What is the intuitive of the perspective of a function

Question

I was reading the book convex optimization by Boyd and Vandenberghe,while reading the chapter about convex functions,there is a part about operations that preserves convexity and the perspective of a function is one of those operations,according to the book:

I understand the definition and the proof why it preserves convexity,but in the earlier chapter there is a perspective function

I find these two functions very similar and i like the intuitive it gives about the perspective function,so I just wonder is there an intuitive understanding of the the perspective of a function and is it a generalized form of the perspective function?

Answer 1

If the perspective function $P$ is like the action of a pinhole camera, then the perspective of a function $f$ is the function $g$ whose graph when viewed through a pinhole camera looks like the graph of $f$. This is like the action of a projector, or a laser light show.

Briefly, $(\mathbf x,t,y)$ is in the graph of $g$ if and only if $P(\mathbf x,y,t) = (\mathbf x,y)/t$ is in the graph of $f$. (Note that the coordinates have been rearranged slightly to make sure that perspective is applied along the $t$ axis.)

All this makes more sense if you first define the perspective of a set $\mathcal S\subseteq\mathbb R^n$ as the set of points in $\mathbb R^{n+1}$ which the perspective function maps into $\mathcal S$, show that this operation preserves convexity of sets, and apply it to the epigraph of $f$. This is left as an exercise for the reader.

Answer 2

An alternative description of the function $g$ is the positively homogeneous convex function generated by $h$, where $h(x,t) = \begin{cases} f(x), & t = 1 \\ +\infty, & \text{otherwise}\end{cases}$. (See Rockafellar's "Convex Analysis".)

The operation of generating the positively homogeneous convex function is conceptually straightforward and (in my opinion) clarifies the connection to the perspective view.

The positively homogeneous convex function $g$ generated by a convex function $h$ is created as follows:

First create the convex cone generated by the epigraph of $h$, $H = \operatorname{cone} ( \operatorname{epi} h)$, and then define $g$ as the function whose epigraph is $H$.

Note that $((x,t), \alpha) \in \operatorname{epi} g $ iff there exists $s>0$ such that $st =1$, $f(sx) \le s \alpha$ iff $t f({1 \over t} x) \le \alpha$