  The intrinsic dimensionality refers to the ``true'' dimensionality of the
data, as opposed to the dimensionality of the data representation. For example,
when attributes are highly correlated, the intrinsic dimensionality can be much
lower than the number of variables. Local intrinsic dimensionality refers to
the observation that this property can vary for different parts of the data
set; and intrinsic dimensionality can serve as a proxy for the local difficulty
of the data set.
  Most popular methods for estimating the local intrinsic dimensionality are
based on distances, and the rate at which the distances to the nearest
neighbors increase, a concept known as ``expansion dimension''. In this paper
we introduce an orthogonal concept, which does not use any distances: we use
the distribution of angles between neighbor points. We derive the theoretical
distribution of angles and use this to construct an estimator for intrinsic
dimensionality.
  Experimentally, we verify that this measure behaves similarly, but
complementarily, to existing measures of intrinsic dimensionality. By
introducing a new idea of intrinsic dimensionality to the research community,
we hope to contribute to a better understanding of intrinsic dimensionality and
to spur new research in this direction.
