in the area of image processing, I'm trying out different dimensionality reduction techniques. I use these to reduce intensity images/bitmaps (sliced as a vector, i.e. such a vector has a high dimensionality, e.g. 6400 for a 80x80 image) to a lower, k-dimensional space (e.g. k=12). One of these techniques is PCA, and as you know, there are many others.
- N point clouds (i.e. a set of k-dimensional real-valued coordinates, refered to as "point")
- Each point cloud has some number of points (e.g. n_i)
- No knowledge about the distribution of the points for each point cloud shall be assumed (although it's probably somehow Gaussian-like distributed, but not uni-variate)
Needed: the amount of overlap of these point clouds. Or, in other words, how separable the data is. I really have no clue how intensive of a problem this is. Visually, I conceive the solution like building the k-dimensional convex hull for every point cloud, calculating the volume of total overlap.
If the distribution was just "spherical" (i.e. uni-variate Gaussian) it would be trivial, I could simply divide 'no of points that are further away from their own cloud center than 0.5*distanceToCenterOfOtherClosestPointCloud' by 'total no of points'
I'm also sorry if I should have made use of the wrong tags. I'm by no means a mathematician.