Bijan Afsari

On the Convergence of Gradient Descent for Finding the Riemannian Center of Mass

We study the problem of finding the global Riemannian center of mass of a set of data points on a Riemannian manifold. Specifically, we investigate the convergence of constant step-size gradient descent algorithms for solving this problem. The challenge is that often the underlying cost function is neither globally differentiable nor convex, and despite this one would like to have guaranteed convergence to the global minimizer. After some necessary preparations we state a conjecture which we argue is the best (in a sense described) convergence condition one can hope for. The conjecture specifies conditions on the spread of the data points, step-size range, and the location of the initial condition (i.e., the region of convergence) of the algorithm. These conditions depend on the topology and the curvature of the manifold and can be conveniently described in terms of the injectivity radius and the sectional curvatures of the manifold. For manifolds of constant nonnegative curvature (e.g., the sphere and the rotation group in

Riemannian Consensus for Manifolds With Bounded Curvature

\mathbb{R}^{3}

Intrinsic consensus on SO(3) with almost-global convergence

) we show that the conjecture holds true (we do this by proving and using a comparison theorem which seems to be of a different nature from the standard comparison theorems in Riemannian geometry). For manifolds of arbitrary curvature we prove convergence results which are weaker than the conjectured one (but still superior over the available results). We also briefly study the effect of the configuration of the data points on the speed of convergence.

Group action induced distances for averaging and clustering Linear Dynamical Systems with applications to the analysis of dynamic scenes

Consensus algorithms are popular distributed algorithms for computing aggregate quantities, such as averages, in ad-hoc wireless networks. However, existing algorithms mostly address the case where the measurements lie in Euclidean space. In this work we propose Riemannian consensus, a natural extension of existing averaging consensus algorithms to the case of Riemannian manifolds. Unlike previous generalizations, our algorithm is intrinsic and, in principle, can be applied to any complete Riemannian manifold. We give sufficient convergence conditions on Riemannian manifolds with bounded curvature and we analyze the differences with respect to the Euclidean case. We test the proposed algorithms on synthetic data sampled from the space of rotations, the sphere and the Grassmann manifold.

Approximate Joint Diagonalization and Geometric Mean of Symmetric Positive Definite Matrices

In this paper we propose a discrete time protocol to align the states of a network of agents evolving in the space of rotations SO(3). The starting point of our work is Riemannian consensus, a general and intrinsic extension of classical consensus algorithms to Riemannian manifolds. Unfortunately, this algorithm is guaranteed to align the states only when the initial states are not too far apart. We show how to modify Riemannian consensus so that the states of the agents can be aligned, in practice, from almost any initial condition. While we focus on the specific case of SO(3), we hope that this work will represent the first step toward more general results.

Average consensus on Riemannian manifolds with bounded curvature

We introduce a framework for defining a distance on the (non-Euclidean) space of Linear Dynamical Systems (LDSs). The proposed distance is induced by the action of the group of orthogonal matrices on the space of statespace realizations of LDSs. This distance can be efficiently computed for large-scale problems, hence it is suitable for applications in the analysis of dynamic visual scenes and other high dimensional time series. Based on this distance we devise a simple LDS averaging algorithm, which can be used for classification and clustering of time-series data. We test the validity as well as the performance of our group-action based distance on synthetic as well as real data and provide comparison with state-of-the-art methods.

Group action induced averaging for HARDI processing

We explore the connection between two problems that have arisen independently in the signal processing and related fields: the estimation of the geometric mean of a set of symmetric positive definite (SPD) matrices and their approximate joint diagonalization (AJD). Today there is a considerable interest in estimating the geometric mean of a SPD matrix set in the manifold of SPD matrices endowed with the Fisher information metric. The resulting mean has several important invariance properties and has proven very useful in diverse engineering applications such as biomedical and image data processing. While for two SPD matrices the mean has an algebraic closed form solution, for a set of more than two SPD matrices it can only be estimated by iterative algorithms. However, none of the existing iterative algorithms feature at the same time fast convergence, low computational complexity per iteration and guarantee of convergence. For this reason, recently other definitions of geometric mean based on symmetric divergence measures, such as the Bhattacharyya divergence, have been considered. The resulting means, although possibly useful in practice, do not satisfy all desirable invariance properties. In this paper we consider geometric means of covariance matrices estimated on high-dimensional time-series, assuming that the data is generated according to an instantaneous mixing model, which is very common in signal processing. We show that in these circumstances we can approximate the Fisher information geometric mean by employing an efficient AJD algorithm. Our approximation is in general much closer to the Fisher information geometric mean as compared to its competitors and verifies many invariance properties. Furthermore, convergence is guaranteed, the computational complexity is low and the convergence rate is quadratic. The accuracy of this new geometric mean approximation is demonstrated by means of simulations.

Rotation invariant features for HARDI

Consensus algorithms are a popular choice for computing averages and other similar quantities in ad-hoc wireless networks. However, existing algorithms mostly address the case where the measurements live in a Euclidean space. In this paper, we propose distributed algorithms for averaging measurements lying in a Riemannian manifold. We first propose a direct extension of the classical average consensus algorithm and derive sufficient conditions for its convergence to a consensus configuration. Such conditions depend on the network connectivity, the geometric configuration of the measurements and the curvature of the manifold. However, the consensus configuration to which the algorithm converges may not coincide with the Fréchet mean of the measurements. We thus propose a second algorithm that performs consensus in the tangent space. This algorithm is guaranteed to converge to the Fréchet mean of the measurements, but needs to be initialized at a consensus configuration. By combining these two methods, we obtain a distributed algorithm that converges to the Fréchet mean of the measurements. We test the proposed algorithms on synthetic data sampled from manifolds such as the space of rotations, the sphere and the Grassmann manifold.

The alignment distance on Spaces of Linear Dynamical Systems

We consider the problem of processing high angular resolution diffusion images described by orientation distribution functions (ODFs). Prior work showed that several processing operations, e.g., averaging, interpolation and filtering, can be reduced to averaging in the space of ODFs. However, this approach leads to anatomically erroneous results when the ODFs to be processed have very different orientations. To address this issue, we propose a group action induced distance for averaging ODFs, which leads to a novel processing framework on the spaces of orientation (the space of 3D rotations) and shape (the space of ODFs with the same orientation). Experiments demonstrate that our framework produces anatomically meaningful results.

Group Action Induced Distances on Spaces of High-Dimensional Linear Stochastic Processes

Reducing the amount of information stored in diffusion MRI (dMRI) data to a set of meaningful and representative scalar values is a goal of much interest in medical imaging. Such features can have far reaching applications in segmentation, registration, and statistical characterization of regions of interest in the brain, as in comparing features between control and diseased patients. Currently, however, the number of biologically relevant features in dMRI is very limited. Moreover, existing features discard much of the information inherent in dMRI and embody several theoretical shortcomings. This paper proposes a new family of rotation invariant scalar features for dMRI based on the spherical harmonic (SH) representation of high angular resolution diffusion images (HARDI). These features describe the shape of the orientation distribution function extracted from HARDI data and are applicable to any reconstruction method that represents HARDI signals in terms of an SH basis. We further illustrate their significance in white matter characterization of synthetic, phantom and real HARDI brain datasets.