Decision boundary

# Decision boundary

Discussion

Encyclopedia
In a statistical-classification problem with two classes, a decision boundary or decision surface is a hypersurface
Hypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...

that partitions the underlying vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

into two sets, one for each class. The classifier will classify all the points on one side of the decision boundary as belonging to one class and all those on the other side as belonging to the other class.

If the decision surface is a hyperplane
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

, then the classification problem is linear, and the classes are linearly separable
Linearly separable
In geometry, two sets of points in a two-dimensional space are linearly separable if they can be completely separated by a single line. In general, two point sets are linearly separable in n-dimensional space if they can be separated by a hyperplane....

.

Decision boundaries are not always clear cut. That is, the transition from one class in the feature space to another is not discontinuous, but gradual. This effect is common in fuzzy logic based classification algorithms, where membership in one class or another is ambiguous.

## In ANNs and SVMs

In the case of backpropagation
Backpropagation
Backpropagation is a common method of teaching artificial neural networks how to perform a given task. Arthur E. Bryson and Yu-Chi Ho described it as a multi-stage dynamic system optimization method in 1969 . It wasn't until 1974 and later, when applied in the context of neural networks and...

based artificial neural network
Artificial neural network
An artificial neural network , usually called neural network , is a mathematical model or computational model that is inspired by the structure and/or functional aspects of biological neural networks. A neural network consists of an interconnected group of artificial neurons, and it processes...

s or perceptron
Perceptron
The perceptron is a type of artificial neural network invented in 1957 at the Cornell Aeronautical Laboratory by Frank Rosenblatt. It can be seen as the simplest kind of feedforward neural network: a linear classifier.- Definition :...

s, the type of decision boundary that the network can learn is determined by the number of hidden layers the network has.
If it has no hidden layers, then it can only learn linear problems. If it has one hidden layer, then it can learn problems with convex decision boundaries (and some concave decision boundaries). The network can learn more complex problems if it has two or more hidden layers.

In particular, support vector machine
Support vector machine
A support vector machine is a concept in statistics and computer science for a set of related supervised learning methods that analyze data and recognize patterns, used for classification and regression analysis...

s find a hyperplane
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

that separates the feature space into two classes with the maximum margin
Maximum-margin hyperplane
In geometry, a maximum-margin hyperplane is a hyperplane which separates two 'clouds' of points and is at equal distance from the two. The margin between the hyperplane and the clouds is maximal. See the article on Support Vector Machines for more details....

. If the problem is not originally linearly separable, the kernel trick
Kernel trick
For machine learning algorithms, the kernel trick is a way of mapping observations from a general set S into an inner product space V , without ever having to compute the mapping explicitly, in the hope that the observations will gain meaningful linear structure in V...

is used to turn it into a linearly separable one, by increasing the number of dimensions. Thus a general hypersurface in a small dimension space is turned into a hyperplane in a space with much larger dimensions.

Neural networks try to learn the decision boundary which minimizes the empirical error, while support vector machines try to learn the decision boundary which gives the best generalization.