I'm trying to follow a research paper

Question

The paper shows an equation to minimize. That makes perfect sense. Then, the paper says:

"The optimal solution to the least squares problem [above] is found by differentiation as a solution of a linear system of equations."

I am very familiar with traditional linear algebra least squares:

y = xb + e
solve for coefficients b by:
b = (x^T * x)^-1 * x^T * y

I understand the equation to be minimized but I don't understand the formulas that follow, and I think the paper is using some alternative least squares approach that I am not familiar with. It doesn't look like the paper is doing regular least squares (like I know). Any ideas what the paper is referring to? How do you use differentiation as a solution to a linear system of equations?

Answer 1

Differentiation is used when the equation is nonlinear. To parameterize the equation, partial derivatives are used to construct the design matrix (in your case, that would be the b matrix.) x would be the unknown vector, e would be the residual vector and y would be your observation vector.