Prev: hpgcc and rpl-stack
Next: A bug and a related mini-challenge re LSQ [what's in a word?]
From: John H Meyers on 4 May 2008 13:56
My stab at a hasty "review of thread"
(at least the part following acknowledgment of LSQ bug),
from my own end of the telescope:

> LSQ generalizes the idea of solving an equation set,
> by seeking to minimize the Euclidean norm of 'A*X-B',
> which is always possible,
> rather than the goal to satisfy 'A*X=B',
> which is not always possible.

The analogy to "least squares line" (LR) comes to mind
(may be special case of LSQ, but no matter,
it has its own command).

We have this set of points. Sometimes they all
lie on the same straight line, but when they don't,
we'd like to find one line which "best fits" the collection.

The thought to make that line minimize the Euclidean norm
of the deviations occurred to someone. The resulting line
may not pass through any of the points at all,
but it does have some "neat" properties,
such as passing through a point whose cordinates
are the arithmetic averages of all the individual coordinates.

An input set could exist where every point except one
does lie on one straight line, with one "outlier,"
but LR will still choose a line which passes through none,
though is "close" to most.

One may analogously have some set of equations, rather than points.
There may be no X which "passes exactly through"
the constraints of all the equations,
but one can find an X (sometimes many)
which minimize some similar norm,
and this also may or may not "pass through" (exactly solve)
any of the individual equations, etc.

The AUR, with reference to LSQ:

It uses the word "solution" in contradictory ways,
speaking both of "minimizing Euclidean norm"
and "solution to equations where A*X=B" on the same page,
as if LR had said "a line minimizing Euclidean norm of differences"
_and_ "passing through all the points" at the same time,
which it would seem most propitious to resolve
by striking out the latter erroneously restrictive phrase,
which applies only to a special case,
whereas LR (or LSQ) are meant to address more general problems,
sometimes reducing to the special case.

Other proposals to resolve LSQ contradiction:

Say that the word "solution" is _undefined_ in LSQ,
instead define four new _two-word_ phrases,
each containing the undefined word "solution,"
each with its own special definition (trying hard not to
accidentally use undefined original word "solution" in definitions).

Or, strike out the AUR's erroneously restrictive phrase
"solution to equations where A*X=B" on LSQ page,
give "solution" the precise "minimize Euclidean norm"
_definition_ (analogous to LR) that otherwise appears
three times as a "qualifier" to the phrase "solution."

When the latter is done, the set of four different two-word phrases
that each contained the word "solution" are no longer necessary,
at least not to define LSQ. I say that the AUR does not need
to prefix "solution" with different words in different places
(any more than LR needs anything like this hair-splitting)
but just to use the appropriate precise definition
in the first place -- or else don't use the word at all :)

The result delivered by LSQ can be characterized
in one general sentence,
without even using the word "solution," perhaps as can LR,
but much more can be explored about the nature of various results
which arise from various different classes of inputs,
which can offer a more complete understanding
of how the "minimal norm" problem (LSQ)
relates to the more restricted "solving equations" problem,
much as LR relates to the more restricted "draw one 2D line
through all these 1D points" (or "draw one hyperplane
through all these n-dimensional points" etc.)

[r->] [OFF]
 | 
Pages: 1
Prev: hpgcc and rpl-stack
Next: A bug and a related mini-challenge re LSQ [what's in a word?]