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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↗
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