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?