Simon Bortz, Max Engelstein, Max Goering, Tatiana Toro, Zihui Zhao
We provide a potential theoretic characterization of vanishing chord-arc domains under minimal assumptions. In particular we show that, if a domain has Ahlfors regular boundary, the oscillation of the logarithm of the interior and exterior Poisson kernels yields a great deal of geometric information about the domain. We use techniques from the classical calculus of variations, potential theory, quantitative geometric measure theory to accomplish this. One feature of this work, compared to Bortz-Hofmann PAMS 16 and Kenig-Toro Crelle 06, is that a priori we only require that the domains in question are connected.
Joseph Feneuil, Svitlana Mayboroda, Zihui Zhao
The present paper pioneers the study of the Dirichlet problem with $L^q$ boundary data for second order operators with complex coefficients in domains with lower dimensional boundaries, e.g., in $Ω:= \mathbb R^n \setminus \mathbb R^d$ with $d<n-1$. Following the first results of Guy David and the two first authors, the article introduces an appropriate degenerate elliptic operator and show that the Dirichlet problem is solvable for all $q>1$ provided that the coefficients satisfy the small Carleson norm condition. Even in the context of the classical case $d=n-1$, (the analogues of) our results are new. The conditions on the coefficients are more relaxed than the previously known ones (most notably, we do not impose any restrictions whatsoever on the first $n-1$ rows of the matrix of coefficients) and the results are more general. We establish local rather than global estimates between the square function and the non-tangential maximal function and, perhaps even more importantly, we establish new Moser-type estimates at the boundary and improve the interior ones.
Mingming Cao, Pablo Hidalgo-Palencia, José María Martell, Cruz Prisuelos-Arribas, Zihui Zhao
In this paper we study the Dirichlet problem for real-valued second order divergence form elliptic operators with boundary data in Hölder spaces. Our context is that of open sets $Ω\subset \mathbb{R}^{n+1}$, $n \ge 2$, satisfying the capacity density condition, without any further topological assumptions. Our main result states that if $Ω$ is either bounded, or unbounded with unbounded boundary, then the corresponding Dirichlet boundary value problem is well-posed; when $Ω$ is unbounded with bounded boundary, we establish that solutions exist, but they fail to be unique in general. These results are optimal in the sense that solvability of the Dirichlet problem in Hölder spaces is shown to imply the capacity density condition. As a consequence of the main result, we present a characterization of the Hölder spaces in terms of the boundary traces of solutions, and obtain well-posedness of several related Dirichlet boundary value problems. All the results above are new even for 1-sided chord-arc domains, and can be extended to generalized Hölder spaces associated with a natural class of growth functions.
Steve Hofmann, José María Martell, Svitlana Mayboroda, Tatiana Toro, Zihui Zhao
The present paper establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular $L^p$ data case. We split our proof on two main steps. In the first one we considered the case in which the desired Carleson measure condition on the coefficients holds with "sufficiently small constant", using a novel application of techniques developed in geometric measure theory. In the second step we establish the final result, that is, the "large constant case". The key elements are a powerful extrapolation argument, which provides a general pathway to self-improve scale-invariant small constant estimates, and a new mechanism to transfer quantitative absolute continuity of elliptic measure between a domain and its subdomains.
Carlos Kenig, Zihui Zhao
Let $u$ be a harmonic function in a $C^1$-Dini domain, such that $u$ vanishes on an open set of the boundary. We show that near every point in the open set, $u$ can be written uniquely as the sum of a non-trivial homogeneous harmonic polynomial and an error term of higher degree (depending on the Dini parameter). In particular, this implies that $u$ has a unique tangent function at every such point, and that the convergence rate to the tangent function can be estimated. We also study the relationship of tangent functions at nearby points in a special case.
Carlos Kenig, Zihui Zhao
Let $u$ be a harmonic function in a $C^1$-Dini domain $D$ such that $u$ vanishes on a boundary surface ball $\partial D \cap B_{5R}(0)$. We consider an effective version of its singular set (up to boundary) $\mathcal{S}(u):=\{X\in \overline{D}: u(X) = |\nabla u(X)| = 0\} $ and give an estimate of its $(d-2)$-dimensional Minkowski content, which only depends on the upper bound of some modified frequency function of $u$ centered at $0$. Such results are already known in the interior and at the boundary of convex domains, when the standard frequency function is monotone at every point. The novelty of our work on Dini domains is how to compensate for the lack of such monotone quantities at boundary as well as interior points.
Carlos Kenig, Zihui Zhao
Let $u$ be a harmonic function in a $C^1$ domain $D\subset \mathbb{R}^d$, which vanishes on an open subset of the boundary. In this note we study its critical set $\{x \in \overline{D}: \nabla u(x) = 0 \}$. When $D$ is a $C^{1,α}$ domain for some $α\in (0,1]$, we give an upper bound on the $(d-2)$-dimensional Hausdorff measure of the critical set by the frequency function. We also discuss possible ways to extend such estimate to all $C^1$-Dini domains, the optimal class of domains for which analogous estimates have been shown to hold for the singular set $\{x \in \overline{D}: u(x) = 0 = |\nabla u(x)| \}$ (see [KZ1, KZ2]).
Zihui Zhao
We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular boundary, the BMO solvability of such problems is equivalent to a quantitative absolute continuity of the elliptic measure with respect to the surface measure, i.e. $ω_L\in A_{\infty}(σ)$. This generalizes a previous result on Lipschitz domains by Dindos, Kenig and Pipher.
Tatiana Toro, Zihui Zhao
We consider second order divergence form elliptic operators with $W^{1,1}$ coefficients, in a uniform domain $Ω$ with Ahlfors regular boundary. We show that the $A_\infty$ property of the elliptic measure associated to any such operator implies that $Ω$ is a set of locally finite perimeter whose boundary, $\partialΩ$, is rectifiable. As a corollary we show that for this type of operators, absolute continuity of the surface measure with respect to the elliptic measure is enough to guarantee rectifiability of the boundary. In the case that the coefficients are continuous we obtain additional information about $Ω$.
Zihui Zhao, Yingxin Li, Yang Li
Multimodal Large Language Models (MLLMs) have demonstrated exceptional success in various multimodal tasks, yet their deployment is frequently limited by substantial computational demands and prolonged inference times. Given that the vision modality typically contains more comprehensive information than the text modality, resulting in encoded representations comprising an extensive number of tokens, leading to significant computational overhead due to the quadratic complexity of the attention mechanism. Current token reduction methods are typically restricted to specific model architectures and often necessitate extensive retraining or fine-tuning, restricting their applicability to many state-of-the-art models. In this paper, we introduce a learning-free token reduction (LFTR) method designed for MLLMs. LFTR can be seamlessly integrated into most open-source MLLM architectures without requiring additional fine-tuning. By capitalizing on the redundancy in visual representations, our approach effectively reduces tokens while preserving the general inference performance of MLLMs. We conduct experiments on multiple MLLM architectures (LLaVA, MiniGPT, QwenVL), and our results show that LFTR achieves up to a $16\times$ reduction of visual tokens while maintaining or even enhancing performance on mainstream vision question-answering benchmarks, all in a learning-free setting. Additionally, LFTR is complementary to other acceleration techniques, such as vision encoder compression and post-training quantization, further promoting the efficient deployment of MLLMs. Our project is available at https://anonymous.4open.science/r/LFTR-AAAI-0528.
Zihui Zhao, Yuanbo Tang, Jieyu Ren, Xiaoping Zhang, Yang Li
Dictionary learning is traditionally formulated as an $L_1$-regularized signal reconstruction problem. While recent developments have incorporated discriminative, hierarchical, or generative structures, most approaches rely on encouraging representation sparsity over individual samples that overlook how atoms are shared across samples, resulting in redundant and sub-optimal dictionaries. We introduce a parsimony promoting regularizer based on the row-wise $L_\infty$ norm of the coefficient matrix. This additional penalty encourages entire rows of the coefficient matrix to vanish, thereby reducing the number of dictionary atoms activated across the dataset. We derive the formulation from a probabilistic model with Beta-Bernoulli priors, which provides a Bayesian interpretation linking the regularization parameters to prior distributions. We further establish theoretical calculation for optimal hyperparameter selection and connect our formulation to both Minimum Description Length, Bayesian model selection and pathlet learning. Extensive experiments on benchmark datasets demonstrate that our method achieves substantially improved reconstruction quality (with a 20\% reduction in RMSE) and enhanced representation sparsity, utilizing fewer than one-tenth of the available dictionary atoms, while empirically validating our theoretical analysis.
Camillo De Lellis, Zihui Zhao
In this paper, we consider multi-valued graphs with a prescribed real analytic interface that minimize the Dirichlet energy. Such objects arise as a linearized model of area minimizing currents with real analytic boundaries and our main result is that their singular set is discrete in 2 dimensions. This confirms (and provides a first step to) a conjecture by B. White \cite{White97} that area minimizing $2$-dimensional currents with real analytic boundaries have a finite number of singularities. We also show that, in any dimension, Dirichlet energy-minimizers with a $C^1$ boundary interface are Hölder continuous at the interface.
Simon Bortz, Tatiana Toro, Zihui Zhao
We show that if $Ω$ is a vanishing chord-arc domain and $L$ is a divergence-form elliptic operator with Hölder-continuous coefficient matrix, then $\log k_L \in VMO$, where $k_L$ is the elliptic kernel for $L$ in the domain $Ω$. This extends the previous work of Kenig and Toro in the case of the Laplacian.
Svitlana Mayboroda, Zihui Zhao
In the recent work [DFM1, DFM2] G. David, J. Feneuil, and the first author have launched a program devoted to an analogue of harmonic measure for lower-dimensional sets. A relevant class of partial differential equations, analogous to the class of elliptic PDEs in the classical context, is given by linear degenerate equations with the degeneracy suitably depending on the distance to the boundary. The present paper continues this line of research and focuses on the criteria of quantitative absolute continuity of the newly defined harmonic measure with respect to the Hausdorff measure, $ω\in A_\infty(σ)$, in terms of solvability of boundary value problems. The authors establish, in particular, square function estimates and solvability of the Dirichlet problem in BMO for domains with lower-dimensional boundaries under the underlying assumption $ω\in A_\infty(σ)$. More generally, it is proved that in all domains with Ahlfors regular boundaries the BMO solvability of the Dirichlet problem is necessary and sufficient for the absolute continuity of the harmonic measure.
Carlos Kenig, Zihui Zhao
Let $u$ be a non-trivial harmonic function in a domain $D\subset \mathbb{R}^d$ which vanishes on an open set of the boundary. In a recent paper, we showed that if $D$ is a $C^1$-Dini domain, then within the open set the singular set of $u$, defined as $\{X\in \overline{D}: u(X) = 0 = |\nabla u(X)|\} $, has finite $(d-2)$-dimensional Hausdorff measure. In this paper, we show that the assumption of $C^1$-Dini domains is sharp, by constructing a large class of non-Dini (but almost Dini) domains whose \textit{singular sets} have infinite $\mathcal{H}^{d-2}$-measures.
Zihui Zhao, Yifei Zhang, Zheng Wang, Yang Li, Kui Jiang, Zihan Geng, Chia-Wen Lin
The raw depth images captured by RGB-D cameras using Time-of-Flight (TOF) or structured light often suffer from incomplete depth values due to weak reflections, boundary shadows, and artifacts, which limit their applications in downstream vision tasks. Existing methods address this problem through depth completion in the image domain, but they overlook the physical characteristics of raw depth images. It has been observed that the presence of invalid depth areas alters the frequency distribution pattern. In this work, we propose a Spatio-Spectral Mutual Learning framework (S2ML) to harmonize the advantages of both spatial and frequency domains for depth completion. Specifically, we consider the distinct properties of amplitude and phase spectra and devise a dedicated spectral fusion module. Meanwhile, the local and global correlations between spatial-domain and frequency-domain features are calculated in a unified embedding space. The gradual mutual representation and refinement encourage the network to fully explore complementary physical characteristics and priors for more accurate depth completion. Extensive experiments demonstrate the effectiveness of our proposed S2ML method, outperforming the state-of-the-art method CFormer by 0.828 dB and 0.834 dB on the NYU-Depth V2 and SUN RGB-D datasets, respectively.
Zihui Zhao, Zechang Li
Direct Preference Optimization (DPO) has emerged as a lightweight and effective alternative to Reinforcement Learning from Human Feedback (RLHF) and Reinforcement Learning with AI Feedback (RLAIF) for aligning large language and vision-language models. However, the standard DPO formulation, in which both the chosen and rejected responses are generated by the same policy, suffers from a weak learning signal because the two responses often share similar errors and exhibit small Kullback-Leibler (KL) divergence. This leads to slow and unstable convergence. To address this limitation, we introduce Reflective Preference Optimization (RPO), a new framework that incorporates hint-guided reflection into the DPO paradigm. RPO uses external models to identify hallucination sources and generate concise reflective hints, enabling the construction of on-policy preference pairs with stronger contrastiveness and clearer preference signals. We theoretically show that conditioning on hints increases the expected preference margin through mutual information and improves sample efficiency while remaining within the policy distribution family. Empirically, RPO achieves superior alignment with fewer training samples and iterations, substantially reducing hallucination rates and delivering state-of-the-art performance across multimodal benchmarks.
Steve Hofmann, José María Martell, Svitlana Mayboroda, Tatiana Toro, Zihui Zhao
The present paper, along with its companion [Hofmann, Martell, Mayboroda, Toro, Zhao, arXiv:1710.06157], establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular $L^p$ data case. The first step in this direction was taken in our previous paper [Hofmann, Martell, Mayboroda, Toro, Zhao, arXiv:1710.06157], where we considered the case in which the desired Carleson measure condition on the coefficients holds with sufficiently small constant. In this paper we establish the final, general result, that is, the "large constant case". The key elements of our approach are a powerful extrapolation argument, which provides a general pathway to self-improve scale-invariant small constant estimates, as well as a new mechanism to transfer quantitative absolute continuity of elliptic measure between a domain and its subdomains.
Simon Bortz, Tatiana Toro, Zihui Zhao
Questions concerning quantitative and asymptotic properties of the elliptic measure corresponding to a uniformly elliptic divergence form operator have been the focus of recent studies. In this setting we show that the elliptic measure of an operator with coefficients satisfying a vanishing Carleson condition in the upper half space is an asymptotically optimal $A_\infty$ weight. In particular, for such operators the logarithm of the elliptic kernel is in the space of (locally) vanishing mean oscillation. To achieve this, we prove local, quantitative estimates on a quantity (introduced by Fefferman, Kenig and Pipher) that controls the $A_\infty$ constant. Our work uses recent results obtained by David, Li and Mayboroda. These quantitative estimates may offer a new framework to approach similar problems.
Steve Hofmann, José María Martell, Svitlana Mayboroda, Tatiana Toro, Zihui Zhao
The present paper, along with its sequel, establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular $L^p$ data case. This paper addresses the free boundary problem under the assumption of smallness of the Carleson measure of the coefficients. Part II of this work develops an extrapolation argument to bootstrap this result to the general case. The ideas in Part I constitute a novel application of techniques developed in geometric measure theory. They highlight the synergy between several areas. The ideas developed in this paper are well suited to study singularities arising in variational problems in a geometric setting.