arXiv subjects
Publications and source records attributed to Jingwei Guo.
We prove the boundedness of multilinear Hilbert transforms along moment curves by establishing uniform estimates for both translated and untranslated multilinear paraproducts and by applying a Sobolev smoothing inequality. As a byproduct of our method, we obtain the same bounds for multilinear maximal functions along moment curves.
Multimodal Large Language Models (MLLMs) have shown strong performance on public medical benchmarks, yet existing evaluations often remain weak proxies for clinical use, relying on isolated inputs and simplified recognition-style tasks. We introduce CardioLens, a leakage-resistant evaluation testbed for multi-sequence Cardiovascular Magnetic Resonance (CMR), constructed from private hospital archives through a rigorous report-to-QA construction and verification pipeline. CardioLens contains 473,896 slices and 13,494 verified QA pairs across 4D Cine, LGE, perfusion, and T2-weighted imaging, and evaluates three stages of CMR interpretation: image understanding, report generation, and disease diagnosis. Across 24 state-of-the-art MLLMs, CardioLens reveals a substantial clinical reality gap: models perform poorly overall, with performance degrading along the real CMR workflow. Confusion analysis further shows a category-collapse failure mode, where models default to frequent abnormal categories rather than distinguishing clinically distinct findings. To rule out MLLM-compatible input construction as the primary cause, we compare random, clinically motivated, and data-driven slice selection protocols under different slice budgets; performance changes only marginally, typically by about 1%. Explicit reasoning prompts also fail to rescue performance, often making models more conservative rather than improving visual evidence use. These results show that current MLLMs remain far from reliable CMR interpretation, where clinical decisions require integrating distributed evidence across sequences, views, and temporal phases. CardioLens provides a clinically grounded testbed for developing next-generation MLLMs toward real-world clinical deployment.
For any integer $n \geq 2$, let $(m_{1},\ldots,m_{n})$ be a strictly increasing $n$-tuple of positive integers. We show that any subset $A\subset [N]^n$ of density at least $(\log N)^{-c}$ contains a nontrivial configuration of the form \begin{equation*} \boldsymbol{x},\boldsymbol{x}+r^{m_{1}}\boldsymbol{e_{1}},\ldots,\boldsymbol{x}+r^{m_{n}}\boldsymbol{e_{n}}, \end{equation*} where $c=c(n,m_{1},\ldots,m_{n} )$ is a positive constant. This quantitative multidimensional Szemer\'{e}di theorem extends a recent two-dimensional result of Peluse, Prendiville, and Shao concerning the configuration of the form $(x,y),(x+r,y),\left(x,y+r^{2}\right)$. The theorem is obtained as a consequence of an effective ``popular'' version.
P\'{o}lya's conjecture on the eigenvalues of the Laplacian has been one of the core problems in spectral geometry. Building upon the recent breakthrough works on P\'{o}lya's conjecture for balls and annuli by Filonov, Levitin, Polterovich and Sher, we study several aspects of P\'{o}lya's conjecture for balls and cylinders: by refining the purely analytical portion of the proof in [2] for the Neumann P\'{o}lya's conjecture for the disk, we extend the regime of the spectral parameter that can be established without computer assistance; we obtain improvement of P\'{o}lya's conjecture for disks and balls; we obtain improvement of P\'{o}lya's conjecture for cylinders and confirm the Neumann P\'{o}lya's conjecture for cylinders in $\mathbb{R}^3$. As a supplementary effort, we study Weyl's law for cylinders.
The purpose of this paper is twofold. One is to investigate the properties of the zeros of cross-products of Bessel functions or derivatives of ultraspherical Bessel functions, as well as the properties of the zeros of the derivative of the first-kind ultraspherical Bessel function. The properties we study include asymptotics (with uniform and nonuniform remainder estimates), upper and lower bounds and so on. In addition, we provide the number of zeros of a certain cross-product within a large circle and show that all its zeros are real and simple. These results may be of independent interest. The other is to investigate the Dirichlet/Neumann Laplacian on balls and spherical shells in $\mathbb{R}^d$ ($d\geq 2$) and the remainder of the associated Weyl's law. We obtain new upper bounds in all dimensions, both in the Dirichlet and Neumann cases. The proof relies on our studies of Bessel functions and the latest development in the Gauss circle problem, which was driven by the application of the emerging decoupling theory of harmonic analysis.
Deploying deep models in real-world scenarios remains challenging due to significant performance drops under distribution shifts between training and deployment environments. Test-Time Adaptation (TTA) has recently emerged as a promising solution, enabling on-the-fly model adaptation. However, its effectiveness deteriorates in the presence of mixed distribution shifts -- common in practical settings -- where multiple target domains coexist. In this paper, we study TTA under mixed distribution shifts and move beyond conventional whole-batch adaptation paradigms. By revisiting distribution shifts from a spectral perspective, we find that the heterogeneity across latent domains is often pronounced in Fourier space. In particular, high-frequency components encode domain-specific variations, which facilitates clearer separation of samples from different distributions. Motivated by this observation, we propose to un-mix heterogeneous data streams using high-frequency domain cues, making diverse shift patterns more tractable. To this end, we propose Frequency-based Decentralized Adaptation (FreDA), a novel framework that decomposes globally heterogeneous data stream into locally homogeneous clusters in the Fourier space. It leverages decentralized learning and augmentation strategies to robustly adapt under mixed domain shifts. Extensive experiments across various environments (corrupted, natural, and medical) show the superiority of our method over the state-of-the-arts.
Medical Image Analysis (MedIA) has become indispensable in modern healthcare, enhancing clinical diagnostics and personalized treatment. Despite the remarkable advancements supported by deep learning (DL) technologies, their practical deployment faces challenges posed by distribution shifts, where models trained on specific datasets underperform on others from varying hospitals, or patient populations. To address this issue, researchers have been actively developing strategies to increase the adaptability of DL models, enabling their effective use in unfamiliar environments. This paper systematically reviews approaches that apply DL techniques to MedIA systems affected by distribution shifts. Rather than organizing existing methods by technical characteristics, we explicitly bridge real-world clinical constraints -- such as limited data accessibility, strict privacy requirements, and heterogeneous collaboration protocols -- with the technical paradigms able to address them. By establishing this connection between operational constraints and methodological evolution, we categorize existing works into Joint Training, Federated Learning, Fine-tuning, and Domain Generalization, each aligned with specific healthcare scenarios. Beyond this taxonomy, our empirical analysis suggests that, as domain information becomes progressively less accessible across these paradigms, performance improvements become increasingly constrained, and further uncovers a gradual shift in methodological focus from explicit distribution alignment toward uncertainty-aware modeling, ultimately pointing to the need for more deployability-aware design in real-world MedIA.
Cyclists face a disproportionate risk of injury, yet conventional crash records are too sparse to identify risk factors at fine spatial and temporal scales. Recently, naturalistic studies have used video data to capture the complex behavioural and infrastructural risk factors. A promising format is panoramic video, which can record 360$^\circ$ views around a rider. However, its use is limited by distortions, large numbers of small objects, and boundary continuity, which cannot be handled using existing computer vision models. This research proposes a novel three-step framework: (1) enhancing object detection accuracy on panoramic imagery by segmenting and projecting the original 360$^\circ$ images into sub-images; (2) modifying multi-object tracking models to incorporate boundary continuity and object category information; and (3) validating through a real-world application of vehicle overtaking detection. The methodology is evaluated using panoramic videos recorded by cyclists on London's roadways under diverse conditions. Experimental results demonstrate improvements over baselines, achieving higher average precision across varying image resolutions. Moreover, the enhanced tracking approach yields a 10.0% decrease in identification switches and a 2.7% improvement in identification precision. The overtaking detection task achieves a high F-score of 0.82, illustrating the practical effectiveness of the proposed method in real-world cycling safety scenarios.
Whilst spectral Graph Neural Networks (GNNs) are theoretically well-founded in the spectral domain, their practical reliance on polynomial approximation implies a profound linkage to the spatial domain. As previous studies rarely examine spectral GNNs from the spatial perspective, their spatial-domain interpretability remains elusive, e.g., what information is essentially encoded by spectral GNNs in the spatial domain? In this paper, to answer this question, we establish a theoretical connection between spectral filtering and spatial aggregation, unveiling an intrinsic interaction that spectral filtering implicitly leads the original graph to an adapted new graph, explicitly computed for spatial aggregation. Both theoretical and empirical investigations reveal that the adapted new graph not only exhibits non-locality but also accommodates signed edge weights to reflect label consistency among nodes. These findings thus highlight the interpretable role of spectral GNNs in the spatial domain and inspire us to rethink graph spectral filters beyond the fixed-order polynomials, which neglect global information. Built upon the theoretical findings, we revisit the state-of-the-art spectral GNNs and propose a novel Spatially Adaptive Filtering (SAF) framework, which leverages the adapted new graph by spectral filtering for an auxiliary non-local aggregation. Notably, our proposed SAF comprehensively models both node similarity and dissimilarity from a global perspective, therefore alleviating persistent deficiencies of GNNs related to long-range dependencies and graph heterophily. Extensive experiments over 13 node classification benchmarks demonstrate the superiority of our proposed framework to the state-of-the-art models.
While recent test-time adaptations exhibit efficacy by adjusting batch normalization to narrow domain disparities, their effectiveness diminishes with realistic mini-batches due to inaccurate target estimation. As previous attempts merely introduce source statistics to mitigate this issue, the fundamental problem of inaccurate target estimation still persists, leaving the intrinsic test-time domain shifts unresolved. This paper delves into the problem of mini-batch degradation. By unraveling batch normalization, we discover that the inexact target statistics largely stem from the substantially reduced class diversity in batch. Drawing upon this insight, we introduce a straightforward tool, Test-time Exponential Moving Average (TEMA), to bridge the class diversity gap between training and testing batches. Importantly, our TEMA adaptively extends the scope of typical methods beyond the current batch to incorporate a diverse set of class information, which in turn boosts an accurate target estimation. Built upon this foundation, we further design a novel layer-wise rectification strategy to consistently promote test-time performance. Our proposed method enjoys a unique advantage as it requires neither training nor tuning parameters, offering a truly hassle-free solution. It significantly enhances model robustness against shifted domains and maintains resilience in diverse real-world scenarios with various batch sizes, achieving state-of-the-art performance on several major benchmarks. Code is available at \url{https://github.com/kiwi12138/RealisticTTA}.
Spectral Graph Neural Networks (GNNs) have achieved tremendous success in graph machine learning, with polynomial filters applied for graph convolutions, where all nodes share the identical filter weights to mine their local contexts. Despite the success, existing spectral GNNs usually fail to deal with complex networks (e.g., WWW) due to such homogeneous spectral filtering setting that ignores the regional heterogeneity as typically seen in real-world networks. To tackle this issue, we propose a novel diverse spectral filtering (DSF) framework, which automatically learns node-specific filter weights to exploit the varying local structure properly. Particularly, the diverse filter weights consist of two components -- A global one shared among all nodes, and a local one that varies along network edges to reflect node difference arising from distinct graph parts -- to balance between local and global information. As such, not only can the global graph characteristics be captured, but also the diverse local patterns can be mined with awareness of different node positions. Interestingly, we formulate a novel optimization problem to assist in learning diverse filters, which also enables us to enhance any spectral GNNs with our DSF framework. We showcase the proposed framework on three state-of-the-arts including GPR-GNN, BernNet, and JacobiConv. Extensive experiments over 10 benchmark datasets demonstrate that our framework can consistently boost model performance by up to 4.92% in node classification tasks, producing diverse filters with enhanced interpretability. Code is available at \url{https://github.com/jingweio/DSF}.
For symbol $a\in S^{n(ρ-1)/2}_{ρ,1}$ the pseudo-differential operator $T_a$ may not be $L^2$ bounded. However, under some mild extra assumptions on $a$, we show that $T_a$ is bounded from $L^{\infty}$ to $BMO$ and on $L^p$ for $2\leq p<\infty$. A key ingredient in our proof of the $L^{\infty}$-$BMO$ boundedness is that we decompose a cube, use $x$-regularity of the symbol and combine certain $L^2$, $L^\infty$ and $L^{\infty}$-$BMO$ boundedness. We use an almost orthogonality argument to prove an $L^2$ boundedness and then interpolation to obtain the desired $L^p$ boundedness.
We prove a quantitative Roth-type theorem for polynomial corners in $\mathbb{R}^2$. Let $P_1$ and $P_2$ be two linearly independent polynomials with zero constant term. We show that any measurable subset of $[0,1]^2$ with positive measure contains three points $(x,y)$, $(x+P_1(t),y)$, $(x,y+P_2(t))$ with a gap estimate on $t$. We also prove boundedness results for a variant of the triangular Hilbert transform involving two polynomials and its associated maximal function. These results extend some earlier work of Christ, Durcik and Roos. The key of the proof is to establish certain smoothing inequalities involving two polynomials. To accomplish that we give sublevel set estimates with general polynomials, explicit exponents and simplified proofs.
While Graph Neural Networks (GNNs) have achieved enormous success in multiple graph analytical tasks, modern variants mostly rely on the strong inductive bias of homophily. However, real-world networks typically exhibit both homophilic and heterophilic linking patterns, wherein adjacent nodes may share dissimilar attributes and distinct labels. Therefore, GNNs smoothing node proximity holistically may aggregate both task-relevant and irrelevant (even harmful) information, limiting their ability to generalize to heterophilic graphs and potentially causing non-robustness. In this work, we propose a novel Edge Splitting GNN (ES-GNN) framework to adaptively distinguish between graph edges either relevant or irrelevant to learning tasks. This essentially transfers the original graph into two subgraphs with the same node set but complementary edge sets dynamically. Given that, information propagation separately on these subgraphs and edge splitting are alternatively conducted, thus disentangling the task-relevant and irrelevant features. Theoretically, we show that our ES-GNN can be regarded as a solution to a disentangled graph denoising problem, which further illustrates our motivations and interprets the improved generalization beyond homophily. Extensive experiments over 11 benchmark and 1 synthetic datasets not only demonstrate the effective performance of ES-GNN but also highlight its robustness to adversarial graphs and mitigation of the over-smoothing problem.
We study the pseudo-differential operator \begin{equation*} T_a f\left(x\right)=\int_{\mathbb{R}^n}e^{ix\cdotξ}a\left(x,ξ\right)\widehat{f}\left(ξ\right)\,\textrm{d}ξ, \end{equation*} where the symbol $a$ is in the Hörmander class $S^{m}_{ρ,1}$ or more generally in the rough Hörmander class $L^{\infty}S^{m}_ρ$ with $m\in\mathbb{R}$ and $ρ\in [0,1]$. It is known that $T_a$ is bounded on $L^1(\mathbb{R}^n)$ for $m<n(ρ-1)$. In this paper we mainly investigate its boundedness properties when $m$ is equal to the critical index $n(ρ-1)$. For any $0\leq ρ\leq 1$ we construct a symbol $a\in S^{n(ρ-1)}_{ρ,1}$ such that $T_a$ is unbounded on $L^1$ and furthermore it is not of weak type $(1,1)$ if $ρ=0$. On the other hand we prove that $T_a$ is bounded from $H^1$ to $L^1$ if $0\leq ρ<1$ and construct a symbol $a\in S^0_{1,1}$ such that $T_a$ is unbounded from $H^1$ to $L^1$. Finally, as a complement, for any $1<p<\infty$ we give an example $a\in S^{-1/p}_{0,1}$ such that $T_a$ is unbounded on $L^p(\mathbb{R})$.
Disentangled Graph Convolutional Network (DisenGCN) is an encouraging framework to disentangle the latent factors arising in a real-world graph. However, it relies on disentangling information heavily from a local range (i.e., a node and its 1-hop neighbors), while the local information in many cases can be uneven and incomplete, hindering the interpretabiliy power and model performance of DisenGCN. In this paper\footnote{This paper is a lighter version of \href{https://jingweio.github.io/assets/pdf/tnnls22.pdf}{"Learning Disentangled Graph Convolutional Networks Locally and Globally"} where the results and analysis have been reworked substantially. Digital Object Identifier \url{https://doi.org/10.1109/TNNLS.2022.3195336}.}, we introduce a novel Local and Global Disentangled Graph Convolutional Network (LGD-GCN) to capture both local and global information for graph disentanglement. LGD-GCN performs a statistical mixture modeling to derive a factor-aware latent continuous space, and then constructs different structures w.r.t. different factors from the revealed space. In this way, the global factor-specific information can be efficiently and selectively encoded via a message passing along these built structures, strengthening the intra-factor consistency. We also propose a novel diversity promoting regularizer employed with the latent space modeling, to encourage inter-factor diversity. Evaluations of the proposed LGD-GCN on the synthetic and real-world datasets show a better interpretability and improved performance in node classification over the existing competitive models. Code is available at \url{https://github.com/jingweio/LGD-GCN}.
We give two Roth theorems, related to the nonlinear configuration $x$, $x+P_1(t)$, $x+P_2(t)$ involving two polynomials, for sets in $\mathbb{R}$ of positive density and of fractional dimensions. The proof uses Fourier analysis.
Let $\mathscr{B}=\{x\in\mathbb{R}^d : |x|<R \}$ ($d\geq 3$) be a ball. We consider the Dirichlet Laplacian associated with $\mathscr{B}$ and prove that its eigenvalue counting function has an asymptotics \begin{equation*} \mathscr{N}_\mathscr{B}(μ)=C_d vol(\mathscr{B})μ^d-C'_d vol(\partial \mathscr{B})μ^{d-1}+O\left(μ^{d-2+\frac{131}{208}}(\log μ)^{\frac{18627}{8320}}\right) \end{equation*} as $μ\rightarrow \infty$.