Function Approximation through Basis Functions
Overview
The basis function approximation seeks to approximate a given function {% f(x) %}
through a series of basis functions {% f_i %} as in the following
{% f(x) = \sum_{j=1}^m c_j f_j(x) %}
The coefficients {% c_i %} are sought to minimize a given los
loss function.
For this analysis, we take the loss function to be the squared error.
{% loss = \sum_{i=1}^n (y_i - f(x_i))^2 %}
Now define the matrix {% B %} as
{% B_{i,j} = f_j(x_i) %}
The vector {% \vec{c} %} that minimizes the squred error loss function above is given by
{% \vec{c} = B^{\dagger} \vec{y} %}
where {% B^{\dagger} %} is the Moore-Penrose inverse.
