RSS Discussion: Statistical Exploration of the Manifold Hypothesis

[Adam M Johansen]

Dear All,

The next RSS discussion paper meeting may be of interest to readers of
this list. It will take place online at 4pm (BST) on October 8th.

Preprint and further details available at
https://rss.org.uk/training-events/events/discussion-papers/

*Paper: *‘Statistical exploration of the Manifold Hypothesis’
*Authors: *Nick Whiteley, University of Bristol, UK, Annie Gray,
University of Bristol, UK, Patrick Rubin-Delanchy, University of
Edinburgh, UK.

*Abstract*: The Manifold Hypothesis is a widely accepted tenet of
Machine Learning which asserts that nominally high-dimensional data are
in fact concentrated near a low-dimensional manifold, embedded in
high-dimensional space. This phenomenon is observed empirically in many
real world situations, has led to development of a wide range of
statistical methods in the last few decades, and has been suggested as a
key factor in the success of modern AI technologies. We show that rich
and sometimes intricate manifold structure in data can emerge from a
generic and remarkably simple statistical model — the Latent Metric
Model — via elementary concepts such as latent variables, correlation
and stationarity. This establishes a general statistical explanation for
why the Manifold Hypothesis seems to hold in so many situations.
Informed by the Latent Metric Model we derive procedures to discover and
interpret the geometry of high-dimensional data, and explore hypotheses
about the data generating mechanism. These procedures operate under
minimal assumptions and make use of well-known graph-analytic algorithms.

Best Wishes,

Adam


--
Prof. Adam M. Johansen, Department of Statistics, University of Warwick, CV4 7AL
We stand with Ukraine.