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Huan Xiong

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math.CO Artificial Intelligence Computer Vision Machine Learning math.OC

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preprint2026arXiv

New Insight of Variance reduce in Zero-Order Hard-Thresholding: Mitigating Gradient Error and Expansivity Contradictions

Hard-thresholding is an important type of algorithm in machine learning that is used to solve $\ell_0$ constrained optimization problems. However, the true gradient of the objective function can be difficult to access in certain scenarios, which normally can be approximated by zeroth-order (ZO) methods. The SZOHT algorithm is the only algorithm tackling $\ell_0$ sparsity constraints with ZO gradients so far. Unfortunately, SZOHT has a notable limitation on the number of random directions % in ZO gradients due to the inherent conflict between the deviation of ZO gradients and the expansivity of the hard-thresholding operator. This paper approaches this problem by considering the role of variance and provides a new insight into variance reduction: mitigating the unique conflicts between ZO gradients and hard-thresholding. Under this perspective, we propose a generalized variance reduced ZO hard-thresholding algorithm as well as the generalized convergence analysis under standard assumptions. The theoretical results demonstrate the new algorithm eliminates the restrictions on the number of random directions, leading to improved convergence rates and broader applicability compared with SZOHT. Finally, we illustrate the utility of our method on a ridge regression problem as well as black-box adversarial attacks.

preprint2026arXiv

Unleashing LLMs in Bayesian Optimization: Preference-Guided Framework for Scientific Discovery

Scientific discovery is increasingly constrained by costly experiments and limited resources, underscoring the need for efficient optimization in AI for science. Bayesian Optimization (BO), though widely adopted for balancing exploration and exploitation, often exhibits slow cold-start performance and poor scalability in high-dimensional settings, limiting its applicability in real-world scientific problems. To overcome these challenges, we propose LLM-Guided Bayesian Optimization (LGBO), the first LLM preference-guided BO framework that continuously integrates the semantic reasoning of large language models (LLMs) into the optimization loop. Unlike prior works that use LLMs only for warm-start initialization or candidate generation, LGBO introduces a region-lifted preference mechanism that embeds LLM-driven preferences into every iteration, shifting the surrogate mean in a stable and controllable way. Theoretically, we prove that LGBO does not perform significantly worse than standard BO in the worst case, while achieving significantly faster convergence when preferences align with the objective. Empirically, LGBO consistently outperforms existing methods across diverse dry benchmarks in physics, chemistry, biology, and materials science. Most notably, in a new wet-lab optimization of Fe-Cr battery electrolytes, LGBO attains \textbf{90\% of the best observed value within 6 iterations}, whereas standard BO and existing LLM-augmented baselines require more than 10. Together, these results suggest that LGBO offers a promising direction for integrating LLMs into scientific optimization workflows.

preprint2022arXiv

Balanced Self-Paced Learning for AUC Maximization

Learning to improve AUC performance is an important topic in machine learning. However, AUC maximization algorithms may decrease generalization performance due to the noisy data. Self-paced learning is an effective method for handling noisy data. However, existing self-paced learning methods are limited to pointwise learning, while AUC maximization is a pairwise learning problem. To solve this challenging problem, we innovatively propose a balanced self-paced AUC maximization algorithm (BSPAUC). Specifically, we first provide a statistical objective for self-paced AUC. Based on this, we propose our self-paced AUC maximization formulation, where a novel balanced self-paced regularization term is embedded to ensure that the selected positive and negative samples have proper proportions. Specially, the sub-problem with respect to all weight variables may be non-convex in our formulation, while the one is normally convex in existing self-paced problems. To address this, we propose a doubly cyclic block coordinate descent method. More importantly, we prove that the sub-problem with respect to all weight variables converges to a stationary point on the basis of closed-form solutions, and our BSPAUC converges to a stationary point of our fixed optimization objective under a mild assumption. Considering both the deep learning and kernel-based implementations, experimental results on several large-scale datasets demonstrate that our BSPAUC has a better generalization performance than existing state-of-the-art AUC maximization methods.

preprint2022arXiv

Combinatorics of Integer Partitions With Prescribed Perimeter

We prove that the number of even parts and the number of times that parts are repeated have the same distribution over integer partitions with a fixed perimeter. This refines Straub's analog of Euler's Odd-Distinct partition theorem. We generalize the two concerned statistics to these of the part-difference less than $d$ and the parts not congruent to $1$ modulo $d+1$ and prove a distribution inequality, that has a similar flavor as Alder's ex-conjecture, over partitions with a prescribed perimeter. Both of our results are proved analytically and combinatorially.

preprint2022arXiv

Dynamic Prototype Convolution Network for Few-Shot Semantic Segmentation

The key challenge for few-shot semantic segmentation (FSS) is how to tailor a desirable interaction among support and query features and/or their prototypes, under the episodic training scenario. Most existing FSS methods implement such support-query interactions by solely leveraging plain operations - e.g., cosine similarity and feature concatenation - for segmenting the query objects. However, these interaction approaches usually cannot well capture the intrinsic object details in the query images that are widely encountered in FSS, e.g., if the query object to be segmented has holes and slots, inaccurate segmentation almost always happens. To this end, we propose a dynamic prototype convolution network (DPCN) to fully capture the aforementioned intrinsic details for accurate FSS. Specifically, in DPCN, a dynamic convolution module (DCM) is firstly proposed to generate dynamic kernels from support foreground, then information interaction is achieved by convolution operations over query features using these kernels. Moreover, we equip DPCN with a support activation module (SAM) and a feature filtering module (FFM) to generate pseudo mask and filter out background information for the query images, respectively. SAM and FFM together can mine enriched context information from the query features. Our DPCN is also flexible and efficient under the k-shot FSS setting. Extensive experiments on PASCAL-5i and COCO-20i show that DPCN yields superior performances under both 1-shot and 5-shot settings.

preprint2022arXiv

Implicit Motion Handling for Video Camouflaged Object Detection

We propose a new video camouflaged object detection (VCOD) framework that can exploit both short-term dynamics and long-term temporal consistency to detect camouflaged objects from video frames. An essential property of camouflaged objects is that they usually exhibit patterns similar to the background and thus make them hard to identify from still images. Therefore, effectively handling temporal dynamics in videos becomes the key for the VCOD task as the camouflaged objects will be noticeable when they move. However, current VCOD methods often leverage homography or optical flows to represent motions, where the detection error may accumulate from both the motion estimation error and the segmentation error. On the other hand, our method unifies motion estimation and object segmentation within a single optimization framework. Specifically, we build a dense correlation volume to implicitly capture motions between neighbouring frames and utilize the final segmentation supervision to optimize the implicit motion estimation and segmentation jointly. Furthermore, to enforce temporal consistency within a video sequence, we jointly utilize a spatio-temporal transformer to refine the short-term predictions. Extensive experiments on VCOD benchmarks demonstrate the architectural effectiveness of our approach. We also provide a large-scale VCOD dataset named MoCA-Mask with pixel-level handcrafted ground-truth masks and construct a comprehensive VCOD benchmark with previous methods to facilitate research in this direction. Dataset Link: https://xueliancheng.github.io/SLT-Net-project.

preprint2022arXiv

Lower and Upper Bounds for Numbers of Linear Regions of Graph Convolutional Networks

The research for characterizing GNN expressiveness attracts much attention as graph neural networks achieve a champion in the last five years. The number of linear regions has been considered a good measure for the expressivity of neural networks with piecewise linear activation. In this paper, we present some estimates for the number of linear regions of the classic graph convolutional networks (GCNs) with one layer and multiple-layer scenarios. In particular, we obtain an optimal upper bound for the maximum number of linear regions for one-layer GCNs, and the upper and lower bounds for multi-layer GCNs. The simulated estimate shows that the true maximum number of linear regions is possibly closer to our estimated lower bound. These results imply that the number of linear regions of multi-layer GCNs is exponentially greater than one-layer GCNs per parameter in general. This suggests that deeper GCNs have more expressivity than shallow GCNs.

preprint2020arXiv

Learning to Learn with Variational Information Bottleneck for Domain Generalization

Domain generalization models learn to generalize to previously unseen domains, but suffer from prediction uncertainty and domain shift. In this paper, we address both problems. We introduce a probabilistic meta-learning model for domain generalization, in which classifier parameters shared across domains are modeled as distributions. This enables better handling of prediction uncertainty on unseen domains. To deal with domain shift, we learn domain-invariant representations by the proposed principle of meta variational information bottleneck, we call MetaVIB. MetaVIB is derived from novel variational bounds of mutual information, by leveraging the meta-learning setting of domain generalization. Through episodic training, MetaVIB learns to gradually narrow domain gaps to establish domain-invariant representations, while simultaneously maximizing prediction accuracy. We conduct experiments on three benchmarks for cross-domain visual recognition. Comprehensive ablation studies validate the benefits of MetaVIB for domain generalization. The comparison results demonstrate our method outperforms previous approaches consistently.

preprint2016arXiv

New hook-content formulas for strict partitions

We introduce the difference operator for functions defined on strict partitions and prove a polynomiality property for a summation involving the hook length and content statistics. As an application, several new hook-content formulas for strict partitions are derived.

preprint2016arXiv

Polynomiality of some hook-content summations for doubled distinct and self-conjugate partitions

In 2009, the first author proved the Nekrasov-Okounkov formula on hook lengths for integer partitions by using an identity of Macdonald in the framework of type $\widetilde A$ affine root systems, and conjectured that some summations over the set of all partitions of size $n$ are always polynomials in $n$. This conjecture was generalized and proved by Stanley. Recently, Pétréolle derived two Nekrasov-Okounkov type formulas for $\widetilde C$ and $\widetilde C\,\check{}$ which involve doubled distinct and self-conjugate partitions. Inspired by all those previous works, we establish the polynomiality of some hook-content summations for doubled distinct and self-conjugate partitions.

preprint2015arXiv

On the largest size of $(t,t+1,..., t+p)$-core partitions

In this paper we prove that Amdeberhan's conjecture on the largest size of $(t, t+1, t+2)$-core partitions is true. We also show that the number of $(t, t + 1, t + 2)$-core partitions with the largest size is $1$ or $2$ based on the parity of $t$. More generally, the largest size of $(t,t+1,..., t+p)$-core partitions and the number of such partitions with the largest size are determined.

preprint2015arXiv

Partitions with the same hook multiset

It is well-known that two conjugate partitions have the same hook multiset. But two different partitions with the same hook multiset may not be conjugate to each other. In $1977$, Herman and Chung proposed the following question: What are the necessary and sufficient conditions for partitions to be determined by their hook multisets up to conjugation? In this paper, we will answer this question by giving a criterion to determine whether two different partitions with the same hook multiset are conjugate to each other.

preprint2014arXiv

The number of simultaneous core partitions

Amdeberhan conjectured that the number of $(t,t+1, t+2)$-core partitions is $\sum_{0\leq k\leq [\frac{t}{2}]}\frac{1}{k+1}\binom{t}{2k}\binom{2k}{k}$. In this paper, we obtain the generating function of the numbers $f_t$ of $(t, t + 1, ..., t + p)$-core partitions. In particular, this verifies that Amdeberhan's conjecture is true. We also prove that the number of $(t_1,t_2,..., t_m)$-core partitions is finite if and only if gcd$(t_1,t_2,..., t_m)=1,$ which extends Anderson's result on the finiteness of the number of $(t_1,t_2)$-core partitions for coprime positive integers $t_1$ and $t_2$ and thus rediscover a result of Keith and Nath with a different proof.

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