L-Norms as Loss Function

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L1-Norm loss function is known as least absolute deviations (LAD).

 S = \sum_{i=1}^n|y_i -f(x_i)|

It is basically minimizing the sum of the absolute differences S between the target value Y_i and the estimated values f(x_i).

L2-Norm loss function is known as least squares error (LSE).

S = \sum_{i=1}^n(y_i -f(x_i))^2

It is basically minimizing the sum of the square of the differences S between the target value Y_i and the estimated values f(x_i)

Differences between L1-L2 norm

The differences of L1-norm and L2-norm as a loss function are the following.

L1-normL2-norm
RobustNot robust
Unstable solutionStable solution
Possible multiple solutionsOnly one solution

 

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