A variational model for data fitting on manifolds by minimizing the acceleration of a B\'ezier curve

We derive a variational model to fit a composite B\'ezier curve to a set of data points on a Riemannian manifold. The resulting curve is obtained in such a way that its mean squared acceleration is minimal in addition to remaining close the data points. We approximate the acceleration by discretizing the squared second order derivative along the curve. We derive a closed-form, numerically stable and efficient algorithm to compute the gradient of a B\'ezier curve on manifolds with respect to its control points, expressed as a concatenation of so-called adjoint Jacobi fields. Several examples illustrate the capabilites and validity of this approach both for interpolation and approximation. The examples also illustrate that the approach outperforms previous works tackling this problem

Manifolds.jl: An Extensible Julia Framework for Data Analysis on Manifolds

For data given on a nonlinear space, like angles, symmetric positive matrices, the sphere, or the hyperbolic space, there is often enough structure to form a Riemannian manifold. We present the Julia package Manifolds.jl, providing a fast and easy to use library of Riemannian manifolds and Lie groups. We introduce a common interface, available in ManifoldsBase.jl, with which new manifolds, applications, and algorithms can be implemented. We demonstrate the utility of Manifolds.jl using B\'ezier splines, an optimization task on manifolds, and a principal component analysis on nonlinear data. In a benchmark, Manifolds.jl outperforms existing packages in Matlab or Python by several orders of magnitude and is about twice as fast as a comparable package implemented in C++

Manifold-valued Image Generation with Wasserstein Generative Adversarial Nets

Generative modeling over natural images is one of the most fundamental machine learning problems. However, few modern generative models, including Wasserstein Generative Adversarial Nets (WGANs), are studied on manifold-valued images that are frequently encountered in real-world applications. To fill the gap, this paper first formulates the problem of generating manifold-valued images and exploits three typical instances: hue-saturation-value (HSV) color image generation, chromaticity-brightness (CB) color image generation, and diffusion-tensor (DT) image generation. For the proposed generative modeling problem, we then introduce a theorem of optimal transport to derive a new Wasserstein distance of data distributions on complete manifolds, enabling us to achieve a tractable objective under the WGAN framework. In addition, we recommend three benchmark datasets that are CIFAR-10 HSV/CB color images, ImageNet HSV/CB color images, UCL DT image datasets. On the three datasets, we experimentally demonstrate the proposed manifold-aware WGAN model can generate more plausible manifold-valued images than its competitors.Comment: Accepted by AAAI 201