[EM] Method Definition Considerations

[Kristofer Munsterhjelm]

On 05.06.2022 03:14, Forest Simmons wrote:
> Traditional math exposition, whether text book, lecture, journal
> article, or monograph,  tends to have a top down deductive logical
> structure that belies all of the messy trial and error scratch work that
> accompanied the creative process.
> 
> This was the style of Gauss ... unveil the finished work in all of its
> polished, deductive logical glory, without any hint of the inductive
> scaffolding or chisel marks that went into the finished work.

This is a little bit off topic, but one thing I wished someone would've
told me at university is just how difficult it is to come up with the
one trick that makes a proof work.

You tend to see (in mathematics/compsci papers at least) something like:

"We wish to reduce minimizing the spectral norm of this matrix function
of M to a convex second order cone program, which we can do by
augmenting the input matrix M like so

	       0   I
	W =    M   M^T - I

and solving the associated optimization program, minimize f^Tx subject
to ||Wx||_2 < b as follows..."

There's then a brief proof of why just this choice of W works, but the
authors might as well have pulled the transformation out of a hat. The
problem is that this makes everything seem so easy: you just proceed
through the steps and the intended relation falls out at the end.

Then try to do it yourself in a novel setting and it's not so easy anymore.

Maybe students just want to know how, but I would say any aspiring
researcher would also want to know why... or at least in my case, know
that it's perfectly normal to be hitting one's head against a wall for a
long time before finding just *what* trick to use!

More generally, some problems are much easier to phrase than they are to
solve. (Particularly number theory, hence Barry Mazur's quote.) I think
providing an intuition of just what kind of problems are deceptive or
merely hard would be a good idea, although I'm not sure how to do it :-)

-km