L 1 Regression LS (L 2 ) is strongly affected by outliers

1. L1 Regression LS (L2) is strongly affected by outliers If outliers are due to incorrect measurements, the inversion should minimize their effect on the estimated model. Effects of outliers in LS is shown by rapid fall-off of the tails of the Normal Distribution In contrast the Exponential Distribution has a longer tail, implying that the probability of realizing data far from the mean is higher. A few data points several  from <d> is much more probable if drawn from an exponential rather than from a normal distribution. Therefore methods based on exponential distributions are able to handle outliers better than methods based on normal distributions. Such methods are said to be robust.

2. L1 Regression min ∑ [di -(Gm)i]/i = min ||dw-Gwm||1 thus more robust to outliers because error is not squared Example: repeating measurement m times: [1 1 … 1]T m =[d1 d2 … dm]T mL2 = (GTG)-1GTd = m-1∑ di f(m) = ||d-Gm||1 = ∑ |di-m| Non-differentiable if m=di Convex, so local minima=global minima f’(m) = ∑ sgn(di-m), sgn(x)=+1 if x>0, =-1 if x<0, =0 if x=0 =0 if half is +, half is - <d>est = median, where 1/2 of data is < <d>est, 1/2 > <d>est

3. L1 Regression Finding min ||dw-Gwm||1is not trivial. Several methods available, such as IRLS, solving a series of LS problems converging to a 1-norm: r=d-Gm f(m) = ||d-Gm||1 = ||r||1 = ∑ |ri| non-differentiable if ri=0. At other points: ∂f(m)/∂mk = - ∑ Gi,k sgn(ri) = -∑ Gi,k ri/|ri| f(m) = -GTRr = -GTR(d-Gm) Ri,i=1/|ri| f(m) = -GTR(d-Gm) = 0 GTRGm = GTRd R depends on m, nonlinear system :( IRLS!