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Smoothing splines on Riemannian manifolds, with applications to 3D shape space (2020)
Journal Article
Kim, K. R., Dryden, I. L., Le, H., & Severn, K. E. (2021). Smoothing splines on Riemannian manifolds, with applications to 3D shape space. Journal of the Royal Statistical Society: Series B, 83(1), 108-132. https://doi.org/10.1111/rssb.12402

© 2020 The Authors. Journal of the Royal Statistical Society: Series B (Statistical Methodology) published by John Wiley & Sons Ltd on behalf of Royal Statistical Society There has been increasing interest in statistical analysis of data lying in m... Read More about Smoothing splines on Riemannian manifolds, with applications to 3D shape space.

Principal nested shape space analysis of molecular dynamics data (2019)
Journal Article
Dryden, I. L., Kim, K., Laughton, C. A., & Le, H. (2019). Principal nested shape space analysis of molecular dynamics data. Annals of Applied Statistics, 13(4), 2213-2234. https://doi.org/10.1214/19-AOAS1277

Molecular dynamics simulations produce huge datasets of temporal sequences of molecules. It is of interest to summarize the shape evolution of the molecules in a succinct, low-dimensional representation. However, Euclidean techniques such as principa... Read More about Principal nested shape space analysis of molecular dynamics data.

Bayesian linear size-and-shape regression with applications to face data (2018)
Journal Article
Dryden, I. L., Le, H., & Kim, K. (2019). Bayesian linear size-and-shape regression with applications to face data. Sankhya A, 81(1), 83–103. https://doi.org/10.1007/s13171-018-0136-8

Regression models for size-and-shape analysis are developed, where the model is specified in the Euclidean space of the landmark coordinates. Statistical models in this space (which is known as the top space or ambient space) are often easier for pra... Read More about Bayesian linear size-and-shape regression with applications to face data.

The logarithm map, its limits and Fréchet means in orthant spaces (2018)
Journal Article
Barden, D., & Le, H. (2018). The logarithm map, its limits and Fréchet means in orthant spaces. Proceedings of the London Mathematical Society, 117(4), 751-789. https://doi.org/10.1112/plms.12149

The first part of the paper studies the expression for, and the properties of, the logarithm map on an orthant space, which is a simple stratified space, with the aim of analysing Fréchet means of probability measures on such a space. In the second p... Read More about The logarithm map, its limits and Fréchet means in orthant spaces.

Rate-invariant analysis of covariance trajectories (2018)
Journal Article
Zhang, Z., Su, J., Klassen, E., Le, H., & Srivastava, A. (2018). Rate-invariant analysis of covariance trajectories. Journal of Mathematical Imaging and Vision, 60(8), 1306–1323. https://doi.org/10.1007/s10851-018-0814-0

Statistical analysis of dynamic systems, such as videos and dynamic functional connectivity, is often translated into a problem of analyzing trajectories of relevant features, particularly covariance matrices. As an example, in video-based action rec... Read More about Rate-invariant analysis of covariance trajectories.

Conjugate duality in stochastic controls with delay (2017)
Journal Article
Wang, Z., Hodge, D. J., & Le, H. (2017). Conjugate duality in stochastic controls with delay. Advances in Applied Probability, 49(4), https://doi.org/10.1017/apr.2017.32

This paper uses the method of conjugate duality to investigate a class of stochastic optimal control problems where state systems are described by stochastic differential equations with delay. For this, we first analyse a stochastic convex problem wi... Read More about Conjugate duality in stochastic controls with delay.

Limiting behaviour of Fréchet means in the space of phylogenetic trees (2016)
Journal Article
Barden, D., Le, H., & Owen, M. (2018). Limiting behaviour of Fréchet means in the space of phylogenetic trees. Annals of the Institute of Statistical Mathematics, 70(1), 99-129. https://doi.org/10.1007/s10463-016-0582-9

As demonstrated in our previous work on T4, the space of phylogenetic trees with four leaves, the topological structure of the space plays an important role in the non-classical limiting behaviour of the sample Fréchet means in T4. Nevertheless, the... Read More about Limiting behaviour of Fréchet means in the space of phylogenetic trees.

Markov decision process algorithms for wealth allocation problems with defaultable bonds (2016)
Journal Article
Pérez López, I., Hodge, D., & Le, H. (in press). Markov decision process algorithms for wealth allocation problems with defaultable bonds. Advances in Applied Probability, 48(2), https://doi.org/10.1017/apr.2016.6

This paper is concerned with analysing optimal wealth allocation techniques within a defaultable financial market similar to Bielecki and Jang (2007). It studies a portfolio optimization problem combining a continuous-time jump market and a defaultab... Read More about Markov decision process algorithms for wealth allocation problems with defaultable bonds.

A diffusion process associated with Fréchet means (2015)
Journal Article
Le, H. (2015). A diffusion process associated with Fréchet means. Annals of Applied Probability, 25(6), https://doi.org/10.1214/14-AAP1066

This paper studies rescaled images, under exp−1μ, of the sample Fréchet means of i.i.d. random variables {Xk|k≥1} with Fréchet mean μ on a Rie-mannian manifold. We show that, with appropriate scaling, these images converge weakly to a diffusion proce... Read More about A diffusion process associated with Fréchet means.

Time-randomized stopping problems for a family of utility functions (2015)
Journal Article
Pérez López, I., & Le, H. (2015). Time-randomized stopping problems for a family of utility functions. SIAM Journal on Control and Optimization, 53(3), https://doi.org/10.1137/130946800

This paper studies stopping problems of the form $V=\inf_{0 \leq \tau \leq T} \mathbb{E}[U(\frac{\max_{0\le s \le T} Z_s }{Z_\tau})]$ for strictly concave or convex utility functions U in a family of increasing functions satisfying certain conditions... Read More about Time-randomized stopping problems for a family of utility functions.

On the measure of the cut locus of a Fréchet mean (2014)
Journal Article
Le, H., Le, H., & Barden, D. (2014). On the measure of the cut locus of a Fréchet mean. Bulletin of the London Mathematical Society, 46(4), 698-708. https://doi.org/10.1112/blms/bdu025

One of the fundamental differences between the Central Limit Theorem for empirical Fréchet means obtained in [Kendall and Le, ‘Limit theorems for empirical Fréchet means of independent and non‐identically distributed manifold‐valued random variables’... Read More about On the measure of the cut locus of a Fréchet mean.