In pervious post, we shows the Gauss-Newton method for fitting non-linear function. The disadvantage of that method is that the inverse matrix could be ill-defined. This makes the method unstable.

Back to the basic, we want to minimize the sum of square of residual (SSR). The SSR is,

SSR(\beta) = (Y - f(\beta))^T\cdot (Y-f(\beta))

The derivative on \beta,

\frac{d}{d\beta} SSR(\beta) = -2 (Y-f(\beta))^T \cdot \nabla f(\beta)

Many literatures denote \nabla f = J, which is the Jacobian. The second derivative of f is Hessian matrix H = \nabla^2 f \sim J^T\cdot J.

The Gradient Descent method is that ,

h = \beta - \beta_0 = \alpha J^T \cdot (Y - f(\beta_0))

where \alpha is a step size. The gradient descent changes the SSR using the steepest path. The step size \alpha has to be adjusted. The simplest way to adjust is testing the \delta = SSR(\beta_0 + h) - SSR(\beta_0). If \delta < 0 , the \alpha increases, else decreases. This method is slow but stable. It is slow because of finding the \alpha . It is stable because the method is always computable.

Thus, we have 2 methods, Gradient Descent is stable and slow, Gauss-Newton method is unstable but fast. Levenberg-Marquardt Algorithm combined this 2 methods so that it is stable and fast by solving,

(J^T \cdot J + \lambda I) h = J^T \cdot (Y - f)

where \lambda is an adjustable parameter. When \lambda >> 1 , the J^T\cdot J is neglectable and the method becomes Gradient Descent with small \alpha. When the \lambda << 1, the method becomes Gauss-Newton method.

Usually, the \lambda_0 is small. The Gauss-Newton method is very good near the minimum of SSR, while Gradient Descent is better far away.

When the \delta < 0, \lambda_{i+1} = \lambda_i / 10 , else \lambda_{i+1} = \lambda_i * 10. I don’t know the exact reason for this setting. In fact if you set oppositely, the method is still work in most cases.

The method add \lambda I on the J^T\cdot J , the inverse is always well-define. Therefore, this method is stable.