In which dimension should nearest neighbor distances be calculated? Confusion regarding some concepts

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.

(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. $L>d$.

$R$ 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, alternatively known as the phase space volume? When does a volume become known as phase space volume?

Confusion2: In vol N sphere wiki link The volume of a $d$-dimensional sphere with radius R is

$V_d(r=R)=\frac{\pi^{d/2}}{\Gamma\left(\frac{d}{2}+1\right)}R^d$

This formula looks very similar to the one in paper but I am not sure if $d$ in this formula applies to embedded dimension or the intrinsic dimension. What ever is the dimension, the radius or the distance is measured in that dimension itself. On the contrary, in the paper, the radius is $R_d$ which are the nearest neighbor distances of points embedded in $L$ dimension are measured in the intrinsic dimension $d$ Shall be grateful for an answer.

I am not sure this is entirely a physics question. Suppose you have a set of measured data - Weight and age of a set of adults. Some are big, some are small. If you plot $w$ vs $a$, you might get a fairly uniformly spread out set of points.
Pick a data point. Look at all the neighbors within a distance $\Delta w$ and $\Delta a$, you get some. If you look within $2\Delta w$ and $2\Delta a$, you get 4 times as many because the data points are scattered in a plane.
Suppose your data is weight and height. Suppose w vs h plots as a nice line. If you count neighbors within $\Delta w$ and $\Delta h$, and again $2\Delta w$ and $2\Delta h$, you find 2 times as many.
• Thank you for your response, but I did not ask how the dimension is estimated. Maybe I did not make it clear. I need help to clarify the following terminologies. (1) I want to know the following if the dimension d in the formula of the volume V_d can be any dimension or if its specifically the correlation/ intrinsic dimension. (2) When does volume become known as phase space volume? (3) If d is the intrinsic dimension then is V_d known as the phase space volume? – Srishti M Sep 4 '17 at 17:28