基于能量的函数型数据路径度量倾斜模型
Energy-Based Models for Functional Data using Path Measure Tilting
摘要：
基于能量的模型（EBM）已被证明是一种非常有效的方法来建模密度有限维空间。EBM能够通过组合将特定领域的选择和约束纳入模型的结构中，这使得EBM成为物理学、生物学和计算机视觉以及其他各种领域中应用的有吸引力的候选者。最近，针对\textit{unconditional}可交换数据（例如，点云）。在这项工作中，我们提出了一个新的EBP子类，称为$\mathcal{F}$ -EBM的\textit{conditional}可交换的数据，它能够学习分布的功能（如曲线或曲面）从功能样本评估在许多点。在功能方面出现了两个独特的挑战。首先，训练数据通常不沿着固定的点集进行评估。其次，必须采取措施来控制模型在评估点之间的行为，以减轻过拟合。所提出的模型是一个基于能量的函数空间上的模型，被分解的频谱，其中高斯过程路径测量被用来重新加权的分布，以捕捉平滑特性的底层过程被建模。所得到的模型具有利用不规则采样的训练数据的能力，并且可以以任何分辨率输出预测，提供了一种有效的方法来放大功能数据。我们证明了我们所提出的方法的有效性建模的一系列数据集，包括从标准普尔500（S\&P）和英国国家电网收集的数据。
英文摘要：
Energy-Based Models (EBMs) have proven to be a highly effective approach for modelling densities on finite-dimensional spaces. Their ability to incorporate domain-specific choices and constraints into the structure of the model through composition make EBMs an appealing candidate for applications in physics, biology and computer vision and various other fields. Recently, Energy-Based Processes (EBP) for modelling stochastic processes was proposed for \textit{unconditional} exchangeable data (e.g., point clouds). In this work, we present a novel subclass of EBPs, called $\mathcal{F}$-EBM for \textit{conditional} exchangeable data, which is able to learn distributions of functions (such as curves or surfaces) from functional samples evaluated at finitely many points. Two unique challenges arise in the functional context. Firstly, training data is often not evaluated along a fixed set of points. Secondly, steps must be taken to control the behaviour of the model between evaluation points, to mitigate overfitting. The proposed model is an energy based model on function space that is decomposed spectrally, where a Gaussian Process path measure is used to reweight the distribution to capture smoothness properties of the underlying process being modelled. The resulting model has the ability to utilize irregularly sampled training data and can output predictions at any resolution, providing an effective approach to up-scaling functional data. We demonstrate the efficacy of our proposed approach for modelling a range of datasets, including data collected from Standard and Poor's 500 (S\&P) and UK National grid.
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