How to get a maximum and minimum score of dot product similarity metric?

Question

I use this library (https://www.sbert.net/examples/applications/semantic-search/README.html) to perform an information retrieval task. There are two options for the similarity function: dot product and cos similarity. I currently know that the cos similarity result will range from 0–1. But the problem is with the dot product. Based on what I read from several sources, the result of this dot product can be any real number. I want to create a non-confusing similarity metric for the user with a consistent scale, but I don't know how to scale it consistently using the dot product function (without losing information by not normalizing the vectors). Can anyone help me?

Answer 1

It looks like they are using an inner product space to perform their searches. There are different inner products on different vector spaces, the more common being you multiply the x components to gether, the y components by the y components, and so on, then add up all those products, it's usually indicated as $ \vec{a} \cdot \vec{b}$ where $\vec{a}$ and $\vec{b}$ are vectors. In general, an inner product is a mapping of the cartesian product of two vector spaces onto the reals.

Once you have a dot product, you can define the norm of a vector $|\vec{a}|=\sqrt{\vec{a}\cdot\vec{a}}$. That in turn allows you to specify a vector as a unit vector having its norm equal to $1$. Specifically, $\hat{a}=\frac{\vec{a}}{|\vec{a}|}$

In the case of spatial vectors, $\vec{a} \cdot \vec{b}=|\vec{a}||\vec{b}|\cos{\theta}$, where $\theta$ is the angle between them. This allows you to define an angle between a pair of vectors even if the vectors are more abstract objects.

In particular, vectors can "overlap" in a sense, one vector can be projected onto another. If the vector is the hypotenuse of a right triangle, its length times its cosine with the other vector as an adjacent side gives you the vector projection, i.e. $\vec{a}\cdot \frac{\vec{b}}{|\vec{b}|}=|\vec{a}|\cos{\theta}. $ Notice their is no overlap if the angle between the is ninety degrees, i.e. they are perpendicular.

Looks like the search generalizes these concepts to detect overlaps or cosine similarities.