In this paper,
Abstract—The intrinsic dimensionality of a set of patterns is important in determining an appropriate number of features for representing the data and whether a reasonable two- or three-dimensional representation of the data exists. We propose an intuitively appealing, noniterative estimator for intrinsic dimensionality which is based on near-neighbor information. We give plausible arguments supporting the consistency of this estimator. The method works well in identifying the true dimensionality for a variety of artificial data sets and is fairly insensitive to the number of samples and to the algorithmic parameters. Comparisons between this new method and the global eigenvalue ap-proach demonstrate the utility of our estimator.
From: http://dataclustering.cse.msu.edu/papers/intrinsic_dimen.pdf
(No arXiv source seems immediately availailable)
under Section III there is a definition of volume of the hypersphere given as $V = V_d R_k^d$ where $d$ is the intrinsic dimension and $R_k$ is the nearest neighbor distances of points in $L$ space. $L>d$. Is this $V$ known as the phase space volume?
$R$ is the euclidean distance which is the k-nearest neighbor distance between points. $x_i \in \mathcal{R}^L$ denotes a point inside the object.
The formula can be written in general as V=(dimensionless quantity)*r^(dimension).
I can use an assumption that neighbors in the actual dimension d are mapped to close neighbors in the embedded higher dimension L.
Confusion1: Since, the distance r are calculated between points in that dimension L Is $V$ the volume of a higher dimensional object in the dimension $L$
defined as V=dimensionless quantity*r^L or by V=dimensionless quantity*r^d?
Confusion2: When we talk about higher dimension, $L$ then does the sphere become known as hypersphere?
Shall be grateful for an answer.
