Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics

We study a probabilistic numerical method for the solution of both boundary and initial value problems that returns a joint Gaussian process posterior over the solution. Such methods have concrete value in the statistics on Riemannian manifolds, where non-analytic ordinary differential equations are involved in virtually all computations. The probabilistic formulation permits marginalising the uncertainty of the numerical solution such that statistics are less sensitive to inaccuracies. This leads to new Riemannian algorithms for mean value computations and principal geodesic analysis. Marginalisation also means results can be less precise than point estimates, enabling a noticeable speed-up over the state of the art. Our approach is an argument for a wider point that uncertainty caused by numerical calculations should be tracked throughout the pipeline of machine learning algorithms.Comment: 11 page (9 page conference paper, plus supplements

An Infinitesimal Probabilistic Model for Principal Component Analysis of Manifold Valued Data

We provide a probabilistic and infinitesimal view of how the principal component analysis procedure (PCA) can be generalized to analysis of nonlinear manifold valued data. Starting with the probabilistic PCA interpretation of the Euclidean PCA procedure, we show how PCA can be generalized to manifolds in an intrinsic way that does not resort to linearization of the data space. The underlying probability model is constructed by mapping a Euclidean stochastic process to the manifold using stochastic development of Euclidean semimartingales. The construction uses a connection and bundles of covariant tensors to allow global transport of principal eigenvectors, and the model is thereby an example of how principal fiber bundles can be used to handle the lack of global coordinate system and orientations that characterizes manifold valued statistics. We show how curvature implies non-integrability of the equivalent of Euclidean principal subspaces, and how the stochastic flows provide an alternative to explicit construction of such subspaces. We describe estimation procedures for inference of parameters and prediction of principal components, and we give examples of properties of the model on embedded surfaces

Multivariate texture discrimination using a principal geodesic classifier

A new texture discrimination method is presented for classification and retrieval of colored textures represented in the wavelet domain. The interband correlation structure is modeled by multivariate probability models which constitute a Riemannian manifold. The presented method considers the shape of the class on the manifold by determining the principal geodesic of each class. The method, which we call principal geodesic classification, then determines the shortest distance from a test texture to the principal geodesic of each class. We use the Rao geodesic distance (GD) for calculating distances on the manifold. We compare the performance of the proposed method with distance-to-centroid and knearest neighbor classifiers and of the GD with the Euclidean distance. The principal geodesic classifier coupled with the GD yields better results, indicating the usefulness of effectively and concisely quantifying the variability of the classes in the probabilistic feature space

Bayesian Inference on Matrix Manifolds for Linear Dimensionality Reduction

We reframe linear dimensionality reduction as a problem of Bayesian inference on matrix manifolds. This natural paradigm extends the Bayesian framework to dimensionality reduction tasks in higher dimensions with simpler models at greater speeds. Here an orthogonal basis is treated as a single point on a manifold and is associated with a linear subspace on which observations vary maximally. Throughout this paper, we employ the Grassmann and Stiefel manifolds for various dimensionality reduction problems, explore the connection between the two manifolds, and use Hybrid Monte Carlo for posterior sampling on the Grassmannian for the first time. We delineate in which situations either manifold should be considered. Further, matrix manifold models are used to yield scientific insight in the context of cognitive neuroscience, and we conclude that our methods are suitable for basic inference as well as accurate prediction.Comment: All datasets and computer programs are publicly available at http://www.ics.uci.edu/~babaks/Site/Codes.htm

Multivariate texture discrimination based on geodesics to class centroids on a generalized Gaussian Manifold

A texture discrimination scheme is proposed wherein probability distributions are deployed on a probabilistic manifold for modeling the wavelet statistics of images. We consider the Rao geodesic distance (GD) to the class centroid for texture discrimination in various classification experiments. We compare the performance of GD to class centroid with the Euclidean distance in a similar context, both in terms of accuracy and computational complexity. Also, we compare our proposed classification scheme with the k-nearest neighbor algorithm. Univariate and multivariate Gaussian and Laplace distributions, as well as generalized Gaussian distributions with variable shape parameter are each evaluated as a statistical model for the wavelet coefficients. The GD to the centroid outperforms the Euclidean distance and yields superior discrimination compared to the k-nearest neighbor approach

Statistical Learning in Wasserstein Space

We seek a generalization of regression and principle component analysis (PCA) in a metric space where data points are distributions metrized by the Wasserstein metric. We recast these analyses as multimarginal optimal transport problems. The particular formulation allows efficient computation, ensures existence of optimal solutions, and admits a probabilistic interpretation over the space of paths (line segments). Application of the theory to the interpolation of empirical distributions, images, power spectra, as well as assessing uncertainty in experimental designs, is envisioned

Quantum Populations in Zeno Regions inside Black Holes

Schwarzschild black-hole interiors border on space-like singularities representing classical information leaks. We show that local quantum physics is decoupled from these leaks due to dynamically generated boundaries, called Zeno borders. Beyond Zeno borders black-hole interiors become asymptotically silent, and quantum fields evolve freely towards the geodesic singularity with vanishing probability measure for populating the geodesic boundary. Thus Zeno borders represent a probabilistic completion of Schwarzschild black holes within the semiclassical framework.Comment: 5 pages, 2 figures, more pedagogical presentation of our unchanged results including an introduction to Zeno region