Scale and Direction: Understanding Homogeneous Functions

TL;DR: This post collects useful facts about homogeneous functions—functions satisfying f &#x00028; a x &#x00029; &#x0003D; a r f &#x00028; x &#x00029; . Key insight: they decompose into spherical behavior plus radial scaling. I also connect to two references in ML. A function f is (positively) r-homogeneous if f &#x00028; a x &#x00029; &#x0003D; a r f &#x00028; x &#x00029; for all a &#x02265; 0…