arXiv subjects

Yutong Zhang

Publications and source records attributed to Yutong Zhang.

At least 55 records · Page 3Linked to original sources

arXiv2026

Two determinant evaluations in Sun's conjectures involving Legendre symbols

We prove two determinant evaluations attached to Sun's conjectures on matrices of Legendre symbols. The first one resolves the $p\equiv1\pmod4$ part of Conjecture 4.8(i) by reducing the determinant with four indeterminates to a four-entry inverse package for the adjacent minor $[χ(j-k+1)]_{0\le j,k<(p-1)/2}$. The core evaluation is \[ \det H=\leg{2}{p}(b'_p-a'_p),\qquad U^TH^{-1}U= \begin{pmatrix} \leg{2}{p}\dfrac{pb'_p-a'_p}{b'_p-a'_p}&1\\[2mm] \dfrac{b'_p-a'_p-1}{b'_p-a'_p}&1 \end{pmatrix}, \] where $U=(\mathbf1,η)$ and $η_j=χ(j)$. The proof uses Vsemirnov's factorisation of Chapman's matrix and an adjacent cofactor calculation. The second result gives a uniform exact congruence modulo $p$ for the determinant underlying Sun's Conjecture 4.10(i), valid for any ordered half-system modulo sign and all $u,v\in\mathbb F_p$. Its standard specialization recovers the asserted square class. The square-class assertion itself also follows from Sun's earlier evaluation of $T(d,p)$; the contribution here is an exact and half-system refinement.

math.NT↗

arXiv2026

A Local Valuation Criterion for Quadratic-Permutation Interleaved Zadoff--Chu Sequences

Berggren and Popović introduced quadratic-permutation-polynomial interleaved Zadoff--Chu sequences and, from exhaustive data, conjectured that all normalized QPP-interleaved Zadoff--Chu sequences are inequivalent to ordinary Zadoff--Chu sequences precisely for prime-power lengths $N=p^n$ with $p>3$ and $n>1$. We give an exact local arithmetic criterion. For a normalized QPP $π_{a,b}(k)=ak^2+bk\pmod N$, the interleaved sequence is equivalent, under the standard five CAZAC-preserving operations, to a Zadoff--Chu sequence if and only if, for every prime power $p^α\Vert N$, the valuation of $a$ satisfies \[ ν_p(a)\ge \begin{cases} 0, & p=2,\ α=1,\\ α-1, & p=2,\ α\ge2,\\ α-1, & p=3,\\ α, & p>3. \end{cases} \] The proof is based on a third finite-difference invariant of the lifted Zadoff--Chu phase, namely \[ Δ^3\bigl((ak^2+bk+\varepsilon_N+2q)(ak^2+bk)\bigr) =12a(2ak+3a+b). \] As a consequence, the conjectured prime-power boundary is not correct: the exact non-vacuous condition for all nonzero normalized QPPs to be inequivalent to Zadoff--Chu sequences is that $N$ is odd, $9\nmid N$, and $p^2\mid N$ for at least one prime $p\ge5$. In particular, $N=75=3\cdot5^2$ is the smallest non-prime-power counterexample to the conjectured ``only if'' direction. A second corollary records the corresponding statement for irreducible QPPs.

math.NT↗

arXiv2026

Finite factorization is detected by undermonoids

Let $M$ be a cancellative commutative monoid and call a submonoid $S$ of $M$ an undermonoid if $\G(S)=\G(M)$ inside the Grothendieck group of $M$. Gotti and Li asked whether the finite factorization property is hereditary once it is known on all undermonoids: if every undermonoid of $M$ is a finite factorization monoid, must every submonoid of $M$ be a finite factorization monoid? We give an affirmative answer. Equivalently, for every cancellative commutative monoid $M$, the following two conditions coincide: every submonoid of $M$ is an FFM, and every undermonoid of $M$ is an FFM. The proof isolates a fixed length $\ell$ and an infinite set of length-$\ell$ factorizations of one element $b$. In the non-group case, a divisor-complement ideal $I=\{m\in M:m\nmid_M b\}$ enlarges the bad submonoid to a bad undermonoid while preserving the chosen length-$\ell$ factorizations. In the group case, a maximality argument over submonoids for which these factorizations survive is combined with a two-sided perturbation $S\mapsto S+\Nzero(2b+u)$. The key point is that the perturbation creates no new units and does not split any atom occurring in the fixed factorizations. This yields an undermonoid with infinitely many factorizations of $b$, contradicting the hypothesis.

math.GR↗

arXiv2026

A compensation theorem for the Sylow-integral invariant and counterexamples to an \texorpdfstring{$A_5$}{A5}-characterization conjecture

Let $ν_p(G)$ be the number of Sylow $p$-subgroups of a finite group $G$, let $σ_p(G)$ be their common order, and set \[ γ(G)=\int_0^1\sum_{p\inπ(G)}ν_p(G)x^{σ_p(G)}\,dx =\sum_{p\inπ(G)}\frac{ν_p(G)}{σ_p(G)+1}. \] A recent conjectural extension of the simple-group theorem for this invariant asserted that a nonsolvable finite group has $γ(G)=9/2$ precisely when $G\cong A_5$. We disprove this assertion by a direct and verifiable construction. More generally, we prove an exact direct-product compensation formula for $A_5$ with an arbitrary nilpotent factor. The formula reduces the equality $γ(A_5\times N)=9/2$ to a finite Egyptian-fraction equation in the orders of the Sylow subgroups of $N$. Taking $N=\C_2\times\C_7\times\C_{11}\times\C_{13}\times\C_{17}\times\C_{19}\times\C_{29}\times\C_{71}\times\C_{83}$, the loss in the $2$-Sylow contribution is exactly compensated by the new normal Sylow subgroups. Consequently $G=A_5\times N$ is nonsolvable, is not isomorphic to $A_5$, has solvable radical $N$, and nevertheless satisfies $γ(G)=9/2$. Several further explicit compensation certificates are also recorded.

math.GR↗

arXiv2026

Weighted Uniform Endpoint Majorants for Integrals Involving Modified Bessel Functions

We give an affirmative full-range solution to Gaunt's 2019 Open Problem~2.10. The problem asks whether, for every $ν>-1/2$ and $0<γ<1$, the reciprocal-power integral $\int_0^x e^{-γt}I_ν(t)t^{-ν}\,\dd t$ is bounded by a constant multiple of $e^{-γx}I_{ν+1}(x)x^{-ν}$, uniformly for all $x>0$. Earlier exponential-tilt estimates proved such endpoint majorants only under an additional smallness condition on $γ$. We prove the estimate throughout the natural range $0<γ<1$, with an explicit admissible constant. More generally, if $μ>-1$, $q>-1$, $0<γ<1$, and $w(x)x^{-q}$ is nondecreasing on $(0,\infty)$, then for every $θ\in(γ,1)$, $\int_0^x e^{-γt}w(t)t^{-μ}I_μ(t)\,\dd t$ is controlled by an explicit multiple of $e^{-γx}w(x)x^{-μ}I_{μ+1}(x)$. The case $w\equiv1$, $q=0$, and $μ=ν$ resolves Gaunt's problem. The weighted theorem also yields shifted-order and moment estimates, applies to approximate power weights and monotone regularly varying amplitudes, and provides two-sided estimates under a reversed comparison. We further analyze the sharp power-weighted quotient via endpoint expansions, a stationary equation, and parameter monotonicity.

math.CA↗

arXiv2026

A Uniform Random-Lattice Tail Bound for the SVP Kissing-Profile Parameter

A recent SICOMP paper on classical and quantum algorithms for the shortest vector problem introduced a lattice-dependent parameter $γ(L)$, bounded universally in the exponential sense by $2^{0.402n+o(n)}$, and conjectured that this parameter is $2^{o(n)}$ for most lattices. We prove the Haar--Siegel random-lattice version in a stronger, dimension-uniform form. Let $X_n=\operatorname{SL}_n(\R)/\operatorname{SL}_n(\Z)$, let $μ_n$ be its invariant probability measure, and let $γ(L)=\sup_{r\ge1} N_L(rλ_1(L))/r^n$, where $N_L(R)$ counts nonzero vectors of $L$ of Euclidean norm at most $R$. For every $n\ge3$ and every $T>0$, \[ μ_n\{L\in X_n:γ(L)>T\}\le C T^{-1} \] with an absolute constant $C$. Consequently, for every sequence $a_n\to\infty$, $γ(L_n)\le a_n$ with $μ_n$-probability tending to one; in particular $γ(L_n)=2^{o(n)}$ with high probability. In the product model of independent Haar--Siegel lattices, $γ(L_n)\le \exp(\sqrt n)$ eventually almost surely. The proof uses Rogers's second-moment estimate only through a dyadic self-normalization argument around the random scale $λ_1(L)$.

math.NT↗

arXiv2026

Compositional Consistency-Guided Decoding for Three-Way Logical Question Answering

Three-way logical question answering (QA) assigns one of $\text{True}$, $\text{False}$, or $\text{Unknown}$ to a hypothesis $H$ given a premise set $S$. We study this task as a compact compositional inference problem: predictions for $H$ and for a mechanically negated hypothesis $\neg H$ should agree under a deterministic negation map. Despite this simple structure, large language models (LLMs) can exhibit two practical failure modes: (i) negation inconsistency, where answers to $H$ and $\neg H$ violate the required label mapping, and (ii) epistemic $\text{Unknown}$, where the model abstains even when one side is entailed. We introduce CGD-PD, a lightweight, training-free test-time layer that combines neural 3-way classification, symbolic negation-consistency projection, and targeted binary entailment probes. On one validation split of FOLIO's first-order logic fields, CGD-PD improves accuracy by 4.4 points on GPT-5.2 and 6.8 points on Claude Sonnet 4.5, while reducing $\text{Unknown}$ predictions and epistemic abstention. These results provide a controlled proof of concept that simple logical composition at inference time can help evaluate and improve LLM reasoning reliability; they do not, by themselves, establish robustness beyond this formal benchmark setting.

cs.CL↗

arXiv2026

A Dynamical Model for Spatio-Temporal Processes Motivated by Second-Order Partial Differential Equations

An important class of spatio-temporal models is constructed by leveraging the hierarchical structure of dynamical (or, state-space) models. This paper proposes a new statistical dynamical model for spatio-temporal processes motivated by second-order stochastic partial differential equations (SPDE). In particular, an infinite-dimensional linear state-space representation is obtained where the state transition is governed by a proposed SDE. Then, using the Galerkin's method, a finite-dimensional approximation to the infinite-dimensional SDE is obtained, yielding a dynamical model with finite states that facilitates computation and parameter estimation. The space-time covariance of the approximated dynamical model is obtained, and the error between the approximate and exact covariance matrices is quantified. Comprehensive numerical investigations, including 2D wave equation, seismic wave propagation, advection-diffusion equations and wildfire aerosol propagation processes, are performed to demonstrate the application of the proposed model. Code is available.

stat.ME↗

arXiv2026

Reflections and New Directions for Human-Centered Large Language Models

Large Language Models (LLMs) are increasingly shaping the private and professional lives of users, with numerous applications in business, education, finance, healthcare, law, and science. With this rise in global influence comes greater urgency to build, evaluate, and deploy these systems in a manner that prioritizes not only technical capabilities but also human priorities. This work presents a framework for developing Human-Centered Large Language Models (HCLLMs), which integrates perspectives from Natural Language Processing (NLP), Human-Computer Interaction (HCI), and responsible AI. Considering the ethics, economics, and technical objectives of language modeling, we argue that model developers need to address human concerns, preferences, values, and goals, not only during a cursory post-training stage, but rather with rigor and care at every stage of the pipeline. This paper offers human-centered insights and recommendations for developers at each stage, from system design to data sourcing, model training, evaluation, and responsible deployment. Then we conclude with a case study, applying these insights to understand the future of work with HCLLMs.

cs.CL↗

arXiv2026

The Rise of AI Companions: Interaction with AI Companions and Psychological Well-being

As large language model (LLM)-enhanced chatbots become increasingly expressive and socially responsive, many users begin forming companionship-like bonds with them. This study investigates how using AI companions relates to psychological well-being. We collected self-reported data from 1,131 U.S. adults who use CharacterAI, including survey responses and 4,664 chat sessions (464,687 messages) from 237 participants. By triangulating self-reported usage, relationship descriptions, and real chat histories, we identify patterns of engagement and associated outcomes. Smaller social networks were associated with reporting companionship as the primary chatbot use (beta = -0.03, 95% confidence interval (CI) [-0.05, -0.01]), which in turn was associated with lower well-being (beta = -0.48, 95% CI [-0.70, -0.25]). For self-reported companionship usage, this association was stronger when interactions were intensive (beta = -0.31, 95% CI [-0.56, -0.06]) and highly disclosive (beta = -0.38, 95% CI [-0.63, -0.14]). These results suggest that the association between AI companionship and well-being is not uniform and depends on how chatbots are used and users' offline social environments.

cs.HC↗

arXiv2026

Universal Smoothness via Bernstein Polynomials: A Constructive Approximation Approach for Activation Functions

The efficacy of deep neural networks is heavily reliant on the design of non-linear activation functions, yet existing approaches often struggle to balance optimization stability with computational efficiency. While piecewise linear functions offer inference speed, they suffer from optimization instability due to non-differentiability at the origin, whereas smooth counterparts typically incur significant computational overhead through their reliance on transcendental operations. To address these limitations, this paper proposes a general smoothing framework based on constructive approximation theory and introduces the Bernstein Linear Unit (BerLU). This novel activation function utilizes Bernstein polynomials to construct a differentiable quadratic transition region that effectively eliminates singularities while maintaining a piecewise linear structure. Theoretical analysis demonstrates that the proposed method guarantees strictly continuous differentiability and a non-expansive Lipschitz constant of one, which ensures stable gradient propagation and prevents the gradient explosion problems common in deep architectures. Comprehensive empirical evaluations across representative Vision Transformer and Convolutional Neural Network architectures confirm that this approach consistently outperforms state-of-the-art baselines on standard image classification benchmarks while delivering superior computational and memory efficiency.

cs.AI↗

arXiv2026

Generative Interfaces for Language Models

Large language models (LLMs) are increasingly seen as assistants, copilots, and consultants, capable of supporting a wide range of tasks through natural conversation. However, most systems remain constrained by a linear request-response format that often makes interactions inefficient in multi-turn, information-dense, and exploratory tasks. To address these limitations, we propose Generative Interfaces for Language Models, a paradigm in which LLMs respond to user queries by proactively generating user interfaces (UIs) that enable more adaptive and interactive engagement. Our framework leverages structured interface-specific representations and iterative refinements to translate user queries into task-specific UIs. For systematic evaluation, we introduce a multidimensional assessment framework that compares generative interfaces with traditional chat-based ones across diverse tasks, interaction patterns, and query types, capturing functional, interactive, and emotional aspects of user experience. Results show that generative interfaces consistently outperform conversational ones, with up to a 72% improvement in human preference. These findings clarify when and why users favor generative interfaces, paving the way for future advancements in human-AI interaction. Data and code are available at https://github.com/SALT-NLP/GenUI.

cs.CL↗

arXiv2026

PRISM: Probing Reasoning, Instruction, and Source Memory in LLM Hallucinations

As large language models (LLMs) evolve from conversational assistants into agents capable of handling complex tasks, they are increasingly deployed in high-risk domains. However, existing benchmarks largely rely on mixed queries and posterior evaluation, output-level scoring, which quantifies hallucination severity but offers limited insight into where and why hallucinations arise in the generation pipeline. We therefore reformulate hallucination evaluation as a diagnostic problem and propose PRISM, a controlled benchmark that disentangles hallucinations into four dimensions: knowledge missing, knowledge errors, reasoning errors, and instruction-following errors, grounded in three stages of generation (memory, instruction, and reasoning). PRISM contains 9,448 instances across 65 tasks and supports fine-grained, stage-aware diagnostic evaluation. Evaluating 24 mainstream open-source and proprietary LLMs, we uncover consistent trade-offs across instruction following, memory retrieval, and logical reasoning, showing that mitigation strategies often improve specific dimensions at the expense of others. We hope PRISM provides a framework for understanding the specific mechanisms behind LLMs hallucinations, ultimately accelerating the development of trustworthy large language models.

cs.CL↗

arXiv2026

Memory-Efficient Transfer Learning with Fading Side Networks via Masked Dual Path Distillation

Memory-efficient transfer learning (METL) approaches have recently achieved promising performance in adapting pre-trained models to downstream tasks. They avoid applying gradient backpropagation in large backbones, thus significantly reducing the number of trainable parameters and high memory consumption during fine-tuning. However, since they typically employ a lightweight and learnable side network, these methods inevitably introduce additional memory and time overhead during inference, which contradicts the ultimate goal of efficient transfer learning. To address the above issue, we propose a novel approach dubbed Masked Dual Path Distillation (MDPD) to accelerate inference while retaining parameter and memory efficiency in fine-tuning with fading side networks. Specifically, MDPD develops a framework that enhances the performance by mutually distilling the frozen backbones and learnable side networks in fine-tuning, and discard the side network during inference without sacrificing accuracy. Moreover, we design a novel feature-based knowledge distillation method for the encoder structure with multiple layers. Extensive experiments on distinct backbones across vision/language-only and vision-and-language tasks demonstrate that our method not only accelerates inference by at least 25.2\% while keeping parameter and memory consumption comparable, but also remarkably promotes the accuracy compared to SOTA approaches. The source code is available at https://github.com/Zhang-VKk/MDPD.

cs.CV↗

arXiv2026

MP-ISMoE: Mixed-Precision Interactive Side Mixture-of-Experts for Efficient Transfer Learning

Parameter-efficient transfer learning (PETL) has emerged as a pivotal paradigm for adapting pre-trained foundation models to downstream tasks, significantly reducing trainable parameters yet suffering from substantial memory overhead caused by gradient backpropagation during fine-tuning. While memory-efficient transfer learning (METL) circumvents this challenge by bypassing backbone gradient computation via lightweight small side networks, its stringent memory constraint severely limits learning capacity of side networks, thereby significantly compromising performance. To address these limitations, we propose a novel Mixed-Precision Interactive Side Mixture-of-Experts framework (MP-ISMoE). Specifically, we first propose a Gaussian Noise Perturbed Iterative Quantization (GNP-IQ) scheme to quantize weights into lower-bits while effectively decreasing quantization errors. By leveraging memory conserved from GNP-IQ, we subsequently employ Interactive Side Mixture-of-Experts (ISMoE) to scaling up side networks without sacrificing overall memory efficiency. Different from conventional mixture-of-experts, ISMoE learns to select optimal experts by interacting with salient features from frozen backbones, thus suppressing knowledge forgetting and boosting performance. Extensive experiments across diverse vision-language and language-only tasks demonstrate that MP-ISMoE remarkably promotes accuracy compared to state-of-the-art METL approaches, while maintaining comparable parameter and memory efficiency.

cs.LG↗

arXiv2026

Multimodal Structure Learning: Disentangling Shared and Specific Topology via Cross-Modal Graphical Lasso

Learning interpretable multimodal representations inherently relies on uncovering the conditional dependencies between heterogeneous features. However, sparse graph estimation techniques, such as Graphical Lasso (GLasso), to visual-linguistic domains is severely bottlenecked by high-dimensional noise, modality misalignment, and the confounding of shared versus category-specific topologies. In this paper, we propose Cross-Modal Graphical Lasso (CM-GLasso) that overcomes these fundamental limitations. By coupling a novel text-visualization strategy with a unified vision-language encoder, we strictly align multimodal features into a shared latent space. We introduce a cross-attention distillation mechanism that condenses high-dimensional patches into explicit semantic nodes, naturally extracting spatial-aware cross-modal priors. Furthermore, we unify tailored GLasso estimation and Common-Specific Structure Learning (CSSL) into a joint objective optimized via the Alternating Direction Method of Multiplier (ADMM). This formulation guarantees the simultaneous disentanglement of invariant and class-specific precision matrices without multi-step error accumulation. Extensive experiments across eight benchmarks covering both natural and medical domains demonstrate that CM-GLasso establishes a new state-of-the-art in generative classification and dense semantic segmentation tasks.

cs.CV↗

arXiv2026

SAATT Nav: a Socially Aware Autonomous Transparent Transportation Navigation Framework for Wheelchairs

While powered wheelchairs reduce physical fatigue as opposed to manual wheelchairs for individuals with mobility impairment, they demand high cognitive workload due to information processing, decision making and motor coordination. Current autonomous systems lack social awareness in navigation and transparency in decision-making, leading to decreased perceived safety and trust from the user and others in context. This work proposes Socially Aware Autonomous Transparent Transportation (SAATT) Navigation framework for wheelchairs as a potential solution. By implementing a Large Language Model (LLM) informed of user intent and capable of predicting other peoples' intent as a decision-maker for its local controller, it is able to detect and navigate social situations, such as passing pedestrians or a pair conversing. Furthermore, the LLM textually communicates its reasoning at each waypoint for transparency. In this experiment, it is compared against a standard global planner, a representative competing social navigation model, and an Ablation study in three simulated environments varied by social levels in eight metrics categorized under Safety, Social Compliance, Efficiency, and Comfort. Overall, SAATT Nav outperforms in most social situations and equivalently or only slightly worse in the remaining metrics, demonstrating the potential of a socially aware and transparent autonomous navigation system to assist wheelchair users.

cs.RO↗

arXiv2026

MARTI-MARS$^2$: Scaling Multi-Agent Self-Search via Reinforcement Learning for Code Generation

While the complex reasoning capability of Large Language Models (LLMs) has attracted significant attention, single-agent systems often encounter inherent performance ceilings in complex tasks such as code generation. Multi-agent collaboration offers a promising avenue to transcend these boundaries. However, existing frameworks typically rely on prompt-based test-time interactions or multi-role configurations trained with homogeneous parameters, limiting error correction capabilities and strategic diversity. In this paper, we propose a Multi-Agent Reinforced Training and Inference Framework with Self-Search Scaling (MARTI-MARS2), which integrates policy learning with multi-agent tree search by formulating the multi-agent collaborative exploration process as a dynamic and learnable environment. By allowing agents to iteratively explore and refine within the environment, the framework facilitates evolution from parameter-sharing homogeneous multi-role training to heterogeneous multi-agent training, breaking through single-agent capability limits. We also introduce an efficient inference strategy MARTI-MARS2-T+ to fully exploit the scaling potential of multi-agent collaboration at test time. We conduct extensive experiments across varied model scales (8B, 14B, and 32B) on challenging code generation benchmarks. Utilizing two collaborating 32B models, MARTI-MARS2 achieves 77.7%, outperforming strong baselines like GPT-5.1. Furthermore, MARTI-MARS2 reveals a novel scaling law: shifting from single-agent to homogeneous multi-role and ultimately to heterogeneous multi-agent paradigms progressively yields higher RL performance ceilings, robust TTS capabilities, and greater policy diversity, suggesting that policy diversity is critical for scaling intelligence via multi-agent reinforcement learning.

cs.LG↗