# A measure of similarity of real vectors independent of their dimension

I am trying to find a measure of similarity between two vectors that works for any pair of vectors v, w $$\in R^n$$ (for any n).

for example:

v1=(1,2,4) v2=(-2,4,4) -> $$sim(v1,v2) \in R$$

v1'=(0,0,2,0,3) v2'=(2,4,6,1,2) -> $$sim(v1',v2') \in R$$

I want to be able to compare the results sim(v1,v2) and sim(v1',v2); so that for any pair (v1,v2) and (v1',v2'); I can tell which pair is more "similar".

Obviously I tried using the standard norm of the euclidean distance. But I found that the result is not actually working when you compare a distance in $$R^2$$ and a distance in $$R^5$$. It penalyses less the component-wise distances as the dimension grows (see example below)

I am wondering if there is any alternative.

** clarification on why I don't like the standard norm of the euc distance **

PAIR 1) v1 = (0) , v2=(1) ---> |v1-v2| = 1

PAIR 2) v1' = (0,0) , v2'=(1,1) ---> |v1'-v2'| = $$sqrt(2)$$ = 1.41

PAIR 3) v1''= (0,0,0), v2''=(1,1,1) ---> |v1''-v2''| = $$sqrt(3)$$ = 1.73

Which pair is more "alike"? I am not sure if the norm of the euclidean distance is an appropiate metric... I think that they are all as different as two vectors in its respective spaces can be. I think that the norm of the euclidean distance is not "scaled" properly.

Any ideas on how to compare?

• You appear to be familiar with TeX formatting. It works here, too. Just surround the mathematics with dollar signs $ as you would in a normal TeX / LaTeX document. – Xander Henderson Jan 9 '19 at 14:57 • I'm not sure what you mean about euclidean norm. Example:$ v_1 = (2, 4), v_2 = (5, 3)$.$w = v_1 - v_2 = (-3, 1)$has squared length$10$. Put these in 5-space, and you get$ v_1 = (2, 4,0,0,0), v_2 = (5, 3,0,0,0)$.$w = v_1 - v_2 = (-3, 1,0,0,0)$, which also has squared length$10$. – John Hughes Jan 9 '19 at 14:58 • Expanding on what @JohnHughes says: the Euclidean distance between two points in$n\$-space is the Euclidean distance between them in the plane (or possibly line) they span. How is that sensitive to dimension? Perhpas edit the question to tell us more about where it comes from and just why Euclidean distance does not server your needs. – Ethan Bolker Jan 9 '19 at 15:02
• To do what Xander suggested, you can click on the word "edit" just below your question. – John Hughes Jan 9 '19 at 15:23
• I edited the post to be more clear! Thanks for the comments. Let me know if now the problem is easier to understand. – Jeremías Rodríguez Jan 9 '19 at 18:26

## 1 Answer

A standard sort-of solution is the "cosine similarity" (although this is usually defined for unit vectors): You compute the angle between the two vectors, thus: $$d(v_1, v_2) = \cos^{-1} \frac{v_1 \cdot v_2}{\|v_1\|\|v_2\|}$$

If $$v_1, v_2$$ are unit vectors, then you can skip dividing by the lengths, of course. The downside? If $$v_1, v_2$$ point in the same direction, but have different lengths, this "distance" still returns the value $$0$$.

The upside? If $$v_1, v_2 \in \Bbb R^2 \subset \Bbb R^5$$, and you compute the distance, you get the same answer whether your think of them as being in $$\Bbb R^2$$ or in $$\Bbb R^5$$.