## Statistics on Manifolds

*27 Feb 2017 16:30*

Yet Another Inadequate Placeholder. For various reasons, I am currently (2013) interested in what to do when the data live in hyperbolic spaces (e.g., the Poincaré disk). Mostly I am interested in density estimation and comparison of densities, but I could be persuaded to care about means...

(A very simple density estimator, which would be available on any manifold
endowed with a metric, would take a "kernel" of bandwidth \( h \) and center \(
x \) to be a uniform distribution on the ball of radius \( h \) centered at \(
x \), call this \( K(x,h) \). Then the "kernel density estimate" from a sample
\( x_1, x_2, \ldots x_n) would be \( n^{-1} \sum_{1}^{n}{K(x_i,h)} \). In
Euclidean space we can achieve this by convolving the kernel density with the
empirical distribution function, and do fast convolution through Fourier
transforms. The Huckemann et al. paper sets out a *de*-convolutional
method for the hyperbolic plane, using its version of Fourier transforms. Is
that the same as the addition-of-densities? Are they generally?)

The dual to this subject is information
geometry, which concerns itself with manifolds of statistical models.
Also, here I am concerning myself with the case where the manifold and its
geometry are *known*. Discovering
a hidden manifold structure in data is
another, and I'd say generally harder, question.

See also: Statistics;

- Recommended (currently completely inadequate):
- Stephan F. Huckemann, Peter T. Kim, Ja-Yong Koo, Axel Munk, "Möbius deconvolution on the hyperbolic plane with application to impedance density estimation", Annals of Statistics
**38**(2010): 2465--2498, arxiv:1010.4202

- Pride compels me to recommend:
- Dena Marie Asta, "Kernel Density Estimation on Symmetric Spaces", arxiv:1411.4040

- To read:
- Marc Arnaudon, Frédéric Barbaresco, Le Yang, "Medians and means in Riemannian geometry: existence, uniqueness and computation", arxiv:1111.3120
- Anil Aswani, Peter Bickel, and Claire Tomlin, "Regression on manifolds: Estimation of the exterior derivative", Annals of Statistics
**39**(2011): 48--81, arxiv:1103.1457 - Nikolay H. Balov, "Comparing and interpolating distributions on manifold", arxiv:0807.0782
- Abhishek Bhattacharya, Rabi Bhattacharya, "Nonparametric statistics on manifolds with applications to shape spaces", arxiv:0805.3282
- Rabi Bhattacharya and Vic Patrangenaru, "Large sample theory of
intrinsic and extrinsic sample means on manifolds--II", Annals of
Statistics
**33**(2005): 1225--1259, math.ST/0507423 - Ming-yen Cheng and Hau-tieng Wu, "Local Linear Regression on Manifolds and Its Geometric Interpretation", Journal of the American Statistical Association
**108**(2013): 1421--1434, arxiv:1201.0327 - Persi Diaconis, Susan Holmes, Mehrdad Shahshahani, "Sampling From A Manifold", arxiv:1206.6913
- Leif Ellingson, Vic Patrangenaru, Frits Ruymgaart, "Nonparametric Estimation of Means on Hilbert Manifolds and Extrinsic Analysis of Mean Shapes of Contours", arxiv:1302.2126
- Wenceslao Gonzalez-Manteiga, Guillermo Henry, Daniela Rodriguez, "Partially linear models on Riemannian manifolds", arxiv:1003.1573
- Harrie Hendriks, Zinoviy Landsman, "Asymptotic data analysis on manifolds", Annals of Statistics
**35**(2007): 109--131, arxiv:0708.0474 - Guillermo Henry, Daniela Rodriguez, "Robust Estimators in Partly Linear Regression Models on Riemannian Manifolds", arxiv:1008.0446
- Jacob Hinkle, Prasanna Muralidharan, P. Thomas Fletcher, Sarang Joshi, "Polynomial Regression on Riemannian Manifolds", arxiv:1201.2395
- Ian H. Jermyn, "Invariant Bayesian estimation on manifolds", Annals of
Statistics
**33**(2005): 583--605, math.ST/0506296 - Sungkyu Jung, Mark Foskey, J. S. Marron, "Principal arc analysis on direct product manifolds", Annals of Applied Statistics
**5**(2011): 578--603, arxiv:1104.3472 - P. E. Jupp
- "Data-driven Sobolev tests of uniformity on compact Riemannian manifolds", Annals of Statistics
**36**(2008): 1246--1260, arxiv:0806.2939 - "Sobolev tests of goodness of fit of distributions on compact Riemannian manifolds", Annals of Statistics
**33**(2005): 2957--2966, arxiv:math/0603135

- "Data-driven Sobolev tests of uniformity on compact Riemannian manifolds", Annals of Statistics
- Gerard Kerkyacharian, Richard Nickl, Dominique Picard, "Concentration Inequalities and Confidence Bands for Needlet Density Estimators on Compact Homogeneous Manifolds", Probability Theory and Related Fields
**153**(2012) 363--404, arxiv:1102.2450 - Partha Niyogi, "Manifold Regularization and Semi-supervised Learning: Some Theoretical Analyses", Journal of Machine Learning Research
**14**(2013): 1229--1250 - Bruno Pelletier, "Kernel density estimation on Riemannian
manifolds", Statistics and
Probability Letters
**73**(2005): 297--304 - Alain Trouvé, François-Xavier Vialard, "Shape Splines and Stochastic Shape Evolutions: A Second Order Point of View", arxiv:1003.3895