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Clusters

Convex Representation

The Caratheodori Theorem tells us that any point \(x \in \textrm{Conv}(T)\), where \(T \subset \mathcal{R}^n\), can be represented as a convex combination of at most \(n+1\) points.1

So for example let's say that our features \(x \in \mathcal{R}^n\). Then for any point in the convex hull of the training set can be represented as the convex combination of at most ten points

\[x = \sum _{i=1}^{11} \alpha_i(x)x_i^*(x), \quad x \in \mathcal{R}^{10}\]

  1. Reference: See here