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Dynamic Mode Decomposition (DMD) is a popular data-driven analysis technique used to decompose complex, nonlinear systems into a set of modes, revealing underlying patterns and dynamics through spectral analysis. This review presents a comprehensive and pedagogical examination of DMD, emphasizing the role of Koopman operators in transforming complex nonlinear dynamics into a linear framework. A distinctive feature of this review is its focus on the relationship between DMD and the spectral properties of Koopman operators, with particular emphasis on the theory and practice of DMD algorithms for spectral computations. We explore the diverse "multiverse" of DMD methods, categorized into three main areas: linear regression-based methods, Galerkin approximations, and structure-preserving techniques. Each category is studied for its unique contributions and challenges, providing a detailed overview of significant algorithms and their applications as outlined in Table 1. We include a MATLAB package with examples and applications to enhance the practical understanding of these methods. This review serves as both a practical guide and a theoretical reference for various DMD methods, accessible to both experts and newcomers, and enabling readers to delve into their areas of interest in the expansive field of DMD.

Variable curvature modeling tools provide an accurate means of controlling infinite degrees-of-freedom deformable bodies and structures. However, their forward and inverse Newton-Euler dynamics are fraught with high computational costs. Assuming piecewise constant strains across discretized Cosserat rods imposed on the soft material, a composite two time-scale singularly perturbed nonlinear backstepping control scheme is here introduced. This is to alleviate the long computational times of the recursive Newton-Euler dynamics for soft structures. Our contribution is three-pronged: (i) we decompose the system's Newton-Euler dynamics to a two coupled sub-dynamics by introducing a perturbation parameter; (ii) we then prescribe a set of stabilizing controllers for regulating each subsystem's dynamics; and (iii) we study the interconnected singularly perturbed system and analyze its stability.

Large Language Models (LLMs) have revolutionized natural language processing tasks, demonstrating their exceptional capabilities in various domains. However, their potential for behavior graph understanding in job recommendations remains largely unexplored. This paper focuses on unveiling the capability of large language models in understanding behavior graphs and leveraging this understanding to enhance recommendations in online recruitment, including the promotion of out-of-distribution (OOD) application. We present a novel framework that harnesses the rich contextual information and semantic representations provided by large language models to analyze behavior graphs and uncover underlying patterns and relationships. Specifically, we propose a meta-path prompt constructor that leverages LLM recommender to understand behavior graphs for the first time and design a corresponding path augmentation module to alleviate the prompt bias introduced by path-based sequence input. By leveraging this capability, our framework enables personalized and accurate job recommendations for individual users. We evaluate the effectiveness of our approach on a comprehensive dataset and demonstrate its ability to improve the relevance and quality of recommended quality. This research not only sheds light on the untapped potential of large language models but also provides valuable insights for developing advanced recommendation systems in the recruitment market. The findings contribute to the growing field of natural language processing and offer practical implications for enhancing job search experiences. We release the code at //github.com/WLiK/GLRec.

The relational calculus (RC) is a concise, declarative query language. However, existing RC query evaluation approaches are inefficient and often deviate from established algorithms based on finite tables used in database management systems. We devise a new translation of an arbitrary RC query into two safe-range queries, for which the finiteness of the query's evaluation result is guaranteed. Assuming an infinite domain, the two queries have the following meaning: The first is closed and characterizes the original query's relative safety, i.e., whether given a fixed database, the original query evaluates to a finite relation. The second safe-range query is equivalent to the original query, if the latter is relatively safe. We compose our translation with other, more standard ones to ultimately obtain two SQL queries. This allows us to use standard database management systems to evaluate arbitrary RC queries. We show that our translation improves the time complexity over existing approaches, which we also empirically confirm in both realistic and synthetic experiments.

The problem of fairly allocating a set of indivisible items is a well-known challenge in the field of (computational) social choice. In this scenario, there is a fundamental incompatibility between notions of fairness (such as envy-freeness and proportionality) and economic efficiency (such as Pareto-optimality). However, in the real world, items are not always allocated once and for all, but often repeatedly. For example, the items may be recurring chores to distribute in a household. Motivated by this, we initiate the study of the repeated fair division of indivisible goods and chores, and propose a formal model for this scenario. In this paper, we show that, if the number of repetitions is a multiple of the number of agents, there always exists a sequence of allocations that is proportional and Pareto-optimal. On the other hand, irrespective of the number of repetitions, an envy-free and Pareto-optimal sequence of allocations may not exist. For the case of two agents, we show that if the number of repetitions is even, it is always possible to find a sequence of allocations that is overall envy-free and Pareto-optimal. We then prove even stronger fairness guarantees, showing that every allocation in such a sequence satisfies some relaxation of envy-freeness. Finally, in case that the number of repetitions can be chosen freely, we show that envy-free and Pareto-optimal allocations are achievable for any number of agents.

Large Language Models (LLMs) have significantly advanced natural language processing (NLP) with their impressive language understanding and generation capabilities. However, their performance may be suboptimal for long-tail or domain-specific tasks due to limited exposure to domain-specific knowledge and vocabulary. Additionally, the lack of transparency of most state-of-the-art (SOTA) LLMs, which can only be accessed via APIs, impedes further fine-tuning with custom data. Moreover, data privacy is a significant concern. To address these challenges, we propose the novel Parametric Knowledge Guiding (PKG) framework, which equips LLMs with a knowledge-guiding module to access relevant knowledge at runtime without altering the LLMs' parameters. Our PKG is based on open-source "white-box" small language models, allowing offline storage of any knowledge that LLMs require. We demonstrate that our PKG framework can enhance the performance of "black-box" LLMs on a range of long-tail and domain-specific downstream tasks requiring factual, tabular, medical, and multimodal knowledge.

Recent contrastive representation learning methods rely on estimating mutual information (MI) between multiple views of an underlying context. E.g., we can derive multiple views of a given image by applying data augmentation, or we can split a sequence into views comprising the past and future of some step in the sequence. Contrastive lower bounds on MI are easy to optimize, but have a strong underestimation bias when estimating large amounts of MI. We propose decomposing the full MI estimation problem into a sum of smaller estimation problems by splitting one of the views into progressively more informed subviews and by applying the chain rule on MI between the decomposed views. This expression contains a sum of unconditional and conditional MI terms, each measuring modest chunks of the total MI, which facilitates approximation via contrastive bounds. To maximize the sum, we formulate a contrastive lower bound on the conditional MI which can be approximated efficiently. We refer to our general approach as Decomposed Estimation of Mutual Information (DEMI). We show that DEMI can capture a larger amount of MI than standard non-decomposed contrastive bounds in a synthetic setting, and learns better representations in a vision domain and for dialogue generation.

Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.

Attention Model has now become an important concept in neural networks that has been researched within diverse application domains. This survey provides a structured and comprehensive overview of the developments in modeling attention. In particular, we propose a taxonomy which groups existing techniques into coherent categories. We review salient neural architectures in which attention has been incorporated, and discuss applications in which modeling attention has shown a significant impact. Finally, we also describe how attention has been used to improve the interpretability of neural networks. We hope this survey will provide a succinct introduction to attention models and guide practitioners while developing approaches for their applications.

Semantic Role Labeling (SRL) is believed to be a crucial step towards natural language understanding and has been widely studied. Recent years, end-to-end SRL with recurrent neural networks (RNN) has gained increasing attention. However, it remains a major challenge for RNNs to handle structural information and long range dependencies. In this paper, we present a simple and effective architecture for SRL which aims to address these problems. Our model is based on self-attention which can directly capture the relationships between two tokens regardless of their distance. Our single model achieves F$_1=83.4$ on the CoNLL-2005 shared task dataset and F$_1=82.7$ on the CoNLL-2012 shared task dataset, which outperforms the previous state-of-the-art results by $1.8$ and $1.0$ F$_1$ score respectively. Besides, our model is computationally efficient, and the parsing speed is 50K tokens per second on a single Titan X GPU.