## Decision boundaries

Separating positive and negative example with a straight line that is as far as possible from both positive and negative examples, a **median that maximizes the space between positive and negative examples**.

Constraints are applied to build a **support vector** *(u)* and define a constant *b* that allow to sort positive examples from negative ones. The width of a “street” between the positive and negative values is maximized.

Going through the algebra, the resulting equation show that **the optimization depends only on the dot product of pair of samples**.

The **decision rule** that defines if a sample is positive or negative only depends on the dot product of the sample vector and the unknown vector.

### No local maximum

Such support vector algorithm can be proven to be evolving in a convex space, meaning that **it will never be blocked at a local maximum**.

### Non linearity

The algorithm cannot find a median between data which cannot be linearly separable. A **transformation** can however be applied to the space to reorganize the samples so that they can be linearly separable. Certain transformations can however create an **over fitting** model that becomes useless by only sorting the example data.