arXiv subjects

Lingxiao Zhang

Publications and source records attributed to Lingxiao Zhang.

10 recordsLinked to original sources

arXiv2026

A Unified Geometric Perspective on Zygmund's Maximal Function Conjecture and Related Questions for the Stein Hilbert Transform

The Zygmund vector-field maximal function conjecture is a long-standing open problem in harmonic analysis. It is notoriously difficult in part because the vector-field maximal function is non-translation-invariant and lies beyond the scope of finite-type theory. Its resolution would yield a Lebesgue differentiation theorem along vector fields, i.e., pointwise recovery of functions from averages along vector fields. We establish an $L^2$ boundedness criterion that substantially enlarges the class of admissible geometric degeneracies. More precisely, the power-type decay condition in Bourgain's theorem can be replaced by the much weaker logarithmic-polynomial decay $(\log\tfrac{1}τ)^{-p}$, $p>2$. This condition no longer yields the power-type integrability underlying Bourgain's argument, yet $L^2$ boundedness persists, permitting degeneracies far beyond any power-law regime. Our result also provides a unified geometric perspective on finite-type, intermediate-degenerate, and highly non-finite-type vector fields, with their behavior organized through the angular-variation quantity $w_x(t)$. This perspective places several classical boundedness criteria within a common hierarchy. Motivated by Lacey and Li's work on the Stein vector-field Hilbert transform conjecture, we further introduce a non-centered rectangular maximal operator adapted to the variable geometry of vector fields and prove its weak-type $(1,1)$ boundedness. Together, these results provide new tools and a unified geometric framework for the study of two fundamental conjectural problems in non-translation-invariant analysis.

math.CA↗

arXiv2026

OrDA: Orthogonal Disentanglement of Access Habits Framework for Homepage Marketing Block Recommendations

Clicks on homepage marketing blocks are driven by a dual-mechanism of content interest and access habits. However, habitual clicks often create Pseudo-Positives in marketing slots, where position advantage masks mediocre content quality, leading to biased recommendation ecosystems. We propose a framework called Orthogonal Disentanglement of Access habits (OrDA) to purify interest signals. OrDA utilizes a dual-tower structure with a gated allocation layer to adaptively route features and minimize interference. To ensure rigorous separation, we employ orthogonal regularization to constrain the latent interest and habit manifolds to be geometrically perpendicular. OrDA performs causal intervention (do-calculus) during inference to rank items solely by purified interest scores. Empirical online evaluations on large-scale datasets demonstrate that OrDA effectively eliminates access-habit bias, outperforming state-of-the-art methods in predictive accuracy. Online AB test 5.64% shows user click-through rates (UCTR) improvement on the Zhima homepage marketing block, Zhima rent-floor recommendation.

cs.LG↗

arXiv2026

On singular integrals with non-negative kernels in the Heisenberg group

In this paper we revisit nonnegative kernels in the first Heisenberg group $\He$, and in particular we further study the family $$K_α(x,y,z)= \frac{|z|^{α/2}}{\|(x,y,z)\|_{H}^{α+1}}, \quad α>0,$$ which was introduced in \cite{CL}. We first show that if $E \subset \He$ is a $1$-Ahlfors regular set and the SIO associated with the kernel $K_4$ is $L^2(E)$-bounded, then $E$ is contained in a $1$-Ahlfors regular curve. Combined with the converse implication which was obtained by Fässler and Orponen in \cite{FO1dim}, our result provides a characterization of uniform $1$-rectifiability in the Heisenberg group via the $L^2$-boundedness of a singular integral. We also give a negative answer to a question of Fässler and Orponen from \cite{FO1dim} by showing that for any $α\in (0,2)$ there exists a $1$-Ahlfors regular curve $E_a$ such that the operators associated with the kernels $K_α$ are not bounded in $L^2(E_α)$. We finally show that there exists a $1$-Ahlfors regular and purely $1$-unrectifiable set $E$ such that the singular integral associated with $|x| \|(x,y,z)\|^{-2}$ is $L^2(E)$ -bounded.

math.CA↗

arXiv2025

Singular integrals on $C^{1,α}$ intrinsic graphs in step 2 Carnot groups

We study singular integral operators induced by Calderón-Zygmund kernels in any step-$2$ Carnot group $\mathbb{G}$. We show that if such an operator satisfies some natural cancellation conditions then it is $L^2$ bounded on all intrinsic graphs of $C^{1,α}$ functions over vertical hyperplanes that do not have rapid growth at $\infty$. In particular, the result applies to the Riesz operator $\mathcal{R}$ induced by the kernel $$ \mathsf{R}(z)= \nabla_{\mathbb{G}} Γ(z), \quad z\in \mathbb{G}\backslash \{0\}, $$ the horizontal gradient of the fundamental solution of the sub-Laplacian. The $L^2$ boundedness of $\mathcal{R}$ is connected with the question of removability for Lipschitz harmonic functions. As a corollary of our result, we infer that closed subsets of the intrinsic graphs mentioned above are non-removable.

math.CA↗

arXiv2025

Spectral multipliers for maximally subelliptic operators

Consider a non-negative, self-adjoint, maximally subelliptic operator on a compact manifold. We show that the spectral multiplier is a singular integral operator under an appropriate Mihlin-Hörmander type condition. We establish the equivalence between non-isotropic Besov and Triebel-Lizorkin spaces adapted to the operator and those adapted to a Carnot-Carathéodory geometry on the manifold. We also give a Mihlin-Hörmander type condition for the boundedness of the spectral multiplier on non-isotropic $L^p$ Sobolev spaces.

math.FA↗

arXiv2024

Hilbert transforms and maximal functions along flat curves on the Heisenberg group

We establish the $L^p$ boundedness of Hilbert transforms and maximal functions along flat curves in the Heisenberg group. This generalizes the $\mathbb{R}^n$ result by Carbery, Christ, Vance, Wainger, and Watson. What is new about our result compared to the Heisenberg group generalization by Carbery, Wainger, and Wright is that we allow all three components of the curves to vary independently, we keep the original form of the conditions required in the $\mathbb{R}^n$ case, and our method is likely to be generalized to other stratified nilpotent groups.

math.CA↗

arXiv2022

Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition

We study operators of the form $$ Tf(x)= ψ(x) \int f(γ_t(x))K(t)\,dt, $$ where $γ_t(x)$ is a real analytic function of $(t,x)$ mapping from a neighborhood of $(0,0)$ in $\mathbb{R}^N \times \mathbb{R}^n$ into $\mathbb{R}^n$ satisfying $γ_0(x)\equiv x$, $ψ(x) \in C_c^\infty(\mathbb{R}^n)$, and $K(t)$ is a "multi-parameter singular kernel" with compact support in $\mathbb{R}^N$; for example when $K(t)$ is a product singular kernel. The celebrated work of Christ, Nagel, Stein, and Wainger studied such operators with smooth $γ_t(x)$, in the single-parameter case when $K(t)$ is a Calderón-Zygmund kernel. Street and Stein generalized their work to the multi-parameter case, and gave sufficient conditions for the $L^p$-boundedness of such operators. This paper shows that when $γ_t(x)$ is real analytic, the sufficient conditions of Street and Stein are also necessary for the $L^p$-boundedness of $T$, for all such kernels $K$.

math.CA↗

arXiv2021

3D-FRONT: 3D Furnished Rooms with layOuts and semaNTics

We introduce 3D-FRONT (3D Furnished Rooms with layOuts and semaNTics), a new, large-scale, and comprehensive repository of synthetic indoor scenes highlighted by professionally designed layouts and a large number of rooms populated by high-quality textured 3D models with style compatibility. From layout semantics down to texture details of individual objects, our dataset is freely available to the academic community and beyond. Currently, 3D-FRONT contains 18,968 rooms diversely furnished by 3D objects, far surpassing all publicly available scene datasets. In addition, the 13,151 furniture objects all come with high-quality textures. While the floorplans and layout designs are directly sourced from professional creations, the interior designs in terms of furniture styles, color, and textures have been carefully curated based on a recommender system we develop to attain consistent styles as expert designs. Furthermore, we release Trescope, a light-weight rendering tool, to support benchmark rendering of 2D images and annotations from 3D-FRONT. We demonstrate two applications, interior scene synthesis and texture synthesis, that are especially tailored to the strengths of our new dataset. The project page is at: https://tianchi.aliyun.com/specials/promotion/alibaba-3d-scene-dataset.

cs.CV↗

arXiv2020

Match4Rec: A Novel Recommendation Algorithm Based on Bidirectional Encoder Representation with the Matching Task

Characterizing users' interests accurately plays a significant role in an effective recommender system. The sequential recommender system can learn powerful hidden representations of users from successive user-item interactions and dynamic users' preferences. To analyze such sequential data, conventional methods mainly include Markov Chains (MCs) and Recurrent Neural Networks (RNNs). Recently, the use of self-attention mechanisms and bi-directional architectures have gained much attention. However, there still exists a major limitation in previous works that they only model the user's main purposes in the behavioral sequences separately and locally, and they lack the global representation of the user's whole sequential behavior. To address this limitation, we propose a novel bidirectional sequential recommendation algorithm that integrates the user's local purposes with the global preference by additive supervision of the matching task. We combine the mask task with the matching task in the training process of the bidirectional encoder. A new sample production method is also introduced to alleviate the effect of mask noise. Our proposed model can not only learn bidirectional semantics from users' behavioral sequences but also explicitly produces user representations to capture user's global preference. Extensive empirical studies demonstrate our approach considerably outperforms various state-of-the-art models.

cs.SI↗

arXiv2010

Rigorous formulation of oblique incidence scattering from dispersive media

We formulate a finite-difference time-domain (FDTD) approach to simulate electromagnetic wave scattering from scatterers embedded in layered dielectric or dispersive media. At the heart of our approach is a derivation of an equivalent one-dimensional wave propagation equation for dispersive media characterized by a linear sum of Debye-, Drude- and Lorentz-type poles. The derivation is followed by a detailed discussion of the simulation setup and numerical issues. The developed methodology is tested by comparison with analytical reflection and transmission coefficients for scattering from a slab, illustrating good convergence behavior. The case of scattering from a sub-wavelength slit in a dispersive thin film is explored to demonstrate the applicability of our formulation to time- and incident angle-dependent analysis of surface waves generated by an obliquely incident plane wave.

cond-mat.mes-hall↗