In linear algebra, functional analysis, and related areas of mathematics, a norm (l-norms) is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero. A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors (in addition to the zero vector).

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-norm L2-norm
Robust Not robust
Unstable solution Stable solution
Possible multiple solutions Only one solution

 

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