Geometry and Statistics

Overview 

Geometry is inherent in data which is either known or to be learnt which should be incorporated for statistical learning and inference. Modern data of complex data are routinely collected in many areas of science and engineering. Typical examples include imaging data (e.g., DTI, FMRI), shape data, network data and so on. In many cases, the underlying space where these general data object lie form an interesting geometric space, often a manifold whose geometry can be well-characterized. 

One of the actively ongoing research area of our lab is Statistics on Manifolds which deals with statistical inference of complex data of the above kind. This requires creative combinations of tools from statistics, differential geometry and Riemannian geometry. Our lab also works on Optimizations on Manifolds where the optimization is constrained to manifolds, which has vast applications such as in building recommender systems, and geometric control theory. 

 

Related Publications

Niu, M.,Cheung, P., Lin, L., Dai, Z., Lawrence, N. and Dunson, D. B. (2017). Intrinsic Gaussian processes on complex constrained domains. Revision Submitted. Link.

Lin, L., Niu, M., Pokman, C. and Dunson. D.B. (2018). Extrinsic Gaussian process mod- els for regression and classification on manifolds. Bayesian Analysis revision submitted. Link.

Lin, L., Thomas, B., Zhu, H. and Dunson, D.B (2017).   Extrinsic local regression on manifold-valued data. Journal of the American Statistical Association-Theory and Methods. 112(519), 1261-1273. Link.

Bhattacharya, R. and Lin, L., (2017).  Omnibus CLTs for Fr\'echet means and nonparametric inference on non-Euclidean spaces.  Proceedings of American Mathematical Society, Vol. 145, 413-428 . Link.

Lazar, D. and Lin, L. (2017). Scale and curvature effects in principal geodesic analysis.  Journal of the Multivariate Analysis 153, 64--82. Link.

Lin, L., Rao, V.,  and Dunson, D.B (2017).  Bayesian nonparametric inference on Stiefel manifold.   Statistics Sinica, 27,  535--553.  Link.

Bhattacharya, R. and Lin, L., (2017).  Differential geometry for model independent analysis of images and other non-Euclidean data: recent developments. Submitted. Link.