Inference and Mixture Modeling with the Elliptical Gamma Distribution

We study modeling and inference with the Elliptical Gamma Distribution (EGD). We consider maximum likelihood (ML) estimation for EGD scatter matrices, a task for which we develop new fixed-point algorithms. Our algorithms are efficient and converge to global optima despite nonconvexity. Moreover, they turn out to be much faster than both a well-known iterative algorithm of Kent & Tyler (1991) and sophisticated manifold optimization algorithms. Subsequently, we invoke our ML algorithms as subroutines for estimating parameters of a mixture of EGDs. We illustrate our methods by applying them to model natural image statistics---the proposed EGD mixture model yields the most parsimonious model among several competing approaches.Comment: 23 pages, 11 figure

Contractile properties of the pigeon supracoracoideus during different modes of flight

The supracoracoideus (SUPRA) is the primary upstroke muscle for avian flight and is the antagonist to the downstroke muscle, the pectoralis (PECT). We studied in vivo contractile properties and mechanical power output of both muscles during take-off, level and landing flight. We measured muscle length change and activation using sonomicrometry and electromyography, and muscle force development using strain recordings on the humerus. Our results support a hypothesis that the primary role of the SUPRA is to supinate the humerus. Antagonistic forces exerted by the SUPRA and PECT overlap during portions of the wingbeat cycle, thereby offering a potential mechanism for enhancing control of the wing. Among flight modes, muscle strain was approximately the same in the SUPRA (33–40%) and the PECT (35–42%), whereas peak muscle stress was higher in the SUPRA $(85–126 N m^{–2})$ than in the PECT $(50–58 N m^{–2})$. The SUPRA mainly shortened relative to resting length and the PECT mainly lengthened. We estimated that elastic energy storage in the tendon of the SUPRA contributed between 28 and 60% of the net work of the SUPRA and 6–10% of the total net mechanical work of both muscles. Mechanical power output in the SUPRA was congruent with the estimated inertial power required for upstroke, but power output from the PECT was only 42–46% of the estimated aerodynamic power requirements for flight. There was a significant effect of flight mode upon aspects of the contractile behavior of both muscles including strain, strain rate, peak stress, work and power.Organismic and Evolutionary BiologyOther Research Uni

Geometric optimisation on positive definite matrices with application to elliptically contoured distributions

Hermitian positive definite (hpd) matrices recur throughout machine learning, statistics, and optimisation. This paper develops (conic) geometric optimisation on the cone of hpd matrices, which allows us to globally optimise a large class of nonconvex functions of hpd matrices. Specifically, we first use the Riemannian manifold structure of the hpd cone for studying functions that are nonconvex in the Euclidean sense but are geodesically convex (g-convex), hence globally optimisable. We then go beyond g-convexity, and exploit the conic geometry of hpd matrices to identify another class of functions that remain amenable to global optimisation without requiring g-convexity. We present key results that help recognise g-convexity and also the additional structure alluded to above. We illustrate our ideas by applying them to likelihood maximisation for a broad family of elliptically contoured distributions: for this maximisation, we derive novel, parameter free fixed-point algorithms. To our knowledge, ours are the most general results on geometric optimisation of hpd matrices known so far. Experiments show that advantages of using our fixed-point algorithms

An alternative to EM for Gaussian mixture models: batch and stochastic Riemannian optimization

Abstract We consider maximum likelihood estimation for Gaussian Mixture Models (Gmm s). This task is almost invariably solved (in theory and practice) via the Expectation Maximization (EM) algorithm. EM owes its success to various factors, of which is its ability to fulfill positive definiteness constraints in closed form is of key importance. We propose an alternative to EM grounded in the Riemannian geometry of positive definite matrices, using which we cast Gmm parameter estimation as a Riemannian optimization problem. Surprisingly, such an out-of-the-box Riemannian formulation completely fails and proves much inferior to EM. This motivates us to take a closer look at the problem geometry, and derive a better formulation that is much more amenable to Riemannian optimization. We then develop Riemannian batch and stochastic gradient algorithms that outperform EM, often substantially. We provide a non-asymptotic convergence analysis for our stochastic method, which is also the first (to our knowledge) such global analysis for Riemannian stochastic gradient. Numerous empirical results are included to demonstrate the effectiveness of our methods