arXiv subjects

Xiaoda Qu

Publications and source records attributed to Xiaoda Qu.

4 recordsLinked to original sources

arXiv2025

Theory and Applications of Kernel Stein's Method on Riemannian Manifolds

Distributional comparison is a fundamental problem in statistical data analysis with numerous applications in a variety of scientific and engineering fields. Numerous methods exist for distributional comparison but kernel Stein's method has gained significant popularity in recent times. In this paper, we first present a novel mathematically rigorous and consistent generalization of the Stein operator to Riemannian manifolds. Then we show that the kernel Stein discrepancy (KSD) defined via this operator is nearly as strong as the KSD in the Euclidean setting in terms of distinguishing the target distributions from the reference. We investigate the asymptotic properties of the minimum kernel Stein discrepancy estimator (MKSDE), apply it to goodness-of-fit testing, and compare it to the maximum likelihood estimator (MLE) experimentally. We present several examples of our theory applied to commonly encountered Riemannian manifolds in practice namely, the n-sphere, the Grassmann, Stiefel, the manifold of symmetric positive definite matrices and other Riemannian homogeneous spaces. On the aforementioned manifolds, we consider a variety of distributions with intractable normalization constants and derive closed form expressions for the KSD and MKSDE.

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arXiv2025

A Steerable Deep Network for Model-Free Diffusion MRI Registration

Nonrigid registration is vital to medical image analysis but remains challenging for diffusion MRI (dMRI) due to its high-dimensional, orientation-dependent nature. While classical methods are accurate, they are computationally demanding, and deep neural networks, though efficient, have been underexplored for nonrigid dMRI registration compared to structural imaging. We present a novel, deep learning framework for model-free, nonrigid registration of raw diffusion MRI data that does not require explicit reorientation. Unlike previous methods relying on derived representations such as diffusion tensors or fiber orientation distribution functions, in our approach, we formulate the registration as an equivariant diffeomorphism of position-and-orientation space. Central to our method is an $\mathsf{SE}(3)$-equivariant UNet that generates velocity fields while preserving the geometric properties of a raw dMRI's domain. We introduce a new loss function based on the maximum mean discrepancy in Fourier space, implicitly matching ensemble average propagators across images. Experimental results on Human Connectome Project dMRI data demonstrate competitive performance compared to state-of-the-art approaches, with the added advantage of bypassing the overhead for estimating derived representations. This work establishes a foundation for data-driven, geometry-aware dMRI registration directly in the acquisition space.

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arXiv2024

Kernel Stein Discrepancy on Lie Groups: Theory and Applications

Distributional approximation is a fundamental problem in machine learning with numerous applications across all fields of science and engineering and beyond. The key challenge in most approximation methods is the need to tackle the intractable normalization constant pertaining to the parametrized distributions used to model the data. In this paper, we present a novel Stein operator on Lie groups leading to a kernel Stein discrepancy (KSD) which is a normalization-free loss function. We present several theoretical results characterizing the properties of this new KSD on Lie groups and its minimizers namely, the minimum KSD estimator (MKSDE). Proof of several properties of MKSDE are presented, including strong consistency, CLT and a closed form of the MKSDE for the von Mises-Fisher distribution on SO(N). Finally, we present experimental evidence depicting advantages of minimizing KSD over maximum likelihood estimation.

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arXiv2022

A Framework for Improving the Characterization Scope of Stein's Method on Riemannian Manifolds

Stein's method has been widely used to achieve distributional approximations for probability distributions defined in Euclidean spaces. Recently, techniques to extend Stein's method to manifold-valued random variables with distributions defined on the respective manifolds have been reported. However, several of these methods impose strong regularity conditions on the distributions as well as the manifolds and/or consider very special cases. In this paper, we present a novel framework for Stein's method on Riemannian manifolds using the Friedrichs extension technique applied to self-adjoint unbounded operators. This framework is applicable to a variety of conventional and unconventional situations, including but not limited to, intrinsically defined non-smooth distributions, truncated distributions on Riemannian manifolds, distributions on incomplete Riemannian manifolds, etc. Moreover, the stronger the regularity conditions imposed on the manifolds or target distributions, the stronger will be the characterization ability of our novel Stein pair, which facilitates the application of Stein's method to problem domains hitherto uncharted. We present several (non-numeric) examples illustrating the applicability of the presented theory.

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