This post brings forward an inconsistency problem in the study of multidimensional scaling from the perspective of dimensional analysis. This post is rather of philosophical nature. Nevertheless, I find it is worthwhile being aware of this issue

for further theoretical investigations.

Dimensional analysis is a simple and useful rule to check the consistence of a mathematical formula or equation. Roughly speaking, this rule demands that both sides of an equation (that has any physical or chemical bearings) must have the same dimension, or unit of measurement. So for instance the equation E = mC

^{2} is consistent with the dimensional analysis, since both sides of the equation have

Energy as dimension. The equation E=mC is not consistent, since the right side would have

momentum as dimension.

Similarly, if a formula is composed of multiple terms by addition, then all terms must have the same dimension. So for instance, the formula: S:=π·r

^{2} + r

^{2}/2 makes sense, but not S:=π·r

^{2} + r/2.

Now, let's turn to a problem of

multidimensional scaling (MDS) from the perspective of the dimensional analysis. Roughly speaking, MDS maps data points from an abstract space with a distance function to a low dimensional, normally, Euclidean space like R

^{2}. The goal of a MDS algorithm is to construct the map in a way, so that the distance in the low dimensional space approximates the distance in the original abstract space.

More formally, let D

_{ij} be the distance between two data points in the original space, d

_{ij} be the distance low dimensional space after MDS mapping; Then a MDS algorithm tries to minimize the following quantity (often called

stress):

S := Σ ( D

_{ij} - d

_{ij})

^{2}Now, from the view point of dimensional analysis the stress function in its above form is

not consistent, since D

_{ij} and d

_{ij} have arguably different dimensions as they measure distance in different spaces.

One work-around to solve this inconsistency issue is to add a special constant or apply a monotonous function on d

_{ij} like:

S := Σ ( D

_{ij} -

C·d

_{ij})

^{2} or S := Σ ( D

_{ij} -

f(d

_{ij}))

^{2}The constant

C or the function

f should magically convert the dimension of d

_{ij} to that of D

_{ij}. This kind of ad-hoc techniques look pretty artificial. Most scientists probably won't like this method.

It should be noticed that the RPM (relational perspecitve map) algorithm avoided this inconsistency by using the following stress function:

S := Σ D

_{ij}/d

_{ij}The dimensional analysis rule does not prohibit composing terms of different dimensions by multiplication. In above formula, both sides of the equation have a completely new dimension, that is the dimension of D

_{ij} divided by the dimension of d

_{ij}.