling, enhancing flexibility under varying signal-to-noise ratios and degradation conditions.Extensive experiments on seismic denoising, interpolation, compressed sensing, and deconvolution demonstrate that PSF delivers high-quality samples and exhibits robust generalization to out-of-distribution data. These results highlight the potential of SGMs as a universal prior for seismic inverse problems and establish PSF as a flexible framework for unsupervised posterior inference across diverse degradation scenarios.

physics.geo-ph↗
arXiv2026

KoopGen: Koopman Generator Networks for Representing and Predicting Dynamical Systems with Continuous Spectra

Representing and predicting high-dimensional and spatiotemporally chaotic dynamical systems remains a fundamental challenge in dynamical systems and machine learning. Although data-driven models can achieve accurate short-term forecasts, they often lack stability, interpretability, and scalability in regimes dominated by broadband or continuous spectra. Koopman-based approaches provide a principled linear perspective on nonlinear dynamics, but existing methods rely on restrictive finite-dimensional assumptions or explicit spectral parameterizations that degrade in high-dimensional settings. Against these issues, we introduce KoopGen, a generator-based neural Koopman framework that models dynamics through a structured, state-dependent representation of Koopman generators. By exploiting the intrinsic Cartesian decomposition into skew-adjoint and self-adjoint components, KoopGen separates conservative transport from irreversible dissipation while enforcing exact operator-theoretic constraints during learning. Across systems ranging from nonlinear oscillators to high-dimensional chaotic and spatiotemporal dynamics, KoopGen improves prediction accuracy and stability, while clarifying which components of continuous-spectrum dynamics admit interpretable and learnable representations.

cs.LG↗
arXiv2026

Vanilla Group Equivariant Vision Transformer: Simple and Effective

Incorporating symmetry priors as inductive biases to design equivariant Vision Transformers (ViTs) has emerged as a promising avenue for enhancing their performance. However, existing equivariant ViTs often struggle to balance performance with equivariance, primarily due to the challenge of achieving holistic equivariant modifications across the diverse modules in ViTs-particularly in harmonizing the Self-Attention mechanism with Patch Embedding. To address this, we propose a straightforward framework that systematically renders key ViT components, including patch embedding, self-attention, positional encodings, and Down/Up-Sampling, equivariant, thereby constructing ViTs with guaranteed equivariance. The resulting architecture serves as a plug-and-play replacement that is both theoretically grounded and practically versatile, scaling seamlessly even to Swin Transformers. Extensive experiments demonstrate that our equivariant ViTs consistently improve performance and data efficiency across a wide spectrum of vision tasks.

cs.CV↗
arXiv2026

Nonparametric Prior Learning in Differential Equation Modeling

This paper addresses Bayesian inference related to partial differential equations (PDEs), particularly nonparametric regression constrained by PDEs. To effectively encode prior information, we propose a novel framework that learns a prediction function of the prior distribution from historical training datasets. We introduce hyper-prior and hyper-posterior distributions and derive a generalization error estimate, which accommodates data-dependent priors by extending the concept of differential privacy. Some mild conditions are given to validate the error estimate, where various typical PDEs such as diffusion and Darcy flow equations can be integrated. We thus formulate an infinite-dimensional optimization problem to obtain the point estimate of the hyper-posterior. Numerical examples demonstrate the performance of our proposed method in learning the prediction function of priors.

math.ST↗
arXiv2025

Towards Prospective Medical Image Reconstruction via Knowledge-Informed Dynamic Optimal Transport

Medical image reconstruction from measurement data is a vital but challenging inverse problem. Deep learning approaches have achieved promising results, but often requires paired measurement and high-quality images, which is typically simulated through a forward model, i.e., retrospective reconstruction. However, training on simulated pairs commonly leads to performance degradation on real prospective data due to the retrospective-to-prospective gap caused by incomplete imaging knowledge in simulation. To address this challenge, this paper introduces imaging Knowledge-Informed Dynamic Optimal Transport (KIDOT), a novel dynamic optimal transport framework with optimality in the sense of preserving consistency with imaging physics in transport, that conceptualizes reconstruction as finding a dynamic transport path. KIDOT learns from unpaired data by modeling reconstruction as a continuous evolution path from measurements to images, guided by an imaging knowledge-informed cost function and transport equation. This dynamic and knowledge-aware approach enhances robustness and better leverages unpaired data while respecting acquisition physics. Theoretically, we demonstrate that KIDOT naturally generalizes dynamic optimal transport, ensuring its mathematical rationale and solution existence. Extensive experiments on MRI and CT reconstruction demonstrate KIDOT's superior performance.

eess.IV↗
arXiv2025

Feed Two Birds with One Scone: Exploiting Function-Space Regularization for Both OOD Robustness and ID Fine-Tuning Performance

Robust fine-tuning aims to achieve competitive in-distribution (ID) performance while maintaining the out-of-distribution (OOD) robustness of a pre-trained model when transferring it to a downstream task. To remedy this, most robust fine-tuning methods aim to preserve the pretrained weights, features, or logits. However, we find that these methods cannot always improve OOD robustness for different model architectures. This is due to the OOD robustness requiring the model function to produce stable prediction for input information of downstream tasks, while existing methods might serve as a poor proxy for the optimization in the function space. Based on this finding, we propose a novel regularization that constrains the distance of fine-tuning and pre-trained model in the function space with the simulated OOD samples, aiming to preserve the OOD robustness of the pre-trained model. Besides, to further enhance the OOD robustness capability of the fine-tuning model, we introduce an additional consistency regularization to promote stable predictions of perturbed samples. Extensive experiments demonstrate our approach could consistently improve both downstream task ID fine-tuning performance and OOD robustness across a variety of CLIP backbones, outperforming existing regularization-based robust fine-tuning methods.

cs.LG↗
arXiv2025

Rotation Equivariant Arbitrary-scale Image Super-Resolution

The arbitrary-scale image super-resolution (ASISR), a recent popular topic in computer vision, aims to achieve arbitrary-scale high-resolution recoveries from a low-resolution input image. This task is realized by representing the image as a continuous implicit function through two fundamental modules, a deep-network-based encoder and an implicit neural representation (INR) module. Despite achieving notable progress, a crucial challenge of such a highly ill-posed setting is that many common geometric patterns, such as repetitive textures, edges, or shapes, are seriously warped and deformed in the low-resolution images, naturally leading to unexpected artifacts appearing in their high-resolution recoveries. Embedding rotation equivariance into the ASISR network is thus necessary, as it has been widely demonstrated that this enhancement enables the recovery to faithfully maintain the original orientations and structural integrity of geometric patterns underlying the input image. Motivated by this, we make efforts to construct a rotation equivariant ASISR method in this study. Specifically, we elaborately redesign the basic architectures of INR and encoder modules, incorporating intrinsic rotation equivariance capabilities beyond those of conventional ASISR networks. Through such amelioration, the ASISR network can, for the first time, be implemented with end-to-end rotational equivariance maintained from input to output. We also provide a solid theoretical analysis to evaluate its intrinsic equivariance error, demonstrating its inherent nature of embedding such an equivariance structure. The superiority of the proposed method is substantiated by experiments conducted on both simulated and real datasets. We also validate that the proposed framework can be readily integrated into current ASISR methods in a plug \& play manner to further enhance their performance.

cs.CV↗