使用边际乘积基系统进行高效多维函数数据分析
Efficient Multidimensional Functional Data Analysis Using Marginal Product Basis Systems
AI总结：
提出一种基于边际乘积基系统的多维函数数据分析框架，通过张量分解和微分算子惩罚构建最优适应数据的连续表示，有效克服维度灾难，并在模拟和神经影像数据中验证了其优越性。
摘要：
许多现代数据集，来自神经影像学和地质统计学等领域，以张量值数据的随机样本形式出现，可以被理解为平滑多维随机函数的含噪观测。函数数据分析中的大多数传统技术受到维度灾难的困扰，并且随着域维度的增加迅速变得难以处理。在本文中，我们提出了一个框架，用于从多维函数数据样本中学习连续表示，该表示对维度灾难的几种表现形式免疫。这些表示是使用一组可分离基函数构建的，这些基函数被定义为最优适应数据。我们表明，由此产生的估计问题可以通过对观测数据的精心定义的归约变换进行张量分解来高效解决。使用一类基于微分算子的惩罚来结合基于粗糙度的正则化。还建立了相关的理论性质。在模拟研究中展示了我们的方法相对于竞争方法的优势。我们以神经影像学中的实际数据应用作为结论。
英文摘要：
Many modern datasets, from areas such as neuroimaging and geostatistics, come in the form of a random sample of tensor-valued data which can be understood as noisy observations of a smooth multidimensional random function. Most of the traditional techniques from functional data analysis are plagued by the curse of dimensionality and quickly become intractable as the dimension of the domain increases. In this paper, we propose a framework for learning continuous representations from a sample of multidimensional functional data that is immune to several manifestations of the curse. These representations are constructed using a set of separable basis functions that are defined to be optimally adapted to the data. We show that the resulting estimation problem can be solved efficiently by the tensor decomposition of a carefully defined reduction transformation of the observed data. Roughness-based regularization is incorporated using a class of differential operator-based penalties. Relevant theoretical properties are also established. The advantages of our method over competing methods are demonstrated in a simulation study. We conclude with a real data application in neuroimaging.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
