A set $S$ is linearly separable if each $x\in S$ can be separated from the rest of $S$ by a linear functional. We study random $N$-element sets in $\mathbb{R}^n$ for large $n$ and demonstrate that for $N<a\exp(b{n})$ they are linearly separable with probability $p$, $p>1-\vartheta$, for a given (small) $\vartheta>0$. Constants $a,b>0$ depend on the probability distribution and the constant $\vartheta$. The results are important for machine learning in high dimension, especially for correction of unavoidable mistakes of legacy Artificial Intelligence systems.
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