# Topological approach to create a space between clouds

I have a dataset associated with labels. According to https://arxiv.org/pdf/1802.03426.pdf --> UMAP (Uniform Manifold Approximation and Projection) which is a novel manifold learning technique for dimension reduction and the data, I succeeded to create the green and red clouds bellow. The problem I have is they are stick together. For machine learning purposes, it is kinda hard to learn something when the clouds are placed that way.

Is there a topological approach that might be used to create a significant space between clouds?

UPDATE

I would be interested by an analytic approach to separated the two clouds. Each cloud can be seen as a compact space.

Here is an example in 2-D. I would like a way to generalize that concept in z-D, where z would be a finite positive integer.

I want to create an algo which will be used with high-frequency speed. So the algo needs to be fast enough. I believe the "Uniform Manifold Approximation and Projection" to be tweakable so that I can preprocess the data and pass it to an LSTM. The idea are is three steps : 1- Reduce the dimensionality, separated the clouds and then pass the data to a LSTM model.

UPDATE 2

I am trying an approach, but I am far from certain that it's the best solution.

1. Cover each cloud by the smallest possible sphere.
2. Extend the intersection of the spheres by an hyperplane.
3. Taking away the clouds according to the orthogonal vector to the hyperplane by a distance alpha. alpha might be the furthest point on the orthogonal line inside the intersection of the spheres.
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• Before applying novel techniques, have you tried simpler methods such as SVM or SVM+kernel? – Neal Dec 7 at 16:23
• I am looking for an analytics technique because I tried many things so far, i.e. PCA, T-SNE, SVC, UMAP (worked better), using neural network to classified the points. I want a technique that can allow me once I apply UMAP to detach what is stuck together. The next step is really to force separating the clouds with a spectral geometry technique probably. – davegaut Dec 8 at 0:04
• What does 2. mean? And also, if you know which point is red and which is green then I don't see where is a problem. Whereas if you don't know that, I don't see how anything could be done to separate them. – მამუკა ჯიბლაძე Dec 8 at 20:49
• @მამუკაჯიბლაძე The point 2 means that if I take the intersection of the spheres, then I am sure we can form an hyperplane according to that intersection. Suppose $V = \{v_1, v_2, ..., v_n\} \subset \mathbb{R}^3$ and $W = \{w_1, w_2, ..., w_m\} \subset \mathbb{R}^3$. Suppose now the smallest sphere covering $V$ and $W$ by $S(V)$ and $S(W)$. I am sure that $S(V) \cap S(W) \subset \mathbb{R}^2$. We just have to define the hyperplane $\mathbb{H} \supset S(V) \cap S(W)$ to define the appropriate orthogonal vector. – davegaut Dec 8 at 21:06
• If I'm understanding this correctly, calculate the diameter of one of the clouds then translate it? – bianchira Dec 8 at 22:48