26 research outputs found
Riemannian Holonomy Groups of Statistical Manifolds
Normal distribution manifolds play essential roles in the theory of
information geometry, so do holonomy groups in classification of Riemannian
manifolds. After some necessary preliminaries on information geometry and
holonomy groups, it is presented that the corresponding Riemannian holonomy
group of the -dimensional normal distribution is
, for all . As a
generalization on exponential family, a list of holonomy groups follows.Comment: 11 page
Fixed-Domain Inference for Gausian Processes with Mat\'ern Covariogram on Compact Riemannian Manifolds
Gaussian processes are widely employed as versatile modeling and predictive
tools in spatial statistics, functional data analysis, computer modeling and in
diverse applications of machine learning. Such processes have been widely
studied over Euclidean spaces, where they are constructed using specified
covariance functions or covariograms. These functions specify valid stochastic
processes that can be used to model complex dependencies in spatial statistics
and other machine learning contexts. Valid (positive definite) covariance
functions have been extensively studied for Gaussian processes on Euclidean
spaces. Such investigations have focused, among other aspects, on the
identifiability and consistency of covariance parameters as well as the problem
of spatial interpolation and prediction within the fixed-domain or infill
paradigm of asymptotic inference. This manuscript undertakes analogous
theoretical developments for Gaussian processes constructed over Riemannian
manifolds. We begin by establishing formal notions and conditions for the
equivalence of two Gaussian random measures on compact manifolds. We build upon
recently introduced Mat\'ern covariograms on compact Riemannian manifold,
derive the microergodic parameter and formally establish the consistency of
maximum likelihood estimators and the asymptotic optimality of the best linear
unbiased predictor (BLUP). The circle and sphere are studied as two specific
examples of compact Riemannian manifolds with numerical experiments that
illustrate the theory
On the Identifiability and Interpretability of Gaussian Process Models
In this paper, we critically examine the prevalent practice of using additive
mixtures of Mat\'ern kernels in single-output Gaussian process (GP) models and
explore the properties of multiplicative mixtures of Mat\'ern kernels for
multi-output GP models. For the single-output case, we derive a series of
theoretical results showing that the smoothness of a mixture of Mat\'ern
kernels is determined by the least smooth component and that a GP with such a
kernel is effectively equivalent to the least smooth kernel component.
Furthermore, we demonstrate that none of the mixing weights or parameters
within individual kernel components are identifiable. We then turn our
attention to multi-output GP models and analyze the identifiability of the
covariance matrix in the multiplicative kernel , where
is a standard single output kernel such as Mat\'ern. We show that is
identifiable up to a multiplicative constant, suggesting that multiplicative
mixtures are well suited for multi-output tasks. Our findings are supported by
extensive simulations and real applications for both single- and multi-output
settings. This work provides insight into kernel selection and interpretation
for GP models, emphasizing the importance of choosing appropriate kernel
structures for different tasks.Comment: 37th Conference on Neural Information Processing Systems (NeurIPS
2023
Spherical Rotation Dimension Reduction with Geometric Loss Functions
Modern datasets often exhibit high dimensionality, yet the data reside in
low-dimensional manifolds that can reveal underlying geometric structures
critical for data analysis. A prime example of such a dataset is a collection
of cell cycle measurements, where the inherently cyclical nature of the process
can be represented as a circle or sphere. Motivated by the need to analyze
these types of datasets, we propose a nonlinear dimension reduction method,
Spherical Rotation Component Analysis (SRCA), that incorporates geometric
information to better approximate low-dimensional manifolds. SRCA is a
versatile method designed to work in both high-dimensional and small sample
size settings. By employing spheres or ellipsoids, SRCA provides a low-rank
spherical representation of the data with general theoretic guarantees,
effectively retaining the geometric structure of the dataset during
dimensionality reduction. A comprehensive simulation study, along with a
successful application to human cell cycle data, further highlights the
advantages of SRCA compared to state-of-the-art alternatives, demonstrating its
superior performance in approximating the manifold while preserving inherent
geometric structures.Comment: 60 page
Density estimation and modeling on symmetric spaces
In many applications, data and/or parameters are supported on non-Euclidean
manifolds. It is important to take into account the geometric structure of
manifolds in statistical analysis to avoid misleading results. Although there
has been a considerable focus on simple and specific manifolds, there is a lack
of general and easy-to-implement statistical methods for density estimation and
modeling on manifolds. In this article, we consider a very broad class of
manifolds: non-compact Riemannian symmetric spaces. For this class, we provide
a very general mathematical result for easily calculating volume changes of the
exponential and logarithm map between the tangent space and the manifold. This
allows one to define statistical models on the tangent space, push these models
forward onto the manifold, and easily calculate induced distributions by
Jacobians. To illustrate the statistical utility of this theoretical result, we
provide a general method to construct distributions on symmetric spaces. In
particular, we define the log-Gaussian distribution as an analogue of the
multivariate Gaussian distribution in Euclidean space. With these new kernels
on symmetric spaces, we also consider the problem of density estimation. Our
proposed approach can use any existing density estimation approach designed for
Euclidean spaces and push it forward to the manifold with an easy-to-calculate
adjustment. We provide theorems showing that the induced density estimators on
the manifold inherit the statistical optimality properties of the parent
Euclidean density estimator; this holds for both frequentist and Bayesian
nonparametric methods. We illustrate the theory and practical utility of the
proposed approach on the space of positive definite matrices