Positive homogeneous functions are functions $f:\mathbb{R}\to\mathbb{R}$ that satisfy $$ f(cx) = cf(x) $$ for all $x\in \mathbb{R}$ for $c>0$. Now I was reading the paper http://arxiv.org/abs/1611.02862, and there the definition of locally positive homogeneous functions was given as $$ f(cx) = cf(x) $$ for all $x\in\mathbb{R}$ and $|c-1|\leq \epsilon\ll 1$.
An example of a positive homogeneous function is $$ f(x) = \max(0,x) = ReLU(x). $$ What kind of functions are locally positive homogeneous? Can we find examples of these that are not (globally) positive homogeneous functions? In particular, are there any nonlinear ones?
What do I know of these functions:
- $\frac{df(x)}{dx}x=f(x)$
- $f(0)=0$
Property 1) and 2) describe a PDE: $$ \frac{df(x)}{dx} = \frac{f(x)}{x}, \\ f(0) = 0. $$
To find out what kind of functions the locally homogeneous functions are, we can solve the this PDE. This can be done by separation of variables and using logarithmic derivatives. First, separate $f(x)$ and $x$: $$ \frac{1}{f(x)}\frac{df(x)}{dx} = \frac{1}{x}. $$ Then integrate: $$ \log(f(x)) = \log(x)+c_0 $$ where $c_0$ is the integration constant. Exponentiate both sides: $$ f(x) = \exp(\log(x)+c_0) = c_1\exp(\log(x)) = c_1x $$ with $c_1=\exp(c_0)$. So all differentiable univariate locally positive homogeneous functions $f$ are linear functions.
Now property 1) does not apply to the origin, so the function can have a weak derivative there (continuity must hold). Therefore, all univariate locally positive homogeneous functions $f$ are continuous piecewise linear functions.
Both suggest that all locally positive homogeneous functions are globally positive homogeneous functions, and vice versa.