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Yukun Du

Publications and source records attributed to Yukun Du.

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Two-Generator Discrete and Faithful Subgroups of Tree Automorphisms

We present a partial classification of two-generator discrete and faithful subgroups of the trivalent tree automorphism group, specifically for cases where the generators satisfy a restriction on a small geometric quantity. When the restrictions on the geometric quantity or tree valency are relaxed, we discuss the possible reduced quotient graphs for these subgroups and construct infinite families of graphs of groups on each. Additionally, we include a generalized Poincar\'e algorithm that determines whether a given set of tree automorphisms generates a discrete subgroup.

math.GR

Meta-Black-Box Optimization with Ensemble Surrogate Modeling for Robustness-Accuracy Trade-off within SAEA

Surrogate-assisted evolutionary algorithms (SAEAs) have been widely used for expensive black-box optimization problems. However, their reliance on rigid and manually designed components limits their flexibility and generalization across tasks. Meta-black-box optimization (MetaBBO) provides a promising paradigm for adaptively configuring algorithmic components. Nevertheless, existing MetaBBO methods usually control only a single component, and few studies have investigated the unified control of multi-component optimizers such as SAEAs. Moreover, the robustness-accuracy trade-off in surrogate modeling, which is crucial for stable early-stage exploration and accurate late-stage exploitation, has rarely been explicitly considered. To address these issues, we propose AdaE-SAEA, an adaptive ensemble surrogate-assisted evolutionary algorithm for expensive multi-objective optimization. AdaE-SAEA embeds SAEA as the low-level optimizer within the MetaBBO framework and jointly controls the infill criterion and ensemble-based surrogate modeling. Specifically, bagging and boosting are designed as surrogate modeling modules to adaptively balance robustness and accuracy across different search phases, while the meta-policy simultaneously selects the infill criterion to enable adaptive sampling decisions. The meta-policy is trained through reinforcement learning with parallel sampling and centralized training, improving both training efficiency and transferability. Experiments on synthetic and real-world problems demonstrate that AdaE-SAEA outperforms state-of-the-art baselines and MetaBBO-based methods. We further verify the effectiveness of TabPFN as the base surrogate model for ensemble learning. To the best of our knowledge, this is the first work to unify the control of surrogate modeling and infill criteria in SAEAs while explicitly addressing the robustness--accuracy trade-off.

cs.NE

Meta-Black-Box Optimization Can Do Search Guidance for Expensive Constrained Multi-Objective Optimization

Existing Meta-Black-Box Optimization (MetaBBO) methods focus on how to search when controlling optimizers, but largely overlook where to search. We propose MetaSG-SAEA, a bi-level MetaBBO framework for expensive constrained multi-objective optimization problems (ECMOPs), in which a meta-policy provides search guidance to the low-level Surrogate-Assisted Evolutionary Algorithm (SAEA). To achieve this, we introduce Max-Min Constraint-Calibrated Inequality (MM-CCI), a compact, problem-agnostic region abstraction that maps heterogeneous constraint evaluations to an ordered scalar level; we further provide a theoretical analysis of its fundamental properties. Building on this region abstraction, we adopt diffusion-based population initialization to translate the meta-policy's region-level guidance into solution-level priors for the SAEA. To make MetaSG-SAEA scalable, we construct an attention-based state representation across varying problem dimensions, population sizes, and numbers of objectives and constraints. Experimental results demonstrate that MetaSG-SAEA outperforms state-of-the-art baselines across diverse benchmarks and exhibits the ability to generalize across problem distributions.

cs.NE

P-SLCR: Unsupervised Point Cloud Semantic Segmentation via Prototypes Structure Learning and Consistent Reasoning

Current semantic segmentation approaches for point cloud scenes heavily rely on manual labeling, while research on unsupervised semantic segmentation methods specifically for raw point clouds is still in its early stages. Unsupervised point cloud learning poses significant challenges due to the absence of annotation information and the lack of pre-training. The development of effective strategies is crucial in this context. In this paper, we propose a novel prototype library-driven unsupervised point cloud semantic segmentation strategy that utilizes Structure Learning and Consistent Reasoning (P-SLCR). First, we propose a Consistent Structure Learning to establish structural feature learning between consistent points and the library of consistent prototypes by selecting high-quality features. Second, we propose a Semantic Relation Consistent Reasoning that constructs a prototype inter-relation matrix between consistent and ambiguous prototype libraries separately. This process ensures the preservation of semantic consistency by imposing constraints on consistent and ambiguous prototype libraries through the prototype inter-relation matrix. Finally, our method was extensively evaluated on the S3DIS, SemanticKITTI, and Scannet datasets, achieving the best performance compared to unsupervised methods. Specifically, the mIoU of 47.1% is achieved for Area-5 of the S3DIS dataset, surpassing the classical fully supervised method PointNet by 2.5%.

cs.CV

Schottky pairs on Trees via Continued Fractions and Axial Geometry

We give a complete criterion for when two hyperbolic automorphisms of a tree generate a free, discrete subgroup. The decision depends only on three geometric invariants: the translation lengths of the generators and the length of overlap of their axes. This data is organized using the continued-fraction expansion of the translation-length ratio. We extend the result to weighted trees, allowing arbitrary positive real translation lengths under local finiteness. In the irrational case, the exceptional configurations are shown to correspond precisely to the gap lengths in the three-gap theorem.

math.GR

Meta-Black-Box Optimization with Bi-Space Landscape Analysis and Dual-Control Mechanism for SAEA

Surrogate-Assisted Evolutionary Algorithms (SAEAs) are widely used for expensive Black-Box Optimization. However, their reliance on rigid, manually designed components such as infill criteria and evolutionary strategies during the search process limits their flexibility across tasks. To address these limitations, we propose Dual-Control Bi-Space Surrogate-Assisted Evolutionary Algorithm (DB-SAEA), a Meta-Black-Box Optimization (MetaBBO) framework tailored for multi-objective problems. DB-SAEA learns a meta-policy that jointly regulates candidate generation and infill criterion selection, enabling dual control. The bi-space Exploratory Landscape Analysis (ELA) module in DB-SAEA adopts an attention-based architecture to capture optimization states from both true and surrogate evaluation spaces, while ensuring scalability across problem dimensions, population sizes, and objectives. Additionally, we integrate TabPFN as the surrogate model for accurate and efficient prediction with uncertainty estimation. The framework is trained via reinforcement learning, leveraging parallel sampling and centralized training to enhance efficiency and transferability across tasks. Experimental results demonstrate that DB-SAEA not only outperforms state-of-the-art baselines across diverse benchmarks, but also exhibits strong zero-shot transfer to unseen tasks with higher-dimensional settings. This work introduces the first MetaBBO framework with dual-level control over SAEAs and a bi-space ELA that captures surrogate model information.

cs.NE

Busemann-Selberg Functions and Completeness for Dirichlet-Selberg domains in $SL(n,\mathbb{R})/SO(n,\mathbb{R})$

We establish a general completeness criterion for Dirichlet-Selberg domains in the symmetric space $SL(n,\mathbb{R})/SO(n)$. By introducing and analyzing Busemann-Selberg functions - which extend classical Busemann functions and capture asymptotic behavior toward the Satake boundary - we show that every gluing manifold or orbifold produced by Dirichlet-Selberg domain is complete. This result parallels the well-known hyperbolic case and ensures that the key completeness condition in Poincaré's Algorithm always holds in specific cases.

math.GR

A Simple Recursion for the Mirzakhani Volume and its Super Extension

In this paper, we derive a simple recursion formula for the Weil-Petersson volumes of moduli spaces of hyperbolic surfaces with boundaries. This formula demonstrates the polynomiality of the volume functions. By constructing the Laplace transform for both the original and our alternative formulas, we show that they are directly equivalent. By considering the top and lowest degree terms of these formulas, we recover the DVV identity and cohomology class identities for $\mathcal{M}_{g,n}$. Similar conclusions are drawn for the super-analog of these results.

math.AG

Geometry of Selberg's bisectors in the symmetric space $SL(n,\mathbb{R})/SO(n,\mathbb{R})$

We discussed some properties of a family of symmetric spaces, namely $\mathcal{P}(n):=SL(n,\mathbb{R})/SO(n,\mathbb{R})$, where we replace the Riemannian metric on $\mathcal{P}(n)$ with a premetric suggested by Selberg. These properties are generalizations of properties on the hyperbolic spaces $\mathbf{H}^n$ related to Poincaré's fundamental polyhedron theorem.

math.GR