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arXiv2026

Twelve Simple Algorithms to Compute Fibonacci Numbers

The Fibonacci numbers are a sequence of integers in which every number after the first two, 0 and 1, is the sum of the two preceding numbers. These numbers are well known, and the algorithms to compute them are simple enough that they are often used in introductory algorithms courses. In this paper, we present twelve such algorithm together with their time and space complexity analyses. Though very simple, these algorithms illustrate eleven concepts from the algorithms field, ranging from top-down vs. bottom-up dynamic programming to recursion depth, and we say which algorithms illustrate which concept. We also present the results of a small-scale experimental comparison of their runtimes on a personal laptop, where the slowest algorithm takes about four orders of magnitude longer than the fastest. Finally, we provide a list of homework questions for students. We hope that this paper can serve as a useful resource for students learning the basics of algorithms.

cs.DS↗

arXiv2026

LMI Properties and Applications in Systems, Stability, and Control Theory

Linear matrix inequalities (LMIs) commonly appear in systems, stability, and control applications. Many analysis and synthesis problems in these areas can be solved as feasibility or optimization problems subject to LMI constraints. Although most well-known LMI properties and manipulation tricks, such as the Schur complement and the congruence transformation, can be found in standard references, many useful LMI properties are scattered throughout the literature. The purpose of this document is to collect and organize properties, tricks, and applications related to LMIs from a number of references together in a single document. In this sense, the document can be thought of as an "LMI encyclopedia" or "LMI cookbook." Proofs of the properties presented in this document are not included when they can be found in the cited references in the interest of brevity. Illustrative examples are included whenever necessary to fully explain a certain property. Multiple equivalent forms of LMIs are often presented to give the reader a choice of which form may be best suited for a particular problem at hand. The equivalency of some of the LMIs in this document may be straightforward to more experienced readers, but the authors believe that some readers may benefit from the presentation of multiple equivalent LMIs.

eess.SY↗

arXiv2026

Model Selection and Parameter Estimation of One-Dimensional Gaussian Mixture Models

In this paper, we study the problem of learning one-dimensional Gaussian mixture models (GMMs) with a specific focus on estimating both the model order and the mixing distribution from independent and identically distributed (i.i.d.) samples. This paper establishes the optimal sampling complexity for model order estimation in one-dimensional Gaussian mixture models. We prove a fundamental lower bound on the number of samples required to correctly identify the number of components with high probability, showing that this limit depends critically on the separation between component means and the total number of components. We then propose a Fourier-based approach to estimate both the model order and the mixing distribution. Our algorithm utilizes Fourier measurements constructed from the samples, and our analysis demonstrates that its sample complexity matches the established lower bound, thereby confirming its optimality. Numerical experiments further show that our method outperforms conventional techniques in terms of efficiency and accuracy.

stat.ML↗

arXiv2026

OM4OV: Leveraging Ontology Matching for Ontology Versioning

Due to the dynamics of the Semantic Web, version control is necessary to manage changes in widely used ontologies. Despite the long-standing recognition of ontology versioning (OV) as a crucial component of efficient ontology management, many approaches treat OV as similar to ontology matching (OM) and directly reuse OM systems for OV tasks. In this study, we systematically analyse similarities and differences between OM and OV and formalise an OM4OV framework to offer more advanced OV support. The framework is implemented and evaluated in the state-of-the-art OM system Agent-OM. The experimental results indicate that OM systems can be effectively reused for OV tasks, but without the necessary extensions, can produce skewed measurements, poor performance in detecting update entities, and limited explanations of false mappings. To tackle these issues, we propose an optimisation method called the cross-reference (CR) mechanism, which builds on existing OM alignments to reduce the number of matching candidates and to improve overall OV performance.

cs.AI↗

arXiv2026

Stein's method for marginals on large graphical models

Many spatial models exhibit locality structures that effectively reduce their intrinsic dimensionality, enabling efficient approximation and sampling of high-dimensional distributions. However, existing approximation techniques primarily focus on joint distributions and do not provide precise accuracy control for low-dimensional marginals, which are of primary interest in many practical scenarios. By leveraging the locality structures, we establish a dimension independent uniform error bound for the marginals of approximate distributions. Inspired by the Stein's method, we introduce a novel $δ$-locality condition that quantifies the locality in distributions, and link it to the structural assumptions such as the sparse graphical models. The theoretical guarantee motivates the localization of existing sampling methods, as we illustrate through the localized likelihood-informed subspace method and localized score matching. We show that by leveraging the locality structure, these methods greatly reduce the sample complexity and computational cost via localized and parallel implementations.

stat.ML↗

arXiv2026

Learning Personalized Prompts for Healthcare Guidance

The rapid development of large language models (LLMs) has transformed many industries, including healthcare. In practice, hospitals and patients increasingly seek LLM-based systems capable of interpreting personal health records and providing healthcare guidance. However, existing approaches mainly rely on general medical knowledge and often fail to account for individual variability, limiting their ability to provide personalized guidance. To address this, we propose personalized prompt learning (PPL), a framework that learns individualized prompts to guide LLMs in generating personalized healthcare recommendations. PPL constructs initial personalized prompts by leveraging both self-informed patient information and peer-informed signals derived from clinically similar cases. These prompts are then refined using reinforcement learning (RL) to better align the generated responses with physician recommendations written for each patient. PPL operates with hard prompts, enabling seamless integration with proprietary LLMs without modifying the underlying models. We evaluate PPL on real-world obstetrics and gynecology data. The results show that our approach produces more personalized healthcare guidance and wins 97 out of 100 comparisons in expert evaluation, demonstrating its potential for broader healthcare applications. Our code is publicly available at https://github.com/CGCL-codes/PPL.

cs.CL↗

arXiv2026

Perforated Backpropagation: A Neuroscience Inspired Extension to Artificial Neural Networks

The neurons of artificial neural networks were originally invented when much less was known about biological neurons than is known today. Our work explores a modification to the core neuron unit to make it more parallel to a biological neuron. The modification is made with the knowledge that biological dendrites are not simply passive activation funnels, but also compute complex non-linear functions as they transmit activation to the cell body. The paper explores a novel system of ``perforated'' backpropagation empowering the artificial neurons of deep neural networks to achieve better performance coding for the same features they coded for in the original architecture. After an initial network training phase, additional ``dendrite'' nodes are added to the network and separately trained with a different objective: to correlate their output with the remaining error of the original neurons. The trained dendrites are then frozen, and the original neurons are further trained, now taking into account the additional error signals provided by the dendrites. The cycle of training the original neurons and then adding and training dendrites can be repeated several times until satisfactory performance is achieved. Our algorithm was successfully added to modern state-of-the-art PyTorch networks across multiple domains, improving upon original accuracies and allowing for significant model compression without a loss in accuracy.

cs.NE↗

arXiv2026

Optimization of a Triangular Delaunay Mesh Generator using Reinforcement Learning

In this work we introduce a triangular Delaunay mesh generator that can be trained using reinforcement learning to maximize a given mesh quality metric. Our mesh generator consists of a graph neural network that distributes and modifies vertices, and a standard Delaunay algorithm to triangulate the vertices. We explore various design choices and evaluate our mesh generator on various tasks including mesh generation, mesh improvement, and producing variable resolution meshes. The learned mesh generator outputs meshes that are comparable to those produced by Triangle and DistMesh, two popular Delaunay-based mesh generators.

cs.CG↗

arXiv2026

Towards Accurate and Lightweight Peripheral Neuroblastic Tumor Diagnosis via Contrastive Multi-scale Pathological Image Analysis

Peripheral neuroblastic tumors (pNTs) are among the most common extracranial solid tumors in children, and accurate pathological subtyping is important for risk stratification and treatment planning. However, pNT subtyping on hematoxylin-eosin whole-slide images (WSIs) remains challenging because of limited pediatric tumor cohorts, marked histological heterogeneity, inter-observer variability, and the computational burden of existing WSI classifiers. To address these challenges, we propose CoPath, a framework consisting of CoHisNet and PathVote. CoHisNet is a lightweight multi-scale feature-fusion network for patch-level histopathological classification. By replacing the multilayer perceptron components in Swin Transformer blocks and the classification head with Kolmogorov-Arnold Network layers, CoHisNet improves nonlinear feature modeling under a compact architecture. Its multi-scale interaction and contrast-driven feature-enhancement design enables the model to capture both tissue-level structures and fine-grained cellular morphology. PathVote further incorporates pathology-informed tissue-component priors to aggregate patch-level predictions into WSI-level decisions. We validated CoPath on a private two-branch PpNTs cohort and the public BreakHis breast cancer histopathology dataset. Experimental results show that CoPath achieves competitive or superior performance compared with general image classifiers, pathology foundation models under linear probing, and pathology-specific classification models, while maintaining substantially lower computational complexity. The source code is available at https://github.com/JSLiam94/CoPath.

cs.CV↗

arXiv2026

Multilevel lattice-based kernel approximation for elliptic PDEs with random coefficients

This paper introduces a multilevel kernel-based approximation method to estimate efficiently solutions to elliptic partial differential equations (PDEs) with periodic random coefficients. Building upon the work of Kaarnioja, Kazashi, Kuo, Nobile, Sloan (Numer. Math., 2022) on kernel interpolation with quasi-Monte Carlo (QMC) lattice point sets, we leverage multilevel techniques to enhance computational efficiency while maintaining a given level of accuracy. In the function space setting with product-type weight parameters, the single-level approximation can achieve an accuracy of $\varepsilon>0$ with cost $\mathcal{O}(\varepsilon^{-η-ν-θ})$ for positive constants $η, ν, θ$ depending on the rates of convergence associated with dimension truncation, kernel approximation, and finite element approximation, respectively. Our multilevel approximation can achieve the same $\varepsilon$ accuracy at a reduced cost $\mathcal{O}(\varepsilon^{-η-\max(ν,θ)})$. Full regularity theory and error analysis are provided, followed by numerical experiments that validate the efficacy of the proposed multilevel approximation in comparison to the single-level approach.

math.NA↗

arXiv2026

Reliable Proof Generation with LLMs via Analogical Retrieval and Symbolic Verification: A Case Study in Euclidean Geometry

Large language models (LLMs) struggle with formal domains that require rigorous logical deduction and symbolic reasoning, such as mathematical proof generation. We propose a neuro-symbolic approach centered on the hypothesis that structurally analogous problems often admit similar proofs. As a proof-of-concept, we focus on SAT-level geometry problems. Our approach is two-fold: (1) We retrieve analogous problems and use their proofs to guide the LLM, and (2) a formal verifier evaluates the generated proofs and provides feedback, helping the model fix incorrect proofs. Our complete pipeline substantially improves proof accuracy across model families, achieving 68%-96% accuracy compared with 10%-44% for LLM-only baselines that use neither analogy retrieval nor verifier feedback. When comparing against models with the same verifier feedback and inference budget, analogical guidance improves accuracy from 88% to 96% for GPT-5, 78% to 86% for Claude Sonnet 4.6, 72% to 86% for Gemini-Flash-2.5, and 52% to 80% for OpenAI o1. More broadly, shifting to LLMs that generate provably correct conclusions has the potential to dramatically improve their reliability, accuracy and consistency, unlocking complex tasks and critical real-world applications that require trustworthiness.

cs.AI↗

arXiv2026

Reasoning Beyond Language: A Comprehensive Survey on Latent Chain-of-Thought Reasoning

Large Language Models (LLMs) have shown impressive performance on complex tasks through Chain-of-Thought (CoT) reasoning. However, conventional CoT relies on explicitly verbalized intermediate steps, which constrains its broader applicability, particularly in abstract reasoning tasks beyond language. To address this, there has been growing research interest in \textit{latent CoT reasoning}, where the reasoning process is embedded within latent spaces. By decoupling reasoning from explicit language generation, latent CoT offers the promise of richer cognitive representations and facilitates more flexible, faster inference. This paper aims to present a comprehensive overview of this emerging paradigm and establish a systematic taxonomy. We analyze recent advances in methods, categorizing them from token-wise horizontal approaches to layer-wise vertical strategies. We then provide in-depth discussions of these methods, highlighting their design principles, applications, and remaining challenges. We hope that our survey provides a structured foundation for advancing this promising direction in LLM reasoning. The relevant papers will be regularly updated at https://github.com/EIT-NLP/Awesome-Latent-CoT.

cs.CL↗

arXiv2026

Popular but Wrong: Understanding and Mitigating LLM Overconfidence through Knowledge Popularity

Large language models (LLMs) often produce incorrect answers with high confidence, yet the factors associated with such overconfidence remain insufficiently understood. We study this problem through the lens of knowledge popularity. Using entity-centric factual QA derived from Wikidata triplets, we characterize popularity through question entity popularity, answer popularity, and question-answer co-occurrence. We find two consistent patterns. First, hallucinated answers are far from random: compared with ground-truth answers, they tend to be more popular or more frequently associated with the question entity. Second, confidence is strongly tied to the popularity of generated answers: even among incorrect predictions, more popular answers or those with higher question-answer co-occurrence receive higher confidence. Together, these findings suggest that popular but wrong alternatives may contribute to overconfidence. We further show that popularity-related signals can mitigate overconfidence and improve overall confidence estimation. Across six models and three datasets, incorporating knowledge popularity reduces average confidence on incorrect answers from 0.765 to 0.254 and overall ECE from 0.356 to 0.050, while improving Alignment from 77.08% to 83.72%.

cs.CL↗

arXiv2026

A foundation model with multi-variate parallel attention to generate neuronal activity

Learning from multi-variate time-series with heterogeneous channel configurations remains a fundamental challenge for deep neural networks, particularly in clinical domains such as intracranial electroencephalography (iEEG), where channel setups vary widely across subjects. In this work, we introduce multi-variate parallel attention (MVPA), a novel self-attention mechanism that disentangles content, temporal, and spatial attention, enabling flexible, generalizable, and efficient modeling of time-series data with varying channel counts and configurations. We use MVPA to build MVPFormer, a generative foundation model for human electrophysiology, trained to predict the evolution of iEEG signals across subjects. To support this and future efforts by the community, we release the SWEC iEEG dataset, the largest publicly available iEEG dataset to date, comprising nearly 10,000 hours of recordings from heterogeneous clinical sources. MVPFormer leverages MVPA to achieve strong generalization across subjects, demonstrating expert-level performance in several iEEG tasks. MVPFormer surpasses state-of-the-art (SOTA) Transformer baselines in seizure detection across the SWEC, the MAYO, and the FNUSA datasets, while also achieving SOTA performance on four Brain TreeBank iEEG decoding tasks (volume, pitch, onset, and speech). We further validate MVPA on standard time-series forecasting and classification tasks, where it matches or exceeds the performance of existing attention-based models. Together, our contributions establish MVPA as a general-purpose attention mechanism for heterogeneous time-series and MVPFormer as the first open-source, open-weights, and open-data iEEG foundation model with SOTA clinical performance. The code is available at https://github.com/IBM/multi-variate-parallel-transformer. The SWEC iEEG dataset is available at https://huggingface.co/datasets/NeuroTec/SWEC_iEEG_Dataset.

cs.LG↗

arXiv2026

On the Equivalence of the Graph-Structural and Optimization-Based Characterizations of Popular Matchings

Popular matchings provide a model of matching under preferences in which a solution corresponds to a Condorcet winner in voting systems. In a bipartite graph in which the vertices have preferences over their neighbours, a matching is defined to be popular if it does not lose in a majority vote against any matching. In this paper, we study the following three primary problems: only the vertices on one side have preferences; a generalization of this problem allowing ties in the preferences; and the vertices on both sides have preferences. A principal issue in the algorithmic aspects of popular matchings is how to determine the popularity of a matching, because it requires exponential time if the definition is simply applied. In the literature, we have the following two types of characterizations: a graph-structural characterization; and an optimization-based characterization described by maximum-weight matchings. The graph-structural characterizations are specifically designed for each problem and provide a combinatorial structure of the popular matchings. The optimization-based characterizations work in the same manner for all problems, while they do not reveal the structure of the popular matchings. A main contribution of this paper is to provide a direct connection of the above two types of characterizations for all of the three problems. Specifically, we prove that each characterization can be derived from the other, without relying on the fact that they characterize popular matchings. Our proofs offer a comprehensive understanding of the equivalence of the two types of characterizations, and suggest a new interpretation of the graph-structural characterization in terms of the dual optimal solution for the maximum-weight matching problem.

cs.GT↗

arXiv2026

Agnostics: Learning to Code in Any Programming Language via Reinforcement with a Universal Learning Environment

Large language models (LLMs) already excel at writing code in high-resource languages such as Python and JavaScript, yet stumble on low-resource languages that remain essential to science and engineering. Besides the obvious shortage of pre-training data, post-training itself is a bottleneck: every new language seems to require new datasets, test harnesses, and reinforcement-learning (RL) infrastructure. We introduce Agnostics, a language-agnostic post-training pipeline that eliminates this per-language engineering. The key idea is to judge code solely by its externally observable behavior, so a single verifier can test solutions written in any language. Concretely, we (i) use an LLM to rewrite existing unit-test datasets into an I/O format, (ii) supply a short configuration that tells the verifier how to compile and run a target language, and (iii) apply reinforcement learning with verifiable rewards (RLVR) in a robust code execution environment. Applied to five low-resource languages--Lua, Julia, R, OCaml, and Fortran--Agnostics (1) improves Qwen-3 4B to performance that rivals other 16B-70B open-weight models; (2) scales cleanly to larger and diverse model families (Qwen-3 8B, DeepSeek Coder 6.7B Instruct, Phi 4 Mini); and (3) for ${\le} 16$B parameter models, sets new state-of-the-art pass@1 results on MultiPL-E and a new multi-language version of LiveCodeBench that we introduce. We release the language-agnostic training datasets (Ag-MBPP-X, Ag-Codeforces-X, Ag-LiveCodeBench-X), training code, and ready-to-use configurations, making RL post-training in any programming language as simple as editing a short YAML file.

cs.LG↗

arXiv2026

TRACE: Evaluating Execution Efficiency of LLM-Based Code Translation

While Large Language Models (LLMs) have substantially improved the functional correctness of code translation, the critical dimension of execution efficiency remains overlooked. We present \textbf{\textsc{Trace}}, the first benchmark to explicitly assess efficiency in LLM-translated code. \textsc{Trace} includes 1,000 efficiency-critical tasks across C++, Java, and Python, each augmented with stress tests that reveal efficiency disparities often overlooked by small-scale tests. Using \textsc{Trace}, we conduct an extensive evaluation of 28 representative LLMs and highlight several key insights: 1) Correctness and efficiency are often misaligned: the correctness leader Claude-Sonnet-4-Think achieves only moderate time efficiency, outperformed by smaller open-source LLMs such as Qwen2.5-Coder-14B-Instruct. 2) Inefficiency is both prevalent and patterned: 23.5\% of correct translations suffer from notable inefficiency, mainly arising from algorithm implementation discrepancy (11.9\%), language construct mismatch (66.4\%), and resource management inefficiency (21.7\%). 3) Inference-time prompt strategies bring only modest improvements, indicating that simple prompting alone is insufficient to improve translation efficiency. Together, our results establish execution efficiency as an essential dimension of code translation and position \textsc{Trace} as a principled foundation for efficiency-oriented evaluation.

cs.SE↗

arXiv2026

MM-BrowseComp: A Comprehensive Benchmark for Multimodal Browsing Agents

AI agents with advanced reasoning and tool-use capabilities have demonstrated impressive performance in web browsing for deep search. However, existing benchmarks such as BrowseComp primarily focus on textual content, overlooking the prevalence of multimodal content. To bridge this gap, we introduce MM-BrowseComp, a novel benchmark comprising 400 challenging, hand-crafted questions designed to evaluate multimodal retrieval and reasoning capabilities. Unlike prior work, MM-BrowseComp incorporates visual prompts and necessitates the extraction of key evidence from web images and videos to complete questions, rendering text-only approaches insufficient. Additionally, we provide a verified checklist for each question, enabling fine-grained analysis of multimodal dependencies and reasoning paths. Our comprehensive evaluation of 27 state-of-the-art models reveals that even leading models like GPT-5-High with tools achieve only 24.25\% accuracy, highlighting the suboptimal multimodal browsing capabilities, establishing MM-BrowseComp as a rigorous new standard for the field.

cs.CL↗