# Characterize a point cloud

Background: I have multiple point clouds (sets of objects) $$\{S_i\}_{i\in\mathbb{N}}$$ of variable size and "purity" (meaning that some sets contain very similar objects, some show a high diversity; some sets are large, some are small). Each object is characterized by a $$n$$-dimensional continuous feature vector $$v_j \in \mathbb{R}^n$$, each set can therefore be seen as a matrix $$M_i \in \mathbb{R}^{|S_i| \times n}$$.

I want to classify these point clouds using a binary classifier into "pure" and "diverse".

Idea: I was thinking about covariance. This way every point cloud is represented by its $$n \times n$$ covariance matrix.

Question: What other features can I use to characterize the individual point clouds? How else can I transform these matrices of different shape to representations of fixed length representations that can be passed on to a classifier?

• Not sure why this is getting downvoted -- looks like a valid question to me. – Aryeh Oct 22 '19 at 19:02

You can use essentially any centroid-based clustering algorithm to define a notion of "coherence" or "purity" of a cluster. Let's say you've chosen some particular metric $$||\cdot||$$ on $$\mathbb{R}^n$$ (I'm using norm notation but any metric will do). Using your notation, $$S_i$$ is the $$i$$th "cluster". For each $$S_i$$, find the* point $$c_i\in\mathbb{R}^n$$ that minimizes $$F(c):=\sum_{x\in S_i}||x-c||\qquad(*)$$ over all $$c\in\mathbb{R}^n$$. Let $$F(c_i)$$ be the purity score of $$S_i$$; it might actually be better to consider the normalized version $$F(c_i)/|S_i|$$. Smaller values means more homogeneity or "purity".
• For a true centroid you'd need to write $||x-c||^2$ in (*); as stated it's more like a generalized median. – Aryeh Oct 22 '19 at 20:16