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.
We are interested in proposing new theory and algorithms for general optimization problems in particular for integer programming, non-convex optimization and optimization on manifolds.
Related Publications
Niu, M.,Cheung, P., Lin, L., Dai, Z., Lawrence, N. and Dunson, D. B. (2019). Intrinsic Gaussian processes on complex constrained domains. Journal of the Royal Statistical Society, Ser. B, 81: 603-627. Link.
Lin, L., Niu, M., Pokman, C. and Dunson. D.B. (2019). Extrinsic Gaussian process mod- els for regression and classification on manifolds. Bayesian Analysis, Volume 14, Number 3, 887-906. Link.
Bhattacharya, R. and Lin, L., (2019). Differential geometry for model independent analysis of images and other non-Euclidean data: recent developments. In: Sidoravicius V. (eds) Sojourns in Probability Theory and Statistical Physics - II. Springer Proceedings in Mathematics & Statistics, vol 299. Springer. Link.
Sarpabayeva, B., Zhang, M and Lin, L. (2018). Communication efficient parallel optimization algorithms on manifolds. Neural Information Processing Systems 2018.
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.