245 research outputs found
Riemannian Optimization via Frank-Wolfe Methods
We study projection-free methods for constrained Riemannian optimization. In
particular, we propose the Riemannian Frank-Wolfe (RFW) method. We analyze
non-asymptotic convergence rates of RFW to an optimum for (geodesically) convex
problems, and to a critical point for nonconvex objectives. We also present a
practical setting under which RFW can attain a linear convergence rate. As a
concrete example, we specialize Rfw to the manifold of positive definite
matrices and apply it to two tasks: (i) computing the matrix geometric mean
(Riemannian centroid); and (ii) computing the Bures-Wasserstein barycenter.
Both tasks involve geodesically convex interval constraints, for which we show
that the Riemannian "linear oracle" required by RFW admits a closed-form
solution; this result may be of independent interest. We further specialize RFW
to the special orthogonal group and show that here too, the Riemannian "linear
oracle" can be solved in closed form. Here, we describe an application to the
synchronization of data matrices (Procrustes problem). We complement our
theoretical results with an empirical comparison of Rfw against
state-of-the-art Riemannian optimization methods and observe that RFW performs
competitively on the task of computing Riemannian centroids.Comment: Under Review. Largely revised version, including an extended
experimental section and an application to the special orthogonal group and
the Procrustes proble
Barycentric Subspace Analysis on Manifolds
This paper investigates the generalization of Principal Component Analysis
(PCA) to Riemannian manifolds. We first propose a new and general type of
family of subspaces in manifolds that we call barycentric subspaces. They are
implicitly defined as the locus of points which are weighted means of
reference points. As this definition relies on points and not on tangent
vectors, it can also be extended to geodesic spaces which are not Riemannian.
For instance, in stratified spaces, it naturally allows principal subspaces
that span several strata, which is impossible in previous generalizations of
PCA. We show that barycentric subspaces locally define a submanifold of
dimension k which generalizes geodesic subspaces.Second, we rephrase PCA in
Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy
of properly embedded linear subspaces of increasing dimension). We show that
the Euclidean PCA minimizes the Accumulated Unexplained Variances by all the
subspaces of the flag (AUV). Barycentric subspaces are naturally nested,
allowing the construction of hierarchically nested subspaces. Optimizing the
AUV criterion to optimally approximate data points with flags of affine spans
in Riemannian manifolds lead to a particularly appealing generalization of PCA
on manifolds called Barycentric Subspaces Analysis (BSA).Comment: Annals of Statistics, Institute of Mathematical Statistics, A
Para\^itr
Real-time manhattan world rotation estimation in 3D
Drift of the rotation estimate is a well known problem in visual odometry systems as it is the main source of positioning inaccuracy. We propose three novel algorithms to estimate the full 3D rotation to the surrounding Manhattan World (MW) in as short as 20 ms using surface-normals derived from the depth channel of a RGB-D camera. Importantly, this rotation estimate acts as a structure compass which can be used to estimate the bias of an odometry system, such as an inertial measurement unit (IMU), and thus remove its angular drift. We evaluate the run-time as well as the accuracy of the proposed algorithms on groundtruth data. They achieve zerodrift rotation estimation with RMSEs below 3.4° by themselves and below 2.8° when integrated with an IMU in a standard extended Kalman filter (EKF). Additional qualitative results show the accuracy in a large scale indoor environment as well as the ability to handle fast motion. Selected segmentations of scenes from the NYU depth dataset demonstrate the robustness of the inference algorithms to clutter and hint at the usefulness of the segmentation for further processing.United States. Office of Naval Research. Multidisciplinary University Research Initiative6 (Awards N00014-11-1-0688 and N00014-10-1-0936)National Science Foundation (U.S.) (Award IIS-1318392
A survey and comparison of contemporary algorithms for computing the matrix geometric mean
In this paper we present a survey of various algorithms for computing matrix geometric means and derive new second-order optimization algorithms to compute the Karcher mean. These new algorithms are constructed using the standard definition of the Riemannian Hessian. The survey includes the ALM list of desired properties for a geometric mean, the analytical expression for the mean of two matrices, algorithms based on the centroid computation in Euclidean (flat) space, and Riemannian optimization techniques to compute the Karcher mean (preceded by a short introduction into differential geometry). A change of metric is considered in the optimization techniques to reduce the complexity of the structures used in these algorithms. Numerical experiments are presented to compare the existing and the newly developed algorithms. We conclude that currently first-order algorithms are best suited for this optimization problem as the size and/or number of the matrices increase. Copyright © 2012, Kent State University
- …