# Help in understanding a formula used in dimension estimation from nearest neighbor distances

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. $L>d$. 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.

From what I understand is this methods tries to estimate the dimensionality by using the fact that, in $d$-dimensional space, volume $V \propto r^d$.

You are trying to calculate dataset's $d$, so why would you use $L$?
Hypersphere in dimension $L$ is $L-1$ dimensional sphere, that is, the set of points $\{x \in \mathbb{R}^L : \| x - p\| = r\}$ for some $p, r$
• Phase space according to my understanding means that if a particle or a point x is in higher dimensional space L then the phase space is defined as the space of all possible arrangements, all possible values say position, momenta of the point x. And the measure (volume) is the way to count the continuous sets. So the volume in phase space and counting of possibilities is the same thing. This is my understanding of phase space and phase space volume based on the book: Astrophysics Processes: The Physics of Astronomical Phenomena Page 107 By Hale Bradt. Sep 7, 2017 at 16:51
• In the book V_phase = 4/3 pi rho^3 V where V is the actual physical dimensionless volume. My confusion is about the power considered in the definition of V_phase. If rho is considered to be the length or euclidean distance,then which power would it be raised to? Would it be the higher dimension, L in which the distance is calculated or the true (lower dimension also known as the correlation dimension). Sep 7, 2017 at 16:52
• Therefore, based on the formula in the book and from your answer, would V` (the term on right hand side) be known as the phase space volume? Sep 7, 2017 at 16:54