相关论文: Polynomial Regression on Riemannian Manifolds

We apply the theory of Peres and Schlag to obtain estimates for generic Hausdorff dimension distortion under orthogonal projections on simply connected two dimensional Riemannian manifolds of constant curvature. As a conclusion we obtain…

This note is an overview of the Poisson sigma model (PSM) and its applications in deformation quantization. Reduction of coisotropic and pre-Poisson submanifolds, their appearance as branes of the PSM, quantization in terms of L-infinity…

Nonlinear dimensionality reduction methods provide a valuable means to visualize and interpret high-dimensional data. However, many popular methods can fail dramatically, even on simple two-dimensional manifolds, due to problems such as…

Real world data often exhibit low-dimensional geometric structures, and can be viewed as samples near a low-dimensional manifold. This paper studies nonparametric regression of H\"{o}lder functions on low-dimensional manifolds using deep…

A method for the analytical evaluation of layer potentials arising in the collocation boundary element method for the Laplace and Helmholtz equation is developed for piecewise flat boundary elements with polynomial shape functions. The…

Given data, deep generative models, such as variational autoencoders (VAE) and generative adversarial networks (GAN), train a lower dimensional latent representation of the data space. The linear Euclidean geometry of data space pulls back…

The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Rieamannian Lie algebras. The two notions are strongly related:…

We generalize Llarull's scalar curvature comparison to Riemannian manifolds admitting metric connections with parallel and alternating torsion and having a nonnegative curvature operator on 2-vectors. As a byproduct, we show that Euler…

This paper introduces the problem of coresets for regression problems to panel data settings. We first define coresets for several variants of regression problems with panel data and then present efficient algorithms to construct coresets…

Despite the popularity of the manifold hypothesis, current manifold-learning methods do not support machine learning directly on the latent $d$-dimensional data manifold, as they primarily aim to perform dimensionality reduction into…

Given an ensemble of randomized regression trees, it is possible to restructure them as a collection of multilayered neural networks with particular connection weights. Following this principle, we reformulate the random forest method of…

Principal Component Analysis can be performed over small domains of an embedded Riemannian manifold in order to relate the covariance analysis of the underlying point set with the local extrinsic and intrinsic curvature. We show that the…

We develop a Bayesian bivariate spatial model for multivariate regression analysis applicable to studies examining the influence of genetic variation on brain structure. Our model is motivated by an imaging genetics study of the Alzheimer's…

The purpose of this paper is to employ the language of Cartan moving frames to study the geometry of the data manifolds and its Riemannian structure, via the data information metric and its curvature at data points. Using this framework and…

We obtain sampling and interpolation theorems in radial weighted spaces of analytic functions for weights of arbitrary (more rapid than polynomial) growth. We give an application to invariant subspaces of arbitrary index in large weighted…

Correlation functions for matrix ensembles with orthogonal and unitarysymplectic rotation symmetry are more complicated to calculate than in the unitary case. The supersymmetry method and the orthogonal polynomials are two techniques to…

Multilinear embedding estimates for the fractional Laplacian are obtained in terms of functionals defined over a hyperbolic surface. Convolution estimates used in the proof enlarge the classical framework of the convolution algebra for…

Deep Learning architectures, albeit successful in most computer vision tasks, were designed for data with an underlying Euclidean structure, which is not usually fulfilled since pre-processed data may lie on a non-linear space. In this…

In this paper an additive regression model for a symmetric positive-definite matrix valued response and multiple scalar predictors is proposed. The model exploits the abelian group structure inherited from either the Log-Cholesky metric or…

Neural networks are playing a crucial role in everyday life, with the most modern generative models able to achieve impressive results. Nonetheless, their functioning is still not very clear, and several strategies have been adopted to…