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Zihui Zhao

Publications and source records attributed to Zihui Zhao.

At least 19 recordsLinked to original sources
arXiv2026

Singularity formation and self-shrinkers for the parabolic Bernoulli problem

We study singularity formation for classical solutions of the parabolic Bernoulli free boundary problem and the associated self-similar profiles. Inspired by Ricci and mean curvature flows, we introduce a Type I assumption on the blow-up rate at a singular free boundary point, under which the parabolic rescalings are compact and every tangent flow is a non-trivial self-shrinking solution of the elliptic profile equation. The proof establishes convergence of the positivity sets and recovers the Bernoulli condition in the limit, despite the absence of a minimizing structure. In the subcritical regime, every tangent flow is a double plane. We next classify all radial self-shrinkers. Besides the known ball and exterior profiles, there is a unique annular profile. We obtain dimension-uniform bounds and sharp asymptotics for its inner and outer radii, including a quantitative outward bias of its midpoint. These delicate estimates yield a complete classification of the signs of the linear spectrum for the annular solution in every dimension. As a consequence, the ball is dynamically stable modulo ambient symmetries, whereas the annulus is dynamically unstable, with genuine unstable modes of angular degrees $0,1,2,3$. We finally prove, from the spectral nondegeneracy of the compact radial profiles, that whenever one tangent flow is a ball or an annular solution, the tangent flow is unique. One borderline spectral sign in dimensions $2\leq n \leq 4000$ is verified by a rigorous computer-assisted argument using interval arithmetic.

math.AP↗
arXiv2026

On the Sparsity-Storage-Accuracy Tradeoff in Parsimoniously Activated Dictionary Learning

Dictionary learning has long been studied from both optimization and probabilistic perspectives. While formulations with element-wise sparsity regularization (e.g., L1-based sparse coding) admit well-established probabilistic interpretations, many structured variants that impose global constraints lack a clear and tractable generative view. In this paper, we revisit a class of practically effective yet theoretically under-explored dictionary learning methods that impose a simple global regularization on the number of activated dictionary atoms, which we term parsimoniously activated dictionary learning (PADL). We show that PADL admits an equivalent formulation as maximum a posteriori estimation under a structured generative model, with auxiliary latent variables that govern global activation patterns. This formulation allows us to derive generalization guarantees that are difficult to obtain under the original formulation. More importantly, it yields an analytical characterization of the tradeoff between sparsity, storage cost, and reconstruction accuracy, enabling data-driven estimation of optimal hyperparameters. Based on this connection, we develop an efficient and interpretable PADL algorithm that eliminates manual hyperparameter tuning, achieving improved reconstruction performance under comparable sparsity levels on visual benchmarks. We further demonstrate its practical utility in accelerating inference for vision-language models.

cs.LG↗
arXiv2025

Reflective Preference Optimization (RPO): Enhancing On-Policy Alignment via Hint-Guided Reflection

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.

cs.AI↗
arXiv2025

Data-Efficient and Robust Trajectory Generation through Pathlet Dictionary Learning

Trajectory generation has recently drawn growing interest in privacy-preserving urban mobility studies and location-based service applications. Although many studies have used deep learning or generative AI methods to model trajectories and have achieved promising results, the robustness and interpretability of such models are largely unexplored. This limits the application of trajectory generation algorithms on noisy real-world data and their trustworthiness in downstream tasks. To address this issue, we exploit the regular structure in urban trajectories and propose a deep generative model based on the pathlet representation, which encode trajectories with binary vectors associated with a learned dictionary of trajectory segments. Specifically, we introduce a probabilistic graphical model to describe the trajectory generation process, which includes a Variational Autoencoder (VAE) component and a linear decoder component. During training, the model can simultaneously learn the latent embedding of pathlet representations and the pathlet dictionary that captures mobility patterns in the trajectory dataset. The conditional version of our model can also be used to generate customized trajectories based on temporal and spatial constraints. Our model can effectively learn data distribution even using noisy data, achieving relative improvements of $35.4\%$ and $26.3\%$ over strong baselines on two real-world trajectory datasets. Moreover, the generated trajectories can be conveniently utilized for multiple downstream tasks, including trajectory prediction and data denoising. Lastly, the framework design offers a significant efficiency advantage, saving $64.8\%$ of the time and $56.5\%$ of GPU memory compared to previous approaches.

cs.LG↗
arXiv2025

On the asymptotic properties of solutions to one-phase free boundary problems

In this article we study the structure of solutions to the one-phase Bernoulli problem that are modeled either infinitesimally or at infinity by one-homogeneous solutions with an isolated singularity. In particular, we prove a uniqueness of blowups result under a natural symmetry condition on the one-homogeneous solution (\`a la Allard--Almgren) and we prove a rigidity result at infinity (\`a la Simon--Solomon) under additional constraints on the linearized operator around the one-homogeneous solution (which are satisfied by the only known examples of minimizing one-homogeneous solutions). We believe these are the first uniqueness of blow-up/blow-down results at singular points for non-minimizing solutions to the one-phase problem.

math.AP↗
arXiv2025

S2ML: Spatio-Spectral Mutual Learning for Depth Completion

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.

cs.CV↗
arXiv2025

A Unified Probabilistic Framework for Dictionary Learning with Parsimonious Activation

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.

cs.LG↗
arXiv2025

LFTR: Learning-Free Token Reduction for Multimodal Large Language Models

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.

cs.CV↗
arXiv2024

Integer Topological Defects Reveal Anti-Symmetric Forces in Active Nematics

Cell layers are often categorized as contractile or extensile active nematics but recent experiments on neural progenitor cells with induced $+1$ topological defects challenge this classification. In a bottom-up approach, we first study a relevant particle-level model and then analyze a continuous theory derived from it. We show that both model and theory account qualitatively for the main experimental result, i.e. accumulation of cells at the core of any type of +1 defect. We argue that cell accumulation is essentially due to two generally ignored 'effective active forces'. We finally discuss the relevance and consequences of our findings in the context of other cellular active nematics experiments and previously proposed theories.

cond-mat.soft↗
arXiv2024

Experimental identification of force, velocity, and nematic order relationships in active nematic cell monolayers

Cell alignment often forms nematic order, which can lead to anomalous collective cell flow due to the so-called active force. Although it is appreciated that cell migration is driven by traction force, a quantitative evaluation of the relationships between the traction force, the nematic patterning, and the cell flow velocity is still elusive. Here we have found that cellular traction force aligns almost perfectly and is proportional in amplitude to the gradient of the nematic order tensor, not only near the topological defects but also globally. Furthermore, the flow in the monolayer was best described by adding nonlinear forces and a diffusion term derived from symmetry considerations. These nonlinear active forces enhance density instability but suppress bending instability, explaining why cell accumulation and dispersion can occur in neural progenitor cell culture while their ordering pattern is stable.

cond-mat.soft↗
arXiv2024

A note on the critical set of harmonic functions near the boundary

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 $C^{1,\alpha}$ domain for some $\alpha \in (0,1]$, 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]).

math.AP↗
arXiv2023

Elliptic operators in rough sets, and the Dirichlet problem with boundary data in Hölder spaces

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.

math.AP↗
arXiv2022

Examples of non-Dini domains with large singular sets

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.

math.CV↗
arXiv2021

Elliptic measures for Dahlberg-Kenig-Pipher operators: Asymptotically optimal estimates

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.

math.AP↗
arXiv2021

Expansion of harmonic functions near the boundary of Dini domains

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.

math.AP↗
arXiv2021

Boundary unique continuation on $C^1$-Dini domains and the size of the singular set

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.

math.AP↗
arXiv2020

Two Phase Free Boundary Problem for Poisson Kernels

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.

math.CA↗