decisionboundary

A decision boundary is a concept in supervised classification that separates the input feature space into regions associated with different predicted classes. It is the locus of points where the model’s predicted class would change, typically where the model is indifferent between two or more classes or where a probability crosses a chosen threshold.

In binary classification, the decision boundary is often defined by a decision function f(x) such that the predicted class switches when f(x) = 0 or when a probability equals 0.5. A common example is a linear classifier, where the boundary is a hyperplane given by w^T x + b = 0. In two dimensions this is a straight line; in higher dimensions it is a hyperplane.

Nonlinear decision boundaries arise with models that use nonlinear transformations of the input. Kernel methods, neural networks, and decision trees can yield curved or highly irregular boundaries. The underlying idea remains the same: the boundary partitions the space into regions where different class decisions are made.

In multiclass classification, there may be multiple decision boundaries resulting from one-vs-rest or pairwise schemes, or from a model that assigns each point to the class with the highest score. The boundary is where two classes have equal score, or where the argmax changes from one class to another.

Practically, the shape and location of a decision boundary depend on the training data and the chosen model, including regularization and capacity. Boundaries that overfit reflect noise, while overly simple boundaries may underfit. Visualization is common in low-dimensional spaces, but in high dimensions the boundary is inferred indirectly through model parameters and performance metrics.