The quadratic performance surface of an adaptive filter is a paraboloid. Its minimum can be found wherever the gradient is zero. However, since there are two types of paraboloids (elliptical and hyperbolic), is there a way to tell if the minimum detected is a global minimum or just a saddle point?


The quadratic surface is determined by the autocorrelation matrix of the data, which is always positive definite or positive semi-definite. This means that any stationary point is always a minimum. In the worst case, this minimum is not unique if the matrix is singular, but it can never be a saddle point.

  • $\begingroup$ Can you please explain how it can never be a saddle point? I assumed that if the quadratic surface is a hyperbolic paraboloid (in which one parabola is turning upwards and the other downwards), a saddle point exists where the gradient is zero. $\endgroup$ – user1832413 May 3 '15 at 19:54
  • 1
    $\begingroup$ @user1832413: If you have a quadratic form $x^TAx+b^Tx$, and if the matrix $A$ is positive (semi-)definite (as is the case with an autocorrelation matrix), then the function defined by the quadratic form is convex and has a minimum. A saddle point can only occur if the matrix $A$ is indefinite. $\endgroup$ – Matt L. May 3 '15 at 20:00
  • $\begingroup$ for the LMS adaptive filter (Widrow), it can't be a saddle point because the function is magnitude square error which can't go negative. a minimum point close to zero can only go up on the sides, not down. $\endgroup$ – robert bristow-johnson May 4 '15 at 1:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.