Huiling Le

Sticky central limit theorems on open books

Given a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Frechet mean (barycenter) is sticky. This nonclassical phenomenon is quantified by a law of large numbers (LLN) stating that the empirical mean eventually almost surely lies on the (codimension 1 and hence measure 0) spine that is the glued hyperplane, and a central limit theorem (CLT) stating that the limiting distribution is Gaussian and supported on the spine. We also state versions of the LLN and CLT for the cases where the mean is nonsticky (i.e., not lying on the spine) and partly sticky (i.e., is, on the spine but not sticky).

Fitting smoothing splines to time-indexed, noisy points on nonlinear manifolds

We address the problem of estimating full curves/paths on certain nonlinear manifolds using only a set of time-indexed points, for use in interpolation, smoothing, and prediction of dynamic systems. These curves are analogous to smoothing splines in Euclidean spaces as they are optimal under a similar objective function, which is a weighted sum of a fitting-related (data term) and a regularity-related (smoothing term) cost functions. The search for smoothing splines on manifolds is based on a Palais metric-based steepest-decent algorithm developed in Samir et al. [38]. Using three representative manifolds: the rotation group for pose tracking, the space of symmetric positive-definite matrices for DTI image analysis, and Kendalls shape space for video-based activity recognition, we demonstrate the effectiveness of the proposed algorithm for optimal curve fitting. This paper derives certain geometrical elements, namely the exponential map and its inverse, parallel transport of tangents, and the curvature tensor, on these manifolds, that are needed in the gradient-based search for smoothing splines. These ideas are illustrated using experimental results involving both simulated and real data, and comparing the results to some current algorithms such as piecewise geodesic curves and splines on tangent spaces, including the method by Kume et al. [24].

Multidimensional scaling of simplex shapes

Abstract Bookstein (Statist. Sci. 1 (1986) 181–242; Morphometric Tools for Landmark Data: Geometry and Biology, Cambridge University Press, Cambridge, 1991) has proposed a method for the representation of triangle shape as points in the Poincare half plane – a space of constant negative curvature. Small (The Statistical Theory of Shape, Springer, New York, 1996) provided an extension of the Bookstein representation by representing the shapes of n-simplexes on manifolds. These manifolds are quite distinct from those proposed by D.G. Kendall based upon Procrustes arguments. In this paper, we examine the geometrical properties of these simplex shape spaces in greater detail. In particular, explicit formulas are given for the geodesic distance between any two points in these spaces. Such formulas permit the implementation of multidimensional scaling methods for the statistical shape analysis of two- and three-dimensional objects. In addition, the curvatures of the simplex shape spaces are examined. It is shown that the spaces of n-simplex shapes are not of constant curvature unless n =2.

Estimation of Riemannian Barycentres

Using Jacobi field arguments, this paper describes an iterative procedure for finding the Riemannian barycentres of a class of probability measures on complete, simply connected Riemannian manifolds with a finite upper bound on their sectional curvatures. This, in particular, generalises an earlier result of the authors (‘Locating Frechet means with application to shape spaces’, Adv. Appl. Probab. 33 (2001) 324-338).

The Fréchet mean shape and the shape of the means

We identify the Fréchet mean shape with respect to the Riemannian metric of a class of probability measures on Booksteins shape space of labelled triangles and show, in contrast to the case of Kendalls shape space, that the Fréchet mean shape of the probability measure on Booksteins shape space induced from independent normal distributions on vertices, having the same covariance matrix σ2 I 2, is not necessarily the shape of the means.

Detection of shape changes in biological features.

The question of analysing shape changes over time, such as during growth and during the progress of disease, is an important fundamental issue for many applications. Recent mathematical developments in the understanding of the detailed structure of shape spaces have made possible the quantitative study of shape variation. In this paper, we combine the classical multidimensional scaling technique with knowledge of the geometry of shape spaces to examine the role played by the geodesics in shape spaces. We present some promising early results answering questions about shape changes over time.

A Finite Time Horizon Optimal Stopping Problem with Regime Switching

We extend the technique developed in [E. Bayraktar, A Proof of the Smoothness of the Finite Time Horizon American Put Option for Jump Diffusion, http://arxiv.org/abs/math/0703782, 2007] to a class of finite time horizonal optimal stopping problems under regime switching models which includes the pricing of American put options. The construction involved also leads to a computational procedure for the solutions of such optimal stopping problems.

2D Affine and Projective Shape Analysis

Current techniques for shape analysis tend to seek invariance to similarity transformations (rotation, translation, and scale), but certain imaging situations require invariance to larger groups, such as affine or projective groups. Here we present a general Riemannian framework for shape analysis of planar objects where metrics and related quantities are invariant to affine and projective groups. Highlighting two possibilities for representing object boundaries-ordered points (or landmarks) and parameterized curves-we study different combinations of these representations (points and curves) and transformations (affine and projective). Specifically, we provide solutions to three out of four situations and develop algorithms for computing geodesics and intrinsic sample statistics, leading up to Gaussian-type statistical models, and classifying test shapes using such models learned from training data. In the case of parameterized curves, we also achieve the desired goal of invariance to re-parameterizations. The geodesics are constructed by particularizing the path-straightening algorithm to geometries of current manifolds and are used, in turn, to compute shape statistics and Gaussian-type shape models. We demonstrate these ideas using a number of examples from shape and activity recognition.

Parallel Transport of Deformations in Shape Space of Elastic Surfaces

Statistical shape analysis develops methods for comparisons, deformations, summarizations, and modeling of shapes in given data sets. These tasks require a fundamental tool called parallel transport of tangent vectors along arbitrary paths. This tool is essential for: (1) computation of geodesic paths using either shooting or path-straightening method, (2) transferring deformations across objects, and (3) modeling of statistical variability in shapes. Using the square-root normal field (SRNF) representation of parameterized surfaces, we present a method for transporting deformations along paths in the shape space. This is difficult despite the underlying space being a vector space because the chosen (elastic) Riemannian metric is non-standard. Using a finite-basis for representing SRNFs of shapes, we derive expressions for Christoffel symbols that enable parallel transports. We demonstrate this framework using examples from shape analysis of parameterized spherical surfaces, in the three contexts mentioned above.

Limiting behaviour of Fréchet means in the space of phylogenetic trees

As demonstrated in our previous work on