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

现代科学技术中收集到的大规模、复杂的数据对传统的统计方法和理论提出了巨大的挑战。函数数据分析(FDA)以随机函数为单位，对时间、空间、波长等连续分布的数据进行建模，是一个很有前途的研究领域。这样的数据可以被视为随机过程的实现，并且通常可以在许多领域中找到，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。函数数据建模中的一个基本问题源于“维度诅咒”，因为函数对象在概念上被定义为无限维过程。此外，大规模收集的复杂结构的功能性数据也面临着更大的挑战，这自然涉及到最近发生革命性变化的领域，如高维数据分析和压缩传感。因此，这项研究计划的长期目标是*(L1)开发一系列灵活而实用的方法来适应大规模和复杂的功能数据，如大规模抽样方案、超高数量的功能和多种结构；*(L2)建立基本框架、适合这些功能模型的正则化和推理方法，将高维技术的优势与功能方法相结合。*(1)高斯序列与函数数据的等价性、恢复和推断*(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的等价理论在多个高斯序列和函数数据之间建立严格的联系。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相关联。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，这些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。第二个主题涉及到涉及功能对象的回归。我将对函数回归的范围进行扩展，以涵盖具有超高函数预测值的大规模场景，并开发有效的正则化估计和严格的函数推理。进一步的研究将被用来自适应地模拟随机流形嵌入的函数回归。这一主题预计将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.**