Regularised 1 norm and distance

The 1-norm of a vector and the distance between two vectors \(\mathbf{a}\) and \(\mathbf{b}\) in \(\mathbb{R}^n\) are given respectively, see Figure The unitspheres of the three classic norms. Right: comparison with the regularised 1 norm of parameter \epsilon=0.2 by

\[||\mathbf{a}||_1 :=\sum_{i=1}^n|a_i|, \qquad d_1(\mathbf{a},\mathbf{b}):=\sum_{i=1}^n|a_i-b_i|.\]

The unitspheres of the three classic norms. Right: comparison with the regularised 1 norm of parameter \(\epsilon=0.2\)

Their regularised counterpart is made such that it is possible to differentiate the expression and avoid divisions by zero by introducing a parameter \(\epsilon\). The corresponding definitions are

\[||\mathbf{a}||_{1,\epsilon} =\sum_{i=1}^n\dfrac{a_i^2}{\sqrt{a_i^2+\epsilon^2}}, \qquad d_{1,\epsilon}(\mathbf{a},\mathbf{b}) =\sum_{i=1}^n\dfrac{(a_i-b_i)^2}{\sqrt{(a_i-b_i)^2+\epsilon^2}}.\]