Contributions to functional data analysis : change point analysis, differential privacy, and fairness

Driven by technological advancements, functional data analysis (FDA) has gained increasing attention over the past two decades. Accompanying this progress is the growing complexity and diversity of modern data, which has driven a soaring demand for novel FDA techniques capable of handling intricate data structures under contemporary constrained environments. In response, this thesis addresses classical FDA problems (e.g. mean and covariance estimation, varying coefficient model estimation and functional classification) within the context of three pivotal and emerging challenges in machine learning: change point analysis, differential privacy and algorithmic fairness. The main part of the thesis consists of three chapters, with each chapter addressing one of these challenges. To begin with, in Chapter 2, we study a change point analysis problem when the data observed are a sequence of discretely observed fragmented functional data. We introduce a two-step, computationally efficient algorithm that achieves consistent change point localisation rates. Additionally, we derive limiting distributions for each change point estimator under two different regimes and provide practical methods for confidence interval constructions. Chapter 3 investigates the preservation of differential privacy in FDA, beginning with functional mean estimation and extending to varying coefficient model estimation. We introduce a distributed learning framework involving multiple servers, each responsible for collecting several sparsely observed functions. To address this complex hierarchy, we employ minimax theory to reveal several fundamental phenomena: from sparse to dense FDA, from user-level to central and federated differential privacy costs, and the intricate interplay between different regimes of FDA and privacy preservation. In Chapter 4, we study fairness-aware Bayes optimal classification for functional data, ensuring the disparity level of the classifier is controlled below a pre-specified threshold. We propose a unified framework targeting infinite-dimensional functional features and design a post-processing algorithm tailored to homoscedastic Gaussian processes. Under weak structural assumptions on eigenspace, theoretical guarantees on fairness and excess risk controls are established. Our proposed algorithm is further validated through extensive experiments on both synthetic and real datasets. The thesis is finally concluded with Chapter 5, which outlines several future research directions.