Linearly separable

In Euclidean geometry, linear separability is a geometric property of a pair of sets of points. This is most easily visualized in two dimensions (the Euclidean plane) by thinking of one set of points as being colored blue and the other set of points as being colored red. These two sets are linearly separable if there exists at least one line in the plane with all of the blue points on one side of the line and all the red points on the other side. This idea immediately generalizes to higher-dimensional Euclidean spaces if line is replaced by hyperplane.

The problem of determining if a pair of sets is linearly separable and finding a separating hyperplane if they are arises in several areas. In statistics and machine learning, classifying certain types of data is a problem for which good algorithms exist that are based on this concept.

Let ${\displaystyle X_{0}}$ and ${\displaystyle X_{1}}$ be two sets of points in an n-dimensional Euclidean space. Then ${\displaystyle X_{0}}$ and ${\displaystyle X_{1}}$ are linearly separable if there exists n + 1 real numbers ${\displaystyle w_{1},w_{2},..,w_{n},k}$, such that every point ${\displaystyle x\in X_{0}}$ satisfies ${\displaystyle \sum _{i=1}^{n}w_{i}x_{i}>k}$ and every point ${\displaystyle x\in X_{1}}$ satisfies ${\displaystyle \sum _{i=1}^{n}w_{i}x_{i}<k}$, where ${\displaystyle x_{i}}$ is the ${\displaystyle i}$-th component of ${\displaystyle x}$.

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