Large-scale and Complex Functional Data: Foundation, Regression and Inference

现代科学技术所收集的大规模、复杂的数据对传统的统计方法和理论提出了巨大的挑战。功能数据分析（Functional data analysis， FDA）是以随机函数为单位，对分布在时间、空间、波长等连续空间上的数据进行建模，是一个新兴的研究领域。这些数据可以被看作是随机过程的实现，在许多领域都很常见，例如纵向研究、微阵列实验、医学图像、互联网商业和金融市场。建模功能数据的一个基本问题源于“维度的诅咒”，因为功能对象在概念上被框架为无限维的过程。此外，对于大规模和复杂结构收集的功能数据，也出现了更大的挑战，这自然与最近发生革命性变化的领域有关，例如高维数据分析和压缩感知。因此，本研究计划的长期目标是开发一种灵活而实用的方法，以适应大规模和复杂的功能数据，如大规模采样方案，超高数量的函数和流形结构；*** (L2)结合高维技术和函数方法的优势，为这些功能模型建立基础框架、合适的正则化和推理方法。***具体来说，我们的研究主要集中在以下两个主题上：***(1)高斯序列与功能数据：等价、恢复和推理***(2)函数回归：大规模和流形上的估计和推理***在第一个主题中，我们的目标是通过Le Cam的等价理论建立多个高斯序列与功能数据之间的严格联系。这种变革性的方法提供了一个简化但基本的框架，类似于在引入小波收缩时将单个高斯序列与非参数回归联系起来。这将导致一系列新的发展，例如功能对象的最优恢复和自适应推理，这将享有严格的理论保证和大量的计算收益，特别是对于大规模方案。第二个主题涉及涉及功能对象的回归。我将扩展函数回归的范围，以包含具有超高数量的函数预测器的大规模场景，并开发有效的正则化估计和严格的函数推理。一个进一步的研究将进行自适应建模与随机流形嵌入函数回归。该主题有望为各种大规模和复杂的功能回归模型奠定基础，并在FDA和复杂/高维领域产生广泛的影响

Large-scale and complex data collected in modern science and technology impose tremendous challenges for traditional statistical methods and theory. Functional data analysis (FDA) has emerged as a promising field that employs random functions as units and is designed to model data distributed over continua such as time, space, wavelength and so on. Such data may be viewed as realizations of random processes and are commonly found in many fields, e.g., longitudinal studies, microarray experiments, medical images, internet commerce and financial markets. A fundamental issue in modelling functional data arises from the “curse of dimensionality”, as functional objects are conceptually framed as infinite-dimensional processes. Further, greater challenges have emerged for functional data that are collected on a large scale and with complex structures, which naturally relates to the recently revolutionized fields, such as high-dimensional data analysis and compressed sensing. Therefore, the long-term objective of this research program is to *** (L1) develop a wide range of flexible yet practical methods to accommodate the large-scale and complex functional data, such as massive sampling schemes, ultrahigh number of functions and manifold structures; *** (L2) establish foundational frameworks, suitable regularization and inferential methods tailored for such functional models, combining the strength of high-dimensional techniques with functional approaches. *** Specifically, the proposed research focuses on the following two themes:*** (1) Gaussian Sequences to Functional Data: Equivalence, Recovery and Inference*** (2) Functional Regression: Estimation and Inference on Large Scale and Manifolds*** In the first theme, we aim to establish a rigorous connection between multiple Gaussian sequences and functional data through Le Cam's equivalence theory. This transformative approach provides a simplified but foundational framework, similar to relating a single Gaussian sequence to nonparametric regression when wavelet shrinkage was introduced. This will lead to a series of novel developments, e.g., optimal recovery and adaptive inference on functional objects, which will enjoy rigorous theoretical guarantees and substantial computation gains, particularly for massive schemes. The second theme concerns the regression involving functional objects. I will expand on the scope of functional regression to embrace large-scale scenarios with ultrahigh number of functional predictors, and develop effective regularized estimation and rigorous functional inference. A further investigation will be conducted for adaptively modelling functional regression with random manifold embeddings. This theme is expected to set ground for a variety of large-scale and complex functional regression models, and generate a broad impact in both FDA and complex/high-dimensional fields.**