Equivalent Discriminant Function and Examples

The Discriminant function can be modified without changing the classification outcome.

• Scaling by a positive constant

$$g_i(x)\to \alpha g_i(x),\alpha >0$$

• Adding a constant

  $$g_i(x)\to \alpha +g_i(x),\alpha \in \mathbb{R}$$

• Applying a function

  $$g_i(\mathbf{x})\to f\left(g_i(\mathbf{x})\right)$$

These transformations are often used to simplify the mathematical calculations involved in classification. Where $f(\cdot )$ is a monotonically increasing function

Use cases

• Improving the confidence of the decision by having a very high $\alpha$
• Using $\alpha$ in the context of a multi-class problem to normalize each classification.

Q

How can you simplify a discriminant function without changing the classification result?

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