Mixtures of Gaussian process experts is a class of models that can simultaneously address two of the key limitations inherent in standard Gaussian processes: scalability and predictive performance. In particular, models that use Dirichlet processes as gating functions permit straightforward interpretation and automatic selection of the number of experts in a mixture. While the existing models are intuitive and capable of capturing non-stationarity, multi-modality and heteroskedasticity, the simplicity of their gating functions may limit the predictive performance when applied to complex data-generating processes. Capitalising on the recent advancement in the dependent Dirichlet processes literature, we propose a new mixture model of Gaussian process experts based on kernel stick-breaking processes. Our model maintains the intuitive appeal yet improve the performance of the existing models. To make it practical, we design a sampler for posterior computation based on the slice sampling. The model behaviour and improved predictive performance are demonstrated in experiments using six datasets.
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Processing 是一門開源編程語言和與之配套的集成開發環境（IDE）的名稱。Processing 在電子藝術和視覺設計社區被用來教授編程基礎，并運用于大量的新媒體和互動藝術作品中。
Weakly hard real-time systems can, to some degree, tolerate deadline misses, but their schedulability still needs to be analyzed to ensure their quality of service. Such analysis usually occurs at early design stages to provide implementation guidelines to engineers so that they can make better design decisions. Estimating worst-case execution times (WCET) is a key input to schedulability analysis. However, early on during system design, estimating WCET values is challenging and engineers usually determine them as plausible ranges based on their domain knowledge. Our approach aims at finding restricted, safe WCET sub-ranges given a set of ranges initially estimated by experts in the context of weakly hard real-time systems. To this end, we leverage (1) multi-objective search aiming at maximizing the violation of weakly hard constraints in order to find worst-case scheduling scenarios and (2) polynomial logistic regression to infer safe WCET ranges with a probabilistic interpretation. We evaluated our approach by applying it to an industrial system in the satellite domain and several realistic synthetic systems. The results indicate that our approach significantly outperforms a baseline relying on random search without learning, and estimates safe WCET ranges with a high degree of confidence in practical time (< 23h).
Stick-breaking (SB) processes are often adopted in Bayesian mixture models for generating mixing weights. When covariates influence the sizes of clusters, SB mixtures are particularly convenient as they can leverage their connection to binary regression to ease both the specification of covariate effects and posterior computation. Existing SB models are typically constructed based on continually breaking a single remaining piece of the unit stick. We view this from a dyadic tree perspective in terms of a lopsided bifurcating tree that extends only in one side. We show that several unsavory characteristics of SB models are in fact largely due to this lopsided tree structure. We consider a generalized class of SB models with alternative bifurcating tree structures and examine the influence of the underlying tree topology on the resulting Bayesian analysis in terms of prior assumptions, posterior uncertainty, and computational effectiveness. In particular, we provide evidence that a balanced tree topology, which corresponds to continually breaking all remaining pieces of the unit stick, can resolve or mitigate several undesirable properties of SB models that rely on a lopsided tree.
In this study, we focus on the development and implementation of a comprehensive ensemble of numerical time series forecasting models, collectively referred to as the Group of Numerical Time Series Prediction Model (G-NM). This inclusive set comprises traditional models such as Autoregressive Integrated Moving Average (ARIMA), Holt-Winters' method, and Support Vector Regression (SVR), in addition to modern neural network models including Recurrent Neural Network (RNN) and Long Short-Term Memory (LSTM). G-NM is explicitly constructed to augment our predictive capabilities related to patterns and trends inherent in complex natural phenomena. By utilizing time series data relevant to these events, G-NM facilitates the prediction of such phenomena over extended periods. The primary objective of this research is to both advance our understanding of such occurrences and to significantly enhance the accuracy of our forecasts. G-NM encapsulates both linear and non-linear dependencies, seasonalities, and trends present in time series data. Each of these models contributes distinct strengths, from ARIMA's resilience in handling linear trends and seasonality, SVR's proficiency in capturing non-linear patterns, to LSTM's adaptability in modeling various components of time series data. Through the exploitation of the G-NM potential, we strive to advance the state-of-the-art in large-scale time series forecasting models. We anticipate that this research will represent a significant stepping stone in our ongoing endeavor to comprehend and forecast the complex events that constitute the natural world.
Gaussian processes are a powerful framework for quantifying uncertainty and for sequential decision-making but are limited by the requirement of solving linear systems. In general, this has a cubic cost in dataset size and is sensitive to conditioning. We explore stochastic gradient algorithms as a computationally efficient method of approximately solving these linear systems: we develop low-variance optimization objectives for sampling from the posterior and extend these to inducing points. Counterintuitively, stochastic gradient descent often produces accurate predictions, even in cases where it does not converge quickly to the optimum. We explain this through a spectral characterization of the implicit bias from non-convergence. We show that stochastic gradient descent produces predictive distributions close to the true posterior both in regions with sufficient data coverage, and in regions sufficiently far away from the data. Experimentally, stochastic gradient descent achieves state-of-the-art performance on sufficiently large-scale or ill-conditioned regression tasks. Its uncertainty estimates match the performance of significantly more expensive baselines on a large-scale Bayesian~optimization~task.
We consider a setting where multiple active sources send real-time updates over a single-hop wireless broadcast network to a monitoring station. Our goal is to design a scheduling policy that minimizes the time-average of general non-decreasing cost functions of Age of Information. We use a Whittle index based approach to find low complexity scheduling policies that have good performance. We prove that for a system with two sources, having possibly different cost functions and reliable channels, the Whittle index policy is exactly optimal. We derive structural properties of an optimal policy, that suggest that the performance of the Whittle index policy may be close to optimal in general. These results might also be of independent interest in the study of restless multi-armed bandit problems with similar underlying structure. We further establish that minimizing monitoring error for linear time-invariant systems and symmetric Markov chains is equivalent to minimizing appropriately chosen monotone functions of Age of Information. Finally, we provide simulations comparing the Whittle index policy with optimal scheduling policies found using dynamic programming, which support our results.
Multivariate time series data that capture the temporal evolution of interconnected systems are ubiquitous in diverse areas. Understanding the complex relationships and potential dependencies among co-observed variables is crucial for the accurate statistical modelling and analysis of such systems. Here, we introduce kernel-based statistical tests of joint independence in multivariate time series by extending the $d$-variable Hilbert-Schmidt independence criterion (dHSIC) to encompass both stationary and non-stationary processes, thus allowing broader real-world applications. By leveraging resampling techniques tailored for both single- and multiple-realisation time series, we show how the method robustly uncovers significant higher-order dependencies in synthetic examples, including frequency mixing data and logic gates, as well as real-world climate and socioeconomic data. Our method adds to the mathematical toolbox for the analysis of multivariate time series and can aid in uncovering high-order interactions in data.
In real-life decision-making problems, determining the influences of the factors on the decision attribute is one of the primary tasks. To affect the decision attribute most, finding a proper hierarchy among the factors and determining their importance values in the system becomes quite important. Interpretive structural modeling (ISM) is a widely used hierarchy-building method that mines factor inter-influences based on expert opinions. This paper discusses one of the main drawbacks of the conventional ISM method in systems where the factors are densely interrelated. We refer to such systems as "dense systems". We propose a novel iterative hierarchy-building technique, called 'Iterative Hierarchy and Ranking Process'(IHRP) which performs effectively in such dense systems. To take the vagueness of the expert opinions into account, intuitionistic fuzzy linguistics has been used in the research work. In this paper, we propose a two-stage calculation of the relative importance of the factors in the system based on their hierarchical positions and rank the factors accordingly. We have performed a case study on student performance assessment by taking up novel Indian high-school administrative factors' data collected by surveying the experts in this field. A comparative study has been conducted in terms of the correlation of the factor ranking achieved by the proposed method and conventional ISM method with that of standard outranking methods like TOPSIS, and VIKOR. Our proposed IHRP framework achieves an 85-95% correlation compared to a 50-60% correlation for the conventional ISM method. This proves the effectiveness of the proposed method in determining a better hierarchy than the conventional method, especially in dense systems.
Logical query answering over Knowledge Graphs (KGs) is a fundamental yet complex task. A promising approach to achieve this is to embed queries and entities jointly into the same embedding space. Research along this line suggests that using multi-modal distribution to represent answer entities is more suitable than uni-modal distribution, as a single query may contain multiple disjoint answer subsets due to the compositional nature of multi-hop queries and the varying latent semantics of relations. However, existing methods based on multi-modal distribution roughly represent each subset without capturing its accurate cardinality, or even degenerate into uni-modal distribution learning during the reasoning process due to the lack of an effective similarity measure. To better model queries with diversified answers, we propose Query2GMM for answering logical queries over knowledge graphs. In Query2GMM, we present the GMM embedding to represent each query using a univariate Gaussian Mixture Model (GMM). Each subset of a query is encoded by its cardinality, semantic center and dispersion degree, allowing for precise representation of multiple subsets. Then we design specific neural networks for each operator to handle the inherent complexity that comes with multi-modal distribution while alleviating the cascading errors. Last, we define a new similarity measure to assess the relationships between an entity and a query's multi-answer subsets, enabling effective multi-modal distribution learning for reasoning. Comprehensive experimental results show that Query2GMM outperforms the best competitor by an absolute average of $5.5\%$. The source code is available at \url{//anonymous.4open.science/r/Query2GMM-C42F}.
Learning a nonparametric system of ordinary differential equations (ODEs) from $n$ trajectory snapshots in a $d$-dimensional state space requires learning $d$ functions of $d$ variables. Explicit formulations scale quadratically in $d$ unless additional knowledge about system properties, such as sparsity and symmetries, is available. In this work, we propose a linear approach to learning using the implicit formulation provided by vector-valued Reproducing Kernel Hilbert Spaces. By rewriting the ODEs in a weaker integral form, which we subsequently minimize, we derive our learning algorithm. The minimization problem's solution for the vector field relies on multivariate occupation kernel functions associated with the solution trajectories. We validate our approach through experiments on highly nonlinear simulated and real data, where $d$ may exceed 100. We further demonstrate the versatility of the proposed method by learning a nonparametric first order quasilinear partial differential equation.
Reasoning with knowledge expressed in natural language and Knowledge Bases (KBs) is a major challenge for Artificial Intelligence, with applications in machine reading, dialogue, and question answering. General neural architectures that jointly learn representations and transformations of text are very data-inefficient, and it is hard to analyse their reasoning process. These issues are addressed by end-to-end differentiable reasoning systems such as Neural Theorem Provers (NTPs), although they can only be used with small-scale symbolic KBs. In this paper we first propose Greedy NTPs (GNTPs), an extension to NTPs addressing their complexity and scalability limitations, thus making them applicable to real-world datasets. This result is achieved by dynamically constructing the computation graph of NTPs and including only the most promising proof paths during inference, thus obtaining orders of magnitude more efficient models. Then, we propose a novel approach for jointly reasoning over KBs and textual mentions, by embedding logic facts and natural language sentences in a shared embedding space. We show that GNTPs perform on par with NTPs at a fraction of their cost while achieving competitive link prediction results on large datasets, providing explanations for predictions, and inducing interpretable models. Source code, datasets, and supplementary material are available online at //github.com/uclnlp/gntp.
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