The multivariate normal density is a monotonic function of the distance to the mean, and its ellipsoidal shape is due to the underlying Euclidean metric. We suggest to replace this metric with a locally adaptive, smoothly changing (Riemannian) metric that favors regions of high local density. The resulting locally adaptive normal distribution (LAND) is a generalization of the normal distribution to the "manifold" setting, where data is assumed to lie near a potentially low-dimensional manifold embedded in @math . The LAND is parametric, depending only on a mean and a covariance, and is the maximum entropy distribution under the given metric. The underlying metric is, however, non-parametric. We develop a maximum likelihood algorithm to infer the distribution parameters that relies on a combination of gradient descent and Monte Carlo integration. We further extend the LAND to mixture models, and provide the corresponding EM algorithm. We demonstrate the efficiency of the LAND to fit non-trivial probability distributions over both synthetic data, and EEG measurements of human sleep.
We are not the first to consider Riemannian normal distributions, e.g. gives a theoretical analysis of the distribution, and consider the Riemannian counterpart of probabilistic PCA. Both consider the scenario where the manifold is known a priori. We adapt the distribution to the manifold learning'' setting by constructing a Riemannian metric that adapts to the data. This is our overarching contribution. Traditionally, manifold learning is seen as an where a low-dimensional representation of the data is sought. This is useful for visualization , clustering @cite_4 , semi-supervised learning @cite_16 and more. However, in embedding approaches, the relation between a new point and the embedded points are less well-defined, and consequently these approaches are less suited for building generative models. In contrast, the Riemannian approach gives the ability to measure continuous geodesics that follow the structure of the data. This makes the learned Riemannian manifold a suitable space for a generative model.
@cite_16 @cite_4
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Deep generative models learn a mapping from a low-dimensional latent space to a high-dimensional data space. Under certain regularity conditions, these models parameterize nonlinear manifolds in the data space. In this paper, we investigate the Riemannian geometry of these generated manifolds. First, we develop efficient algorithms for computing geodesic curves, which provide an intrinsic notion of distance between points on the manifold. Second, we develop an algorithm for parallel translation of a tangent vector along a path on the manifold. We show how parallel translation can be used to generate analogies, i.e., to transport a change in one data point into a semantically similar change of another data point. Our experiments on real image data show that the manifolds learned by deep generative models, while nonlinear, are surprisingly close to zero curvature. The practical implication is that linear paths in the latent space closely approximate geodesics on the generated manifold. Recently, manifold learning has been widely exploited in pattern recognition, data analysis, and machine learning. This paper presents a novel framework, called Riemannian manifold learning (RML), based on the assumption that the input high-dimensional data lie on an intrinsically low-dimensional Riemannian manifold. The main idea is to formulate the dimensionality reduction problem as a classical problem in Riemannian geometry, that is, how to construct coordinate charts for a given Riemannian manifold? We implement the Riemannian normal coordinate chart, which has been the most widely used in Riemannian geometry, for a set of unorganized data points. First, two input parameters (the neighborhood size k and the intrinsic dimension d) are estimated based on an efficient simplicial reconstruction of the underlying manifold. Then, the normal coordinates are computed to map the input high-dimensional data into a low- dimensional space. Experiments on synthetic data, as well as real-world images, demonstrate that our algorithm can learn intrinsic geometric structures of the data, preserve radial geodesic distances, and yield regular embeddings. 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 to a nonlinear Riemannian geometry on the latent space. The latent space thus provides a low-dimensional nonlinear representation of data and classical linear statistical techniques are no longer applicable. In this paper we show how statistics of data in their latent space representation can be performed using techniques from the field of nonlinear manifold statistics. Nonlinear manifold statistics provide generalizations of Euclidean statistical notions including means, principal component analysis, and maximum likelihood fits of parametric probability distributions. We develop new techniques for maximum likelihood inference in latent space, and adress the computational complexity of using geometric algorithms with high-dimensional data by training a separate neural network to approximate the Riemannian metric and cometric tensor capturing the shape of the learned data manifold. In computer vision applications, features often lie on Riemannian manifolds with known geometry. Popular learning algorithms such as discriminant analysis, partial least squares, support vector machines, etc., are not directly applicable to such features due to the non-Euclidean nature of the underlying spaces. Hence, classification is often performed in an extrinsic manner by mapping the manifolds to Euclidean spaces using kernels. However, for kernel based approaches, poor choice of kernel often results in reduced performance. In this paper, we address the issue of kernel selection for the classification of features that lie on Riemannian manifolds using the kernel learning approach. We propose two criteria for jointly learning the kernel and the classifier using a single optimization problem. Specifically, for the SVM classifier, we formulate the problem of learning a good kernel-classifier combination as a convex optimization problem and solve it efficiently following the multiple kernel learning approach. Experimental results on image set-based classification and activity recognition clearly demonstrate the superiority of the proposed approach over existing methods for classification of manifold features.