absolute homogeneity

Absolute Homogeneity

A function has absolute homogeneity if, for any scalar $\alpha$ and input $x$, the following condition holds:

$$f(\alpha x)=|\alpha |f(x)$$

This means that scaling the input by a factor $\alpha$ results in the output being scaled by the absolute value of that factor.

euclidean norm

$$||\alpha x|{|}_2=\sqrt{\sum _i(\alpha x_i{)}^2}=\sqrt{{\alpha }^2\sum _i{x}_i^2}=|\alpha |\sqrt{\sum _i{x}_i^2}=|\alpha |||x|{|}_2$$