Statistics on Manifolds

09 Dec 2014 00:22

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;

Statistics of structured data
    Pride compels me to recommend:
  • Dena Marie Asta, "Kernel Density Estimation on Symmetric Spaces", arxiv:1411.4040