arXiv subjects

Huan Xiong

Publications and source records attributed to Huan Xiong.

At least 19 recordsLinked to original sources

arXiv2026

The 1/2-Conjecture for $q$-Binomial Coefficients with Fractional Index

For a nonnegative integer $k$ and a rational number $r\in\mathbb{Q}^+$, we define the generalized Gaussian binomial coefficient $\qbinom{r+k}{k} = \frac{(q^{r+1}; q)_k}{(q; q)_k}$. When $r=a/b$ with $a,b$ coprime positive integers and $b\geq 2$, expanding $\qbinom{r+k}{k}$ via the finite $q$-binomial theorem produces fractional powers of $q$, so that $\qbinom{r+k}{k}$ is a \emph{Puiseux series} in $q$ with nonnegative exponents; concretely it lies in $\mathbb{Q}[[q^{1/b}]]$. The notion we single out is the \emph{integer trace} of this expansion, the subseries consisting of those terms $c_r(d)\,q^d$ whose exponent $d$ is an integer, with all fractional powers discarded. This projection is not standard, and there is no a~priori reason for the surviving coefficients to behave coherently as $r$ varies. Nonetheless, ordering the family by the coefficientwise partial order leads to the \emph{$\tfrac{1}{2}$-Conjecture}: among all $r\in\mathbb{Q}^+$, the value $r=\tfrac{1}{2}$ maximizes the integer trace, in the sense that the coefficients of $\qbinom{1/2+k}{k}$ dominate those of $\qbinom{r+k}{k}$ coefficientwise for every $r$. That so elementary a definition should single out $\tfrac{1}{2}$ this cleanly came as a surprise to us. We prove the conjecture in several special cases and provide further computational evidence.

math.CO↗

arXiv2026

Autonomous Chemical Mechanistic Discovery through Agentic Reasoning and Validation

Unraveling reaction mechanisms is central to modern chemistry, yet automating these investigations remains challenging because computational workflows still rely heavily on expert intervention. Here we introduce ARCHE, an autonomous agentic system that integrates a general-purpose reasoning model, a domain-specialized computational chemistry model, and a structured tool registry to transform mechanistic inquiry into a scalable, self-validating process. ARCHE interprets scientific questions, generates and prioritizes mechanistic hypotheses, orchestrates computational workflows, and iteratively refines conclusions based on computed evidence within a closed loop. We validate its capabilities across three increasingly demanding scenarios: reconstructing stereocontrolling transition states and validating the corresponding reaction mechanism in a previously reported asymmetric catalytic reaction; proposing and validating a plausible radical pathway through iterative hypothesis refinement for a recently discovered but unpublished $α$-iodoboronate C-I cleavage reaction; and identifying a chemically interpretable descriptor that governs selectivity in nickel-catalysed migratory cross-coupling reactions. By coupling agentic reasoning with rigorous computational validation, ARCHE advances autonomous mechanistic discovery and establishes a foundation for broader machine-assisted chemical research. The code for ARCHE is publicly available at https://github.com/JetAstra/Arche-Harness.

cs.AI↗

arXiv2026

Reusing Rollouts under Policy Lag: Prefix-Normalized Policy Optimization for LLM Reinforcement Learning

Autoregressive rollout generation is a major computational cost in reinforcement learning for large language models. Reusing each rollout batch for additional learner updates amortizes this cost, but later updates become increasingly off-policy as the learner departs from the behavior policy. At a token position, exact off-policy correction must account for both the current action and the probability of reaching its prefix. The cumulative importance ratio provides this correction, but its product form can produce an unwieldy dynamic range. We study Prefix-Normalized Policy Optimization (PNPO), which replaces the cumulative ratio with the geometric mean of likelihood ratios along each causal prefix, preserving causal-prefix dependence at each position while compressing the log-weight scale. In controlled long-context mathematical reasoning experiments, we induce two off-policy regimes by using one or four policy-update epochs per rollout batch. PNPO does not consistently outperform GSPO with one epoch. With four epochs, it attains the highest observed Avg@32 on each benchmark; the unweighted mean of the three independently selected benchmark peaks is 50.24, 3.00 percentage points above GSPO. Under a matched 2,400-update budget, four-epoch PNPO reaches a final macro Avg@32 of 49.66 after 150 rollout batches, comparable to the 49.56 reached after 600 batches with one epoch. These results provide preliminary evidence that PNPO can be advantageous as training moves further off-policy.

cs.AI↗

arXiv2026

$q$-Derivative Grammar

Context-free grammars, originating in computer science, are related to enumerative combinatorics through two distinct lines of development pioneered by Schützenberger and Chen, respectively. In the framework established by Schützenberger and Delest-Schützenberger-Viennot, unambiguous grammars are translated into functional equations for ordinary generating functions. Inspired by Rota's umbral calculus, Chen later developed a grammatical calculus by associating each context-free grammar with a formal derivative operator. Dumont further developed this method through numerous combinatorial interpretations of grammars with finite and infinite alphabets. Substantial progress in this direction has been achieved over the last decade. In this paper, we introduce a q-analogue of grammatical calculus, which we call the q-derivative grammar. We establish the basic framework of q-grammars and develop the q-grammatical calculus for computing q-exponential generating functions associated with q-grammars. Concrete q-grammars are constructed to study q-Eulerian, q-Roselle and q-André polynomials, including their generating functions and recurrences. This work extends the grammatical method to the q-setting and opens up new research directions.

math.CO↗

arXiv2026

On the distributions of the statistics (des, maj, inv) over several classes of permutations

We investigate the joint distribution of the trivariate statistics (des, maj, inv) on classical permutations, Andre permutations of the first and second kinds, and Simsun permutations. By decomposing permutations according to the position of the smallest element, we obtain explicit recurrence relations for the generating functions of these statistics. In the classical permutation setting, our recurrence relation yields the generating function for the trivariate statistics (des, maj, inv) due to Gessel, which is typically proved using MacMahon's technique.

math.CO↗

arXiv2026

Plug-and-Play Spiking Operators: Breaking the Nonlinearity Bottleneck in Spiking Transformers

ANN-to-SNN conversion offers a practical, training-free route to spiking large language models. However, current pipelines primarily focus on spike-driven realizations for Transformer linear-algebra operations, while providing limited support for key nonlinear operators. This gap limits compatibility with neuromorphic-style execution constraints, where such nonlinearities typically require division, exponentiation, or norm computations that are not naturally supported by standard leaky integrate-and-fire dynamics. To solve this problem, we propose a plug-and-play framework that implements spike-friendly approximations for Transformer nonlinearities and integrates into existing ANN-to-SNN pipelines. Our method decomposes these nonlinear computations into three recurring primitives -- division, exponentiation, and $\ell_2$ norms -- and realizes them via population computation using LIF neuron groups, combined with lightweight bit-shift scaling to avoid floating-point arithmetic. By composing these primitives as modular operator blocks, our framework supports common Transformer nonlinearities (e.g., Softmax, SiLU, and normalization) without any fine-tuning. Experiments on a range of LLMs Transformers show that selectively replacing the targeted nonlinear operators incurs less than a $1\%$ accuracy drop across all evaluated tasks.

cs.LG↗

arXiv2026

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.

cs.AI↗

arXiv2026

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.

cs.AI↗

arXiv2025

Inverse descent statistic for André and simsun permutations

Simsun permutations, André I permutations and André II permutations are three combinatorial models for Euler numbers. It's known that the descent statistic is equidistributed over the set of André I permutations and the set of simsun permutations. In this paper, we prove that the trivariate statistic (ides, des, maj), comprising the inverse descent, descent, and major index, are equidistributed over these three sets. This result is equivalent to showing that the inverse descent is equidistributed over these three sets that share the same tree shape. The proof of the equidistribution of the inverse descent over the set of André I permutations and the set of André II permutations with the same tree shape reduces to establishing new refinements of Stanley's shuffle theorem.

math.CO↗

arXiv2025

Dynamic Spiking Framework for Graph Neural Networks

The integration of Spiking Neural Networks (SNNs) and Graph Neural Networks (GNNs) is gradually attracting attention due to the low power consumption and high efficiency in processing the non-Euclidean data represented by graphs. However, as a common problem, dynamic graph representation learning faces challenges such as high complexity and large memory overheads. Current work often uses SNNs instead of Recurrent Neural Networks (RNNs) by using binary features instead of continuous ones for efficient training, which would overlooks graph structure information and leads to the loss of details during propagation. Additionally, optimizing dynamic spiking models typically requires propagation of information across time steps, which increases memory requirements. To address these challenges, we present a framework named \underline{Dy}namic \underline{S}p\underline{i}king \underline{G}raph \underline{N}eural Networks (\method{}). To mitigate the information loss problem, \method{} propagates early-layer information directly to the last layer for information compensation. To accommodate the memory requirements, we apply the implicit differentiation on the equilibrium state, which does not rely on the exact reverse of the forward computation. While traditional implicit differentiation methods are usually used for static situations, \method{} extends it to the dynamic graph setting. Extensive experiments on three large-scale real-world dynamic graph datasets validate the effectiveness of \method{} on dynamic node classification tasks with lower computational costs.

cs.NE↗

arXiv2025

VideoExpert: Augmented LLM for Temporal-Sensitive Video Understanding

The core challenge in video understanding lies in perceiving dynamic content changes over time. However, multimodal large language models struggle with temporal-sensitive video tasks, which requires generating timestamps to mark the occurrence of specific events. Existing strategies require MLLMs to generate absolute or relative timestamps directly. We have observed that those MLLMs tend to rely more on language patterns than visual cues when generating timestamps, affecting their performance. To address this problem, we propose VideoExpert, a general-purpose MLLM suitable for several temporal-sensitive video tasks. Inspired by the expert concept, VideoExpert integrates two parallel modules: the Temporal Expert and the Spatial Expert. The Temporal Expert is responsible for modeling time sequences and performing temporal grounding. It processes high-frame-rate yet compressed tokens to capture dynamic variations in videos and includes a lightweight prediction head for precise event localization. The Spatial Expert focuses on content detail analysis and instruction following. It handles specially designed spatial tokens and language input, aiming to generate content-related responses. These two experts collaborate seamlessly via a special token, ensuring coordinated temporal grounding and content generation. Notably, the Temporal and Spatial Experts maintain independent parameter sets. By offloading temporal grounding from content generation, VideoExpert prevents text pattern biases in timestamp predictions. Moreover, we introduce a Spatial Compress module to obtain spatial tokens. This module filters and compresses patch tokens while preserving key information, delivering compact yet detail-rich input for the Spatial Expert. Extensive experiments demonstrate the effectiveness and versatility of the VideoExpert.

cs.CV↗

arXiv2025

An Efficient Algorithm for Vertex Enumeration of Arrangement

This paper presents a state-of-the-art algorithm for the vertex enumeration problem of arrangements, which is based on the proposed new pivot rule, called the Zero rule. The Zero rule possesses several desirable properties: i) It gets rid of the objective function; ii) Its terminal satisfies uniqueness; iii) We establish the if-and-only if condition between the Zero rule and its valid reverse, which is not enjoyed by earlier rules; iv) Applying the Zero rule recursively definitely terminates in $d$ steps, where $d$ is the dimension of input variables. Because of so, given an arbitrary arrangement with $v$ vertices of $n$ hyperplanes in $\mathbb{R}^d$, the algorithm's complexity is at most $\mathcal{O}(n^2d^2v)$ and can be as low as $\mathcal{O}(nd^4v)$ if it is a simple arrangement, while Moss' algorithm takes $\mathcal{O}(nd^2v^2)$, and Avis and Fukuda's algorithm goes into a loop or skips vertices because the if-and-only-if condition between the rule they chose and its valid reverse is not fulfilled. Systematic and comprehensive experiments confirm that the Zero rule not only does not fail but also is the most efficient.

math.CO↗

arXiv2025

A Self-Conjugate Partition Analog of $(t,t+1)$-Core Partitions with Distinct Parts

Simultaneous core partitions have been widely studied in the past 20 years. In 2013, Amdeberhan gave several conjectures on the number, the average size, and the largest size of $(t,t+1)$-core partitions with distinct parts, which was proved and generalized by Straub, Xiong, Nath-Sellers, Zaleski-Zeilberger, Paramonov, and many other mathematicians. In this paper, we introduce a proper self-conjugate partition analog of $(t,t+1)$-core partitions with distinct parts, and derive the number, the average size, and the largest size for such core partitions.

math.CO↗

arXiv2025

On Expressivity of Height in Neural Networks

In this work, beyond width and depth, we augment a neural network with a new dimension called height by intra-linking neurons in the same layer to create an intra-layer hierarchy, which gives rise to the notion of height. We call a neural network characterized by width, depth, and height a 3D network. To put a 3D network in perspective, we theoretically and empirically investigate the expressivity of height. We show via bound estimation and explicit construction that given the same number of neurons and parameters, a 3D ReLU network of width $W$, depth $K$, and height $H$ has greater expressive power than a 2D network of width $H\times W$ and depth $K$, \textit{i.e.}, $\mathcal{O}((2^H-1)W)^K)$ vs $\mathcal{O}((HW)^K)$, in terms of generating more pieces in a piecewise linear function. Next, through approximation rate analysis, we show that by introducing intra-layer links into networks, a ReLU network of width $\mathcal{O}(W)$ and depth $\mathcal{O}(K)$ can approximate polynomials in $[0,1]^d$ with error $\mathcal{O}\left(2^{-2WK}\right)$, which improves $\mathcal{O}\left(W^{-K}\right)$ and $\mathcal{O}\left(2^{-K}\right)$ for fixed width networks. Lastly, numerical experiments on 5 synthetic datasets, 15 tabular datasets, and 3 image benchmarks verify that 3D networks can deliver competitive regression and classification performance.

cs.LG↗

arXiv2024

Asymptotic Normality and Concentration Inequalities of Statistics of Core Partitions with Bounded Perimeters

Core partitions have attracted much attention since Anderson's work (2002) on the number of $(s,t)$-core partitions for coprime $s,t$. Recently, there has been a growing interest in studying the limiting distributions of the sizes of random simultaneous core partitions. In this paper, we prove the asymptotic normality of certain statistics of uniform random core partitions with bounded perimeters in the Kolmogorov and Wasserstein $W_1$ distances, including the length and size of a random (strict) $n$-core partition, the length of the Durfee square and the size of a random self-conjugate $n$-core partition. Accordingly, we prove that these statistics are subgaussian. This contrasts with the asymptotic behavior of the size of a random $(s, t)$-core partition for coprime $s,t$ studied by Even-Zohar (2022), which converges in law to Watson's $U^2$ distribution. Our results show that the distribution of the size of a random strict $(n, dn+1)$-core partition is asymptotically normal when $d \ge 3$ is fixed and $n$ tends to infinity, which is an analog of Zaleski's conjecture (2017) and covers Komlós, Sergel, and Tusnády's result (2020) as a special case. Our proof integrates a variety of combinatorial and probabilistic tools, including Stein's method based on Hoeffding decomposition, Hoeffding's combinatorial central limit theorem, the Efron-Stein inequalities on product spaces and slices, and asymptotics of Pólya frequency sequences. Furthermore, our approach is potentially applicable to the study of the asymptotic normality of functionals of random variables with certain global dependence structures that can be decomposed into appropriate mixture forms.

math.PR↗

arXiv2024

Diffusion-Enhanced Test-time Adaptation with Text and Image Augmentation

Existing test-time prompt tuning (TPT) methods focus on single-modality data, primarily enhancing images and using confidence ratings to filter out inaccurate images. However, while image generation models can produce visually diverse images, single-modality data enhancement techniques still fail to capture the comprehensive knowledge provided by different modalities. Additionally, we note that the performance of TPT-based methods drops significantly when the number of augmented images is limited, which is not unusual given the computational expense of generative augmentation. To address these issues, we introduce IT3A, a novel test-time adaptation method that utilizes a pre-trained generative model for multi-modal augmentation of each test sample from unknown new domains. By combining augmented data from pre-trained vision and language models, we enhance the ability of the model to adapt to unknown new test data. Additionally, to ensure that key semantics are accurately retained when generating various visual and text enhancements, we employ cosine similarity filtering between the logits of the enhanced images and text with the original test data. This process allows us to filter out some spurious augmentation and inadequate combinations. To leverage the diverse enhancements provided by the generation model across different modals, we have replaced prompt tuning with an adapter for greater flexibility in utilizing text templates. Our experiments on the test datasets with distribution shifts and domain gaps show that in a zero-shot setting, IT3A outperforms state-of-the-art test-time prompt tuning methods with a 5.50% increase in accuracy.

cs.CV↗

arXiv2024

Deep ReLU Networks Have Surprisingly Simple Polytopes

A ReLU network is a piecewise linear function over polytopes. Figuring out the properties of such polytopes is of fundamental importance for the research and development of neural networks. So far, either theoretical or empirical studies on polytopes only stay at the level of counting their number, which is far from a complete characterization. Here, we propose to study the shapes of polytopes via the number of faces of the polytope. Then, by computing and analyzing the histogram of faces across polytopes, we find that a ReLU network has relatively simple polytopes under both initialization and gradient descent, although these polytopes can be rather diverse and complicated by a specific design. This finding can be appreciated as a kind of generalized implicit bias, subjected to the intrinsic geometric constraint in space partition of a ReLU network. Next, we perform a combinatorial analysis to explain why adding depth does not generate a more complicated polytope by bounding the average number of faces of polytopes with the dimensionality. Our results concretely reveal what kind of simple functions a network learns and what will happen when a network goes deep. Also, by characterizing the shape of polytopes, the number of faces can be a novel leverage for other problems, \textit{e.g.}, serving as a generic tool to explain the power of popular shortcut networks such as ResNet and analyzing the impact of different regularization strategies on a network's space partition.

cs.LG↗

arXiv2024

Effectiveness Assessment of Recent Large Vision-Language Models

The advent of large vision-language models (LVLMs) represents a remarkable advance in the quest for artificial general intelligence. However, the model's effectiveness in both specialized and general tasks warrants further investigation. This paper endeavors to evaluate the competency of popular LVLMs in specialized and general tasks, respectively, aiming to offer a comprehensive understanding of these novel models. To gauge their effectiveness in specialized tasks, we employ six challenging tasks in three different application scenarios: natural, healthcare, and industrial. These six tasks include salient/camouflaged/transparent object detection, as well as polyp detection, skin lesion detection, and industrial anomaly detection. We examine the performance of three recent open-source LVLMs, including MiniGPT-v2, LLaVA-1.5, and Shikra, on both visual recognition and localization in these tasks. Moreover, we conduct empirical investigations utilizing the aforementioned LVLMs together with GPT-4V, assessing their multi-modal understanding capabilities in general tasks including object counting, absurd question answering, affordance reasoning, attribute recognition, and spatial relation reasoning. Our investigations reveal that these LVLMs demonstrate limited proficiency not only in specialized tasks but also in general tasks. We delve deep into this inadequacy and uncover several potential factors, including limited cognition in specialized tasks, object hallucination, text-to-image interference, and decreased robustness in complex problems. We hope that this study can provide useful insights for the future development of LVLMs, helping researchers improve LVLMs for both general and specialized applications.

cs.CV↗