相关论文: Polynomial Regression on Riemannian Manifolds

As a general and robust alternative to traditional mean regression models, quantile regression avoids the assumption of normally distributed errors, making it a versatile choice when modeling outcomes such as cognitive scores that typically…

We propose a natural intrinsic extension of ridge regression from Euclidean spaces to general Riemannian manifolds for time-series prediction. Our approach combines Riemannian least-squares fitting via B\'ezier curves, empirical covariance…

Shape analysis and compuational anatomy both make use of sophisticated tools from infinite-dimensional differential manifolds and Riemannian geometry on spaces of functions. While comprehensive references for the mathematical foundations…

In recent years, manifold learning has become increasingly popular as a tool for performing non-linear dimensionality reduction. This has led to the development of numerous algorithms of varying degrees of complexity that aim to recover man…

We propose regression models for curve-valued responses in two or more dimensions, where only the image but not the parametrization of the curves is of interest. Examples of such data are handwritten letters, movement paths or outlines of…

Many phenomena are naturally characterized by measuring continuous transformations such as shape changes in medicine or articulated systems in robotics. Modeling the variability in such datasets requires performing statistics on Lie groups,…

In image set classification, a considerable advance has been made by modeling the original image sets by second order statistics or linear subspace, which typically lie on the Riemannian manifold. Specifically, they are Symmetric Positive…

We tackle the problem of optimizing over all possible positive definite radial kernels on Riemannian manifolds for classification. Kernel methods on Riemannian manifolds have recently become increasingly popular in computer vision. However,…

Recent methods in geometric deep learning have introduced various neural networks to operate over data that lie on Riemannian manifolds. Such networks are often necessary to learn well over graphs with a hierarchical structure or to learn…

We reveal the phenomenon that ``naive'' multivariate local polynomial regression can adapt to local smooth lower dimensional structure in the sense that it achieves the optimal convergence rate for nonparametric estimation of regression…

It is often of interest to infer lower-dimensional structure underlying complex data. As a flexible class of non-linear structures, it is common to focus on Riemannian manifolds. Most existing manifold learning algorithms replace the…

Shape restricted regressions, including isotonic regression and concave regression as special cases, are studied using priors on Bernstein polynomials and Markov chain Monte Carlo methods. These priors have large supports, select only…

In many applications, geodesic hierarchical models are adequate for the study of temporal observations. We employ such a model derived for manifold-valued data to Kendall's shape space. In particular, instead of the Sasaki metric, we adapt…

Learning in Riemannian orbifolds is motivated by existing machine learning algorithms that directly operate on finite combinatorial structures such as point patterns, trees, and graphs. These methods, however, lack statistical…

In this survey we present classical results on methods to use group actions to collapse manifolds to the orbit spaces while keeping some control on the curvature, and recent extensions of these constructions to the setting of singular…

Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. The strategies employed to investigate their theoretical properties mainly rely…

In this work, we investigate Riemannian geometry based dimensionality reduction methods that respect the underlying manifold structure of the data. In particular, we focus on Principal Geodesic Analysis (PGA) as a nonlinear generalization…

We introduce a wide and deep neural network for prediction of progression from patients with mild cognitive impairment to Alzheimer's disease. Information from anatomical shape and tabular clinical data (demographics, biomarkers) are fused…

In this paper we introduce two new notions of sectional curvature for Riemannian manifolds with density. Under both notions of curvature we classify the constant curvature manifolds. We also prove generalizations of the theorems of…

Kernel methods have had great success in Statistics and Machine Learning. Despite their growing popularity, however, less effort has been drawn towards developing kernel based classification methods on Riemannian manifolds due to difficulty…