Radial Basis Function (RBF) Networks

• a type of instance-based learning and artificial neural network constructed from spatially localized kernel functions. These can be seen as a blend of instance-based approaches (spatially localized influence of each kernel function) and neural network approaches (a global approximation to the target function is formed at training time rather than a local approximation at query time)

• RBF networks have been used successfully in applications such as interpreting visual scenes, in which the assumption of spatially local influences is well-justified

Hypothesis Representation

• 𝑓ˆ(𝑋) = 𝑤ₒ + 𝛴_{1≤𝑖≤𝑘}[𝑤ᵢ * 𝐾ᵢ(𝑑(𝑥ᵢ,𝑥))]

where:

• each 𝑥ᵢ is an instance from 𝑋
• 𝐾ᵢ(𝑑(𝑥ᵢ,𝑥)) is a kernel function that is inversely proportional to distance 𝑑(𝑥ᵢ,𝑥)
• 𝑑 is a user-provided constant

Kernel Function 𝐾ᵢ(𝑑(𝑥ᵢ,𝑥))

can be a gaussian function

• 𝐾ᵢ(𝑑(𝑥ᵢ,𝑥)) = 𝑒^{(1/2𝜎ᵢ²) * 𝑑²(𝑥ᵢ,𝑥)}

where:

• mean = 𝑥ᵢ
• variance = 𝜎ᵢ²