# Are the RLS filter and Kalman filter gradient methods?

I would like to extend my previous question What is difference between LMS and gradient-descent adaptation? with this other question.

I found out, that RLS and Kalman filter learning seems to be somehow similar. My question is: Can be those algorithms called gradient descent methods? If not, how is this kind of algorithms called?

• Recursive least squares (RLS) filters don't use gradient descent. As their name suggests, they use a least-squares fit to determine the optimum coefficients at each time step. Via clever formulation of the filter structure, one can use the calculations done from time step $n$ to recursively calculate the updated coefficients for time step $n+1$ without having to do the full least-squares fit again. – Jason R May 13 '16 at 12:23
• @JasonR Good to know. Does this approach a common name? Can you create an answer from your comment please? – matousc May 23 '16 at 9:27
• Done. I'm not sure what you mean about the name of the approach; it's recursive least squares. – Jason R May 23 '16 at 12:50

## 1 Answer

Recursive least squares (RLS) filters don't use gradient descent. As their name suggests, they use a least-squares fit to determine the optimum coefficients at each time step. Via clever formulation of the filter structure, one can use the calculations done from time step $n$ to recursively calculate the updated coefficients for time step $n+1$ without having to do the full least-squares fit again.

• well, the least square solution actually nullify the gradient at each step. – LJSilver May 23 '16 at 13:53
• //Via clever formulation of the filter structure,// You mean the lattice structure? – Naveen May 23 '16 at 15:08
• Actually, this is far from optimal. And the $\lambda$ dependence turns it a parametric algorithm, such any standard gradient procedure. – Brethlosze Jan 16 '17 at 5:31