A bayesian approach to the estimation of maps between riemannian manifolds, II: examples

A Bayesian approach to the estimation of maps between riemannian manifolds

Let Θbe a smooth compact oriented manifold without boundary, embedded in a euclidean space and let γbe a smooth map Θinto a riemannian manifold Λ. An unknown state θ\in Θis observed via X=θ+εξwhere ε>0 is a small parameter and ξis a white Gaussian noise. For a given smooth prior on Θand smooth estimator g of the map γwe derive a second-order asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of the underlying spaces Θand Λ, in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of γis found based on the modern…

Bayesian Quadrature on Riemannian Data Manifolds

Riemannian manifolds provide a principled way to model nonlinear geometric structure inherent in data. A Riemannian metric on said manifolds determines geometry-aware shortest paths and provides the means to define statistical models accordingly. However, these operations are typically computationally demanding. To ease this computational burden, we advocate probabilistic numerical methods for Riemannian statistics. In particular, we focus on Bayesian quadrature (BQ) to numerically compute integrals over normal laws on Riemannian manifolds learned from data. In this task, each function evaluat…

在流形上的鲁棒贝叶斯推断

Robust Bayesian Inference on Riemannian Submanifold

本文提出了一种在黎曼子流形上进行鲁棒贝叶斯推断的方法，能够有效处理流形值参数的不确定性量化问题。

用于贝叶斯神经网络的黎曼拉普拉斯近似

Riemannian Laplace approximations for Bayesian neural networks

针对贝叶斯神经网络高斯近似后验非高斯性导致性能下降的问题，提出基于对数后验梯度黎曼度量的黎曼拉普拉斯近似，通过求解常微分方程高效采样，性能优于传统方法且对先验选择不敏感。

内蕴贝叶斯 Cramér-Rao 界及其在协方差矩阵估计中的应用

Intrinsic Bayesian Cramér-Rao Bound with an Application to Covariance Matrix Estimation

本文提出适用于 Riemannian 流形参数与先验分布的内蕴贝叶斯 Cramér-Rao 界，并用于 Gaussian 数据、逆 Wishart 先验下的协方差矩阵估计。

A Bayesian Variational principle for dynamic Self Organizing Maps

We propose organisation conditions that yield a method for training SOM with adaptative neighborhood radius in a variational Bayesian framework. This method is validated on a non-stationary setting and compared in an high-dimensional setting with an other adaptative method.