🤖 AI Summary
This work addresses the need for more efficient, robust, and flexible metrics for measuring distances between probability distributions in statistical inference and numerical integration. Centered on kernel methods, we propose an efficient estimator for Maximum Mean Discrepancy (MMD), develop novel MMD-based approaches for conditional expectation estimation and integral calibration, and introduce a new family of distance measures—kernel quantile discrepancies—that effectively overcome MMD’s limitations in tail sensitivity and discriminative power. Both theoretical analysis and empirical experiments demonstrate that the proposed methods offer strong scalability, computational efficiency, and superior performance, thereby providing more powerful and practical kernel-based tools for nonparametric statistics and integration tasks.
📝 Abstract
Representing, comparing, and measuring the distance between probability distributions is a key task in computational statistics and machine learning. The choice of representation and the associated distance determine properties of the methods in which they are used: for example, certain distances can allow one to encode robustness or smoothness of the problem. Kernel methods offer flexible and rich Hilbert space representations of distributions that allow the modeller to enforce properties through the choice of kernel, and estimate associated distances at efficient nonparametric rates. In particular, the maximum mean discrepancy (MMD), a kernel-based distance constructed by comparing Hilbert space mean functions, has received significant attention due to its computational tractability and is favoured by practitioners.
In this thesis, we conduct a thorough study of kernel-based distances with a focus on efficient computation, with core contributions in Chapters 3 to 6. Part I of the thesis is focused on the MMD, specifically on improved MMD estimation. In Chapter 3 we propose a theoretically sound, improved estimator for MMD in simulation-based inference. Then, in Chapter 4, we propose an MMD-based estimator for conditional expectations, a ubiquitous task in statistical computation. Closing Part I, in Chapter 5 we study the problem of calibration when MMD is applied to the task of integration.
In Part II, motivated by the recent developments in kernel embeddings beyond the mean, we introduce a family of novel kernel-based discrepancies: kernel quantile discrepancies. These address some of the pitfalls of MMD, and are shown through both theoretical results and an empirical study to offer a competitive alternative to MMD and its fast approximations. We conclude with a discussion on broader lessons and future work emerging from the thesis.