1
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ Not sure why this is getting downvoted -- looks like a valid question to me. $\endgroup$ – Aryeh Oct 22 '19 at 19:02
1
$\begingroup$

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".

*How to do this computationally? Try (sub)gradient descent. If the minimum is not unique then break ties arbitrarily.

$\endgroup$
  • $\begingroup$ Thanks for your answer! If I chose the euclidean metric, c_i would just be the centroid of the set, right? And the normalized version would be the average distance of every point to this centroid. Although this certainly makes sense, I was looking for sth more sophisticated that is (more) independent of absolute distance and more about the overall structure of the point cloud. $\endgroup$ – moi Oct 22 '19 at 20:09
  • $\begingroup$ For a true centroid you'd need to write $||x-c||^2$ in (*); as stated it's more like a generalized median. $\endgroup$ – Aryeh Oct 22 '19 at 20:16
  • $\begingroup$ I see. Is there a reason why would I approximate this "generalized median" using gradient descend if I could also used the centroid that I can calculate in closed form? $\endgroup$ – moi Oct 23 '19 at 9:07
  • $\begingroup$ Not at all -- feel free to use the centroid IF you believe that the Euclidean metric is the appropriate one for your problem. $\endgroup$ – Aryeh Oct 23 '19 at 9:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.