Finite dynamical systems (FDSs) are commonly used to model systems with a finite number of states that evolve deterministically and at discrete time steps. Considered up to isomorphism, those correspond to functional graphs. As such, FDSs have a sum and product operation, which correspond to the direct sum and direct product of their respective graphs; the collection of FDSs endowed with these operations then forms a semiring. The algebraic structure of the product of FDSs is particularly interesting. For instance, an FDS can be factorised if and only if it is composed of two sub-systems running in parallel. In this work, we further the understanding of the factorisation, division, and root finding problems for FDSs. Firstly, an FDS $A$ is cancellative if one can divide by it unambiguously, i.e. $AX = AY$ implies $X = Y$. We prove that an FDS $A$ is cancellative if and only if it has a fixpoint. Secondly, we prove that if an FDS $A$ has a $k$-th root (i.e. $B$ such that $B^k = A$), then it is unique. Thirdly, unlike integers, the monoid of FDS product does not have unique factorisation into irreducibles. We instead exhibit a large class of monoids of FDSs with unique factorisation. To obtain our main results, we introduce the unrolling of an FDS, which can be viewed as a space-time expansion of the system. This allows us to work with (possibly infinite) trees, where the product is easier to handle than its counterpart for FDSs.
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Many applications, such as system identification, classification of time series, direct and inverse problems in partial differential equations, and uncertainty quantification lead to the question of approximation of a non-linear operator between metric spaces $\mathfrak{X}$ and $\mathfrak{Y}$. We study the problem of determining the degree of approximation of such operators on a compact subset $K_\mathfrak{X}\subset \mathfrak{X}$ using a finite amount of information. If $\mathcal{F}: K_\mathfrak{X}\to K_\mathfrak{Y}$, a well established strategy to approximate $\mathcal{F}(F)$ for some $F\in K_\mathfrak{X}$ is to encode $F$ (respectively, $\mathcal{F}(F)$) in terms of a finite number $d$ (repectively $m$) of real numbers. Together with appropriate reconstruction algorithms (decoders), the problem reduces to the approximation of $m$ functions on a compact subset of a high dimensional Euclidean space $\mathbb{R}^d$, equivalently, the unit sphere $\mathbb{S}^d$ embedded in $\mathbb{R}^{d+1}$. The problem is challenging because $d$, $m$, as well as the complexity of the approximation on $\mathbb{S}^d$ are all large, and it is necessary to estimate the accuracy keeping track of the inter-dependence of all the approximations involved. In this paper, we establish constructive methods to do this efficiently; i.e., with the constants involved in the estimates on the approximation on $\mathbb{S}^d$ being $\mathcal{O}(d^{1/6})$. We study different smoothness classes for the operators, and also propose a method for approximation of $\mathcal{F}(F)$ using only information in a small neighborhood of $F$, resulting in an effective reduction in the number of parameters involved.
Intensive Care Units usually carry patients with a serious risk of mortality. Recent research has shown the ability of Machine Learning to indicate the patients' mortality risk and point physicians toward individuals with a heightened need for care. Nevertheless, healthcare data is often subject to privacy regulations and can therefore not be easily shared in order to build Centralized Machine Learning models that use the combined data of multiple hospitals. Federated Learning is a Machine Learning framework designed for data privacy that can be used to circumvent this problem. In this study, we evaluate the ability of deep Federated Learning to predict the risk of Intensive Care Unit mortality at an early stage. We compare the predictive performance of Federated, Centralized, and Local Machine Learning in terms of AUPRC, F1-score, and AUROC. Our results show that Federated Learning performs equally well as the centralized approach and is substantially better than the local approach, thus providing a viable solution for early Intensive Care Unit mortality prediction. In addition, we show that the prediction performance is higher when the patient history window is closer to discharge or death. Finally, we show that using the F1-score as an early stopping metric can stabilize and increase the performance of our approach for the task at hand.
This paper describes three methods for carrying out non-asymptotic inference on partially identified parameters that are solutions to a class of optimization problems. Applications in which the optimization problems arise include estimation under shape restrictions, estimation of models of discrete games, and estimation based on grouped data. The partially identified parameters are characterized by restrictions that involve the unknown population means of observed random variables in addition to structural parameters. Inference consists of finding confidence intervals for functions of the structural parameters. Our theory provides finite-sample lower bounds on the coverage probabilities of the confidence intervals under three sets of assumptions of increasing strength. With the moderate sample sizes found in most economics applications, the bounds become tighter as the assumptions strengthen. We discuss estimation of population parameters that the bounds depend on and contrast our methods with alternative methods for obtaining confidence intervals for partially identified parameters. The results of Monte Carlo experiments and empirical examples illustrate the usefulness of our method.
We characterize a Hawkes point process with kernel proportional to the probability density function of Mittag-Leffler random variables. This kernel decays as a power law with exponent $\beta +1 \in (1,2]$. Several analytical results can be proved, in particular for the expected intensity of the point process and for the expected number of events of the counting process. These analytical results are used to validate algorithms that numerically invert the Laplace transform of the expected intensity as well as Monte Carlo simulations of the process. Finally, Monte Carlo simulations are used to derive the full distribution of the number of events. The algorithms used for this paper are available at {\tt //github.com/habyarimanacassien/Fractional-Hawkes}.
Goldwasser et al.\ (2021) recently proposed the setting of PAC verification, where a hypothesis (machine learning model) that purportedly satisfies the agnostic PAC learning objective is verified using an interactive proof. In this paper we develop this notion further in a number of ways. First, we prove a lower bound for PAC verification of $\Omega(\sqrt{d})$ i.i.d.\ samples for hypothesis classes of VC dimension $d$. Second, we present a protocol for PAC verification of unions of intervals over $\mathbb{R}$ that improves upon their proposed protocol for that task, and matches our lower bound. Third, we introduce a natural generalization of their definition to verification of general statistical algorithms, which is applicable to a wider variety of practical algorithms beyond agnostic PAC learning. Showcasing our proposed definition, our final result is a protocol for the verification of statistical query algorithms that satisfy a combinatorial constraint on their queries.
Multi-stage ranking pipelines have been a practical solution in modern search systems, where the first-stage retrieval is to return a subset of candidate documents, and latter stages attempt to re-rank those candidates. Unlike re-ranking stages going through quick technique shifts during past decades, the first-stage retrieval has long been dominated by classical term-based models. Unfortunately, these models suffer from the vocabulary mismatch problem, which may block re-ranking stages from relevant documents at the very beginning. Therefore, it has been a long-term desire to build semantic models for the first-stage retrieval that can achieve high recall efficiently. Recently, we have witnessed an explosive growth of research interests on the first-stage semantic retrieval models. We believe it is the right time to survey current status, learn from existing methods, and gain some insights for future development. In this paper, we describe the current landscape of the first-stage retrieval models under a unified framework to clarify the connection between classical term-based retrieval methods, early semantic retrieval methods and neural semantic retrieval methods. Moreover, we identify some open challenges and envision some future directions, with the hope of inspiring more researches on these important yet less investigated topics.
Recommender systems play a fundamental role in web applications in filtering massive information and matching user interests. While many efforts have been devoted to developing more effective models in various scenarios, the exploration on the explainability of recommender systems is running behind. Explanations could help improve user experience and discover system defects. In this paper, after formally introducing the elements that are related to model explainability, we propose a novel explainable recommendation model through improving the transparency of the representation learning process. Specifically, to overcome the representation entangling problem in traditional models, we revise traditional graph convolution to discriminate information from different layers. Also, each representation vector is factorized into several segments, where each segment relates to one semantic aspect in data. Different from previous work, in our model, factor discovery and representation learning are simultaneously conducted, and we are able to handle extra attribute information and knowledge. In this way, the proposed model can learn interpretable and meaningful representations for users and items. Unlike traditional methods that need to make a trade-off between explainability and effectiveness, the performance of our proposed explainable model is not negatively affected after considering explainability. Finally, comprehensive experiments are conducted to validate the performance of our model as well as explanation faithfulness.
To solve the information explosion problem and enhance user experience in various online applications, recommender systems have been developed to model users preferences. Although numerous efforts have been made toward more personalized recommendations, recommender systems still suffer from several challenges, such as data sparsity and cold start. In recent years, generating recommendations with the knowledge graph as side information has attracted considerable interest. Such an approach can not only alleviate the abovementioned issues for a more accurate recommendation, but also provide explanations for recommended items. In this paper, we conduct a systematical survey of knowledge graph-based recommender systems. We collect recently published papers in this field and summarize them from two perspectives. On the one hand, we investigate the proposed algorithms by focusing on how the papers utilize the knowledge graph for accurate and explainable recommendation. On the other hand, we introduce datasets used in these works. Finally, we propose several potential research directions in this field.
In this monograph, I introduce the basic concepts of Online Learning through a modern view of Online Convex Optimization. Here, online learning refers to the framework of regret minimization under worst-case assumptions. I present first-order and second-order algorithms for online learning with convex losses, in Euclidean and non-Euclidean settings. All the algorithms are clearly presented as instantiation of Online Mirror Descent or Follow-The-Regularized-Leader and their variants. Particular attention is given to the issue of tuning the parameters of the algorithms and learning in unbounded domains, through adaptive and parameter-free online learning algorithms. Non-convex losses are dealt through convex surrogate losses and through randomization. The bandit setting is also briefly discussed, touching on the problem of adversarial and stochastic multi-armed bandits. These notes do not require prior knowledge of convex analysis and all the required mathematical tools are rigorously explained. Moreover, all the proofs have been carefully chosen to be as simple and as short as possible.
Explainable recommendation attempts to develop models that generate not only high-quality recommendations but also intuitive explanations. The explanations may either be post-hoc or directly come from an explainable model (also called interpretable or transparent model in some context). Explainable recommendation tries to address the problem of why: by providing explanations to users or system designers, it helps humans to understand why certain items are recommended by the algorithm, where the human can either be users or system designers. Explainable recommendation helps to improve the transparency, persuasiveness, effectiveness, trustworthiness, and satisfaction of recommendation systems. In this survey, we review works on explainable recommendation in or before the year of 2019. We first highlight the position of explainable recommendation in recommender system research by categorizing recommendation problems into the 5W, i.e., what, when, who, where, and why. We then conduct a comprehensive survey of explainable recommendation on three perspectives: 1) We provide a chronological research timeline of explainable recommendation, including user study approaches in the early years and more recent model-based approaches. 2) We provide a two-dimensional taxonomy to classify existing explainable recommendation research: one dimension is the information source (or display style) of the explanations, and the other dimension is the algorithmic mechanism to generate explainable recommendations. 3) We summarize how explainable recommendation applies to different recommendation tasks, such as product recommendation, social recommendation, and POI recommendation. We also devote a section to discuss the explanation perspectives in broader IR and AI/ML research. We end the survey by discussing potential future directions to promote the explainable recommendation research area and beyond.
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