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Dekai Zhang

Publications and source records attributed to Dekai Zhang.

At least 19 recordsLinked to original sources
arXiv2026

Concavity Properties of Robin Solutions on $C^{3,1}$ Uniformly Convex Domains

We prove that, on a bounded uniformly convex domain of class $C^{3,1}$, the first Robin eigenfunction is strictly log-concave and the Robin torsion function is strictly $1/2$-concave for all sufficiently large Robin parameters. This means that Conjecture 1.1 of Andrews, Clutterbuck and Hauer holds for uniformly convex $C^{3,1}$ domains in any dimension, and settles an open problem posed by Crasta and Fragal\`a when the domain has $C^{3,1}$ regularity. Our proof derives suitable uniform $C^2$-decay estimates by maximum principle arguments, thereby removing the imposed higher-order boundary regularity assumption required previously.

math.AP↗
arXiv2026

Limiting Behavior of a Class of Hermitian Yang--Mills Metrics, II: Exponential Approximation

This paper is a sequel to [9], where the first-named author constructed a family of approximate Hermitian Yang--Mills metrics $H_{0,\epsilon}$ on stable rank-two holomorphic vector bundles arising from double spectral covers over the product of two one-dimensional complex tori. We prove that these approximate metrics give an all-order, exponentially accurate asymptotic description of the exact Hermitian Yang--Mills metrics in the large K\"ahler limit. More precisely, the mean curvature of $H_{0,\epsilon}$ decays exponentially in every $C^k$-norm. Moreover, if $H_{1,\epsilon}$ denotes the exact Hermitian Yang--Mills metric and \[ H_\epsilon=H_{0,\epsilon}^{-1}H_{1,\epsilon}, \] then, after normalization, for every nonnegative integer $k$, there exist positive constants $C_k$ and $c_k$ such that \[ \|H_\epsilon- Id\|_{C^k}\leq C_k e^{-\frac{c_{k}}{\epsilon}}. \] The main analytic difficulty lies in the global $C^0$-comparison. Obtaining $C^0$-estimates for the coupled nonlinear Hermitian Yang--Mills system is intrinsically difficult; moreover, the equation controls only the contraction of the curvature, and hence only certain combinations of second derivatives, whereas one needs global control of the full matrix-valued metric.

math.DG↗
arXiv2026

A Numerical Criterion for the 2-Hessian Equation on Compact Kähler Manifolds

We show that a Nakai--Moishezon-type criterion associated with the complex $2$-Hessian equation produces a Gauduchon class. In complex dimension three, this numerical criterion is equivalent to the existence of a smooth $2$-admissible representative and hence to the solvability of the $2$-Hessian equation. As consequences of these results, we prove the corresponding conjectures of Murakami for the complex Hessian equation and of Székelyhidi for the Hessian quotient equation in dimension three. We also establish a boundary version of the above results.

math.DG↗
arXiv2026

The Critical LYZ Equation in Kähler Geometry

We establish the existence of smooth solutions for the LYZ equation at the critical phase $θ=(n-2)\fracπ{2}$, thereby solving the critical case of a problem posed by Collins-Jacob-Yau and Li concerning the solvability for phase $θ\leq (n-2)\fracπ{2}$. As applications, we solve the 3D Hessian equation $σ_2 = 1$ and the 4D Hessian quotient equation $σ_3 = σ_1$ under weaker assumptions than previously required.

math.DG↗
arXiv2025

Meronymic Ontology Extraction via Large Language Models

Ontologies have become essential in today's digital age as a way of organising the vast amount of readily available unstructured text. In providing formal structure to this information, ontologies have immense value and application across various domains, e.g., e-commerce, where countless product listings necessitate proper product organisation. However, the manual construction of these ontologies is a time-consuming, expensive and laborious process. In this paper, we harness the recent advancements in large language models (LLMs) to develop a fully-automated method of extracting product ontologies, in the form of meronymies, from raw review texts. We demonstrate that the ontologies produced by our method surpass an existing, BERT-based baseline when evaluating using an LLM-as-a-judge. Our investigation provides the groundwork for LLMs to be used more generally in (product or otherwise) ontology extraction.

cs.CL↗
arXiv2025

Clustered Federated Learning via Embedding Distributions

Federated learning (FL) is a widely used framework for machine learning in distributed data environments where clients hold data that cannot be easily centralised, such as for data protection reasons. FL, however, is known to be vulnerable to non-IID data. Clustered FL addresses this issue by finding more homogeneous clusters of clients. We propose a novel one-shot clustering method, EMD-CFL, using the Earth Mover's distance (EMD) between data distributions in embedding space. We theoretically motivate the use of EMDs using results from the domain adaptation literature and demonstrate empirically superior clustering performance in extensive comparisons against 16 baselines and on a range of challenging datasets.

cs.LG↗
arXiv2025

XAI-Units: Benchmarking Explainability Methods with Unit Tests

Feature attribution (FA) methods are widely used in explainable AI (XAI) to help users understand how the inputs of a machine learning model contribute to its outputs. However, different FA models often provide disagreeing importance scores for the same model. In the absence of ground truth or in-depth knowledge about the inner workings of the model, it is often difficult to meaningfully determine which of the different FA methods produce more suitable explanations in different contexts. As a step towards addressing this issue, we introduce the open-source XAI-Units benchmark, specifically designed to evaluate FA methods against diverse types of model behaviours, such as feature interactions, cancellations, and discontinuous outputs. Our benchmark provides a set of paired datasets and models with known internal mechanisms, establishing clear expectations for desirable attribution scores. Accompanied by a suite of built-in evaluation metrics, XAI-Units streamlines systematic experimentation and reveals how FA methods perform against distinct, atomic kinds of model reasoning, similar to unit tests in software engineering. Crucially, by using procedurally generated models tied to synthetic datasets, we pave the way towards an objective and reliable comparison of FA methods.

cs.LG↗
arXiv2025

Hidden Conflicts in Neural Networks and Their Implications for Explainability

Artificial Neural Networks (ANNs) often represent conflicts between features, arising naturally during training as the network learns to integrate diverse and potentially disagreeing inputs to better predict the target variable. Despite their relevance to the ``reasoning'' processes of these models, the properties and implications of conflicts for understanding and explaining ANNs remain underexplored. In this paper, we develop a rigorous theory of conflicts in ANNs and demonstrate their impact on ANN explainability through two case studies. In the first case study, we use our theory of conflicts to inspire the design of a novel feature attribution method, which we call Conflict-Aware Feature-wise Explanations (CAFE). CAFE separates the positive and negative influences of features and biases, enabling more faithful explanations for models applied to tabular data. In the second case study, we take preliminary steps towards understanding the role of conflicts in out-of-distribution (OOD) scenarios. Through our experiments, we identify potentially useful connections between model conflicts and different kinds of distributional shifts in tabular and image data. Overall, our findings demonstrate the importance of accounting for conflicts in the development of more reliable explanation methods for AI systems, which are crucial for the beneficial use of these systems in the society.

cs.LG↗
arXiv2025

A new flow solving the LYZ equation in Kähler geometry

We introduced a new flow to the LYZ equation on a compact Kähler manifold. We first show the existence of the longtime solution of the flow. We then show that under the Collins-Jacob-Yau's condition on the subsolution, the longtime solution converges to the solution of the LYZ equation, which was solved by Collins-Jacob-Yau [5] by the continuity method. Moreover, as an application of the flow, we show that on a compact Kähler surface, if there exists a semi-subsolution of the LYZ equation, then our flow converges smoothly to a singular solution to the LYZ equation away from a finite number of curves of negative self-intersection. Such a solution can be viewed as a boundary point of the moduli space of the LYZ solutions for a given Kähler metric.

math.DG↗
arXiv2024

The Neumann problem of special Lagrangian type equations

We study the Neumann problem for special Lagrangian type equations with critical and supercritical phases. These equations naturally generalize the special Lagrangian equation and the k-Hessian equation. By establishing uniform a priori estimates up to the second order, we obtain the existence result using the continuity method. The new technical aspect is our direct proof of boundary double normal derivative estimates. In particular, we directly prove the double normal estimates for the 2-Hessian equation in dimension 3. Moreover, we solve the classical Neumann problem by proving the uniform gradient estimate.

math.AP↗
arXiv2024

Contestable AI needs Computational Argumentation

AI has become pervasive in recent years, but state-of-the-art approaches predominantly neglect the need for AI systems to be contestable. Instead, contestability is advocated by AI guidelines (e.g. by the OECD) and regulation of automated decision-making (e.g. GDPR). In this position paper we explore how contestability can be achieved computationally in and for AI. We argue that contestable AI requires dynamic (human-machine and/or machine-machine) explainability and decision-making processes, whereby machines can (i) interact with humans and/or other machines to progressively explain their outputs and/or their reasoning as well as assess grounds for contestation provided by these humans and/or other machines, and (ii) revise their decision-making processes to redress any issues successfully raised during contestation. Given that much of the current AI landscape is tailored to static AIs, the need to accommodate contestability will require a radical rethinking, that, we argue, computational argumentation is ideally suited to support.

cs.AI↗
arXiv2024

The Neumann Problem for Parabolic Hessian Quotient Equations

In this paper, we consider the Neumann problem for parabolic Hessian quotient equations. We show that the $k$-admissible solution of the parabolic Hessian quotient equation exists for all time and converges to the smooth solution of elliptic Hessian quotient equations. Also the solutions of the classical Neumann problem converge to a translating solution.

math.AP↗
arXiv2024

The exterior Dirichlet problem for the homogeneous $k$-Hessian equation

We study the exterior Dirichlet problem for the homogeneous $k$-Hessian equation. The prescribed asymptotic behavior at infinity of the solution is zero if $k<\frac{n}{2}$, it is $\log|x|+O(1)$ if $k=\frac{n}{2}$ and it is $|x|^{\frac{2k-n}{n}}+O(1)$ if $k>\frac{n}{2}$. By constructing smooth solutions of approximating non-degenerate $k$-Hessian equations with uniform $C^{1,1}$-estimates, we prove the existence part. The uniqueness follows from the comparison theorem and thus the $C^{1,1}$ regularity of the solution of the homogeneous $k$-Hessian equation in the exterior domain is proved. We also prove a uniform positive lower bound of the gradient. As an implication of the $C^{1,1}$ estimates, we derive an almost monotonicity formula along the level set of the approximating solution. In particular, we get an weighted geometric inequality which is a natural generalization of the $k=1$ case.

math.AP↗
arXiv2023

Targeted Activation Penalties Help CNNs Ignore Spurious Signals

Neural networks (NNs) can learn to rely on spurious signals in the training data, leading to poor generalisation. Recent methods tackle this problem by training NNs with additional ground-truth annotations of such signals. These methods may, however, let spurious signals re-emerge in deep convolutional NNs (CNNs). We propose Targeted Activation Penalty (TAP), a new method tackling the same problem by penalising activations to control the re-emergence of spurious signals in deep CNNs, while also lowering training times and memory usage. In addition, ground-truth annotations can be expensive to obtain. We show that TAP still works well with annotations generated by pre-trained models as effective substitutes of ground-truth annotations. We demonstrate the power of TAP against two state-of-the-art baselines on the MNIST benchmark and on two clinical image datasets, using four different CNN architectures.

cs.CV↗
arXiv2023

The Dirichlet problem of homogeneous complex k-Hessian equation in a (k-1)-pesudoconvex domain with isolated singularity

In this paper, we consider the homogeneous complex k-Hessian equation in $Ω\backslash\{0\}$. We prove the existence and uniqueness of the $C^{1,α}$ solution by constructing approximating solutions. The key point for us is to construct the subsolution for approximating problem and establish uniform gradient estimates and complex Hessian estimates which is independent of the approximation.

math.AP↗
arXiv2023

The Dirichlet problem of the homogeneous $k$-Hessian equation in a punctured domain

In this paper, we consider the Dirichlet problem for the homogeneous $k$-Hessian equation with prescribed asymptotic behavior at $0\inΩ$ where $Ω$ is a $(k-1)$-convex bounded domain in the Euclidean space. The prescribed asymptotic behavior at $0$ of the solution is zero if $k>\frac{n}{2}$, it is $\log|x|+O(1)$ if $k=\frac{n}{2}$ and $-|x|^{\frac{2k-n}{n}}+O(1)$ if $k<\frac{n}{2}$. To solve this problem, we consider the Dirichlet problem of the approximating $k$-Hessian equation in $Ω\setminus \overline{B_r(0)}$ with $r$ small. We firstly construct the subsolution of the approximating $k$-Hessian equation. Then we derive the pointwise $C^{2}$-estimates of the approximating equation based on new gradient and second order estimates established previously by the second author and the third author. In addition, we prove a uniform positive lower bound of the gradient if the domain is starshaped with respect to $0$. As an application, we prove an identity along the level set of the approximating solution and obtain a nearly monotonicity formula. In particular, we get a weighted geometric inequality for smoothly and strictly $(k-1)$-convex starshaped closed hypersurface in $\mathbb R^n$ with $\frac{n}{2}\le k<n$.

math.AP↗
arXiv2023

The Monge-Ampère equation for $(n-1)$-quaternionic PSH functions on a hyperKähler manifold

We prove the existence of unique smooth solutions to the quaternionic Monge-Ampère equation for $(n-1)$-quaternionic plurisubharmonic functions on a hyperKähler manifold and thus obtain solutions for the quaternionic form type equation. We derive $C^0$ estimate by establishing a Cherrier-type inequality as in Tosatti and Weinkove [22]. By adopting the approach of Dinew and Sroka [9] to our context, we obtain $C^1$ and $C^2$ estimates without assuming the flatness of underlying hyperKähler metric comparing to previous results [14].

math.DG↗