I think it would help to say uniform is on the Haar measure

is that explicable in less technical language?

very few people will know what it means

or, less intimidatingly, is a distribution invariant under applying any fixed orthogonal matrix to either side

actually, i guess what is important is to emphasize that this is a continuous distribution, and so being uniform is a matter of density

Matrix of reals I assume

I added a quote from the wiki

@HWalters yes they have to be real

unlike in the discrete case, a one-to-one mapping of values from a uniform distribution might not lead to a uniform distribution

would you allow the following strategy: generate random matrices up to machine precision, test if they are orthogonal up to machine precision, accept the first that succeeds?

oh, wait, that's rule out by the running time

it is indeed :)

I did anticipate that

:)

:)

Real numbers can't be represented on a computer, so mathematical definitions using them don't really apply.

What is "orthogonal" or "uniform" for IEEE 754 numbers?

yes, i was wondering if there are practical ambiguities due to machine precision

I don't want to make this the focus. Do I really need to say how close to the identity M^T M is?

I added that I should be the identity to 4 decimal places

I really don't want this to be about machine precision

that's fair

might special orthogonal make for a cleaner challenge?

what is special orthogonal?

determinant 1

is that over a group?

hmm..

it's a subgroup: all orthogonal matrices have determinant 1 or -1

there's 2 identical connected components

it's an interesting question

why would that be cleaner though?

i suspect the most common method to generate an orthogonal matrix would be to generate a special orthogonal one, then with 50% chance flip the first row

Why would you post the question in the middle of a discussion of how to improve it?

@feersum I actually posted it before but the notice came up in the middle

but your question still holds

it probably wasn't the best move.. the problem is I need to go to sleep

i'd suggest at least a note that this is orthogonal, not special orthogonal, so people don't miss the difference

when i googled random orthogonal matrix, some SO methods came out

@xnor done

thanks..it's great there are people here who are good at math!

I did think about having it as fastest-code as that is really my favourite :)

but I think there are very few people interested in those

is a library function that generates a random gaussian matrix ok?

i.e. uniform independent gaussian in each entry

@xnor as long as it isn't orthogonal then yes

how about a function that performs the gram-schmidt decomposition to orthogonalize a matrix?

I think so yes

I did wonder about excluding those

I can still change my mind if you convince me :)

when you say then "which creates orthogonal matrices", you mean that outputs a random orthogonal matrix, rather than one whose output is guaranteed orthogonal?

I meant any function that outputs a matrix which is guaranteed to be orthogonal but I see your point

hmmm

your point is quite subtle ;)

argh

i'm trying to figure out if orthogonalize(random_gaussian(n)) works

not sure if it's uniform

ok so I think I want to ban any function which takes some input and outputs a matrix which guarantees to be orthogonal.. which is not what I said above

what do you think about that rule?

i think it's good

i found a citation for the cheap method, it would indeed work

be sure your wording also excludes matrix decompositions one of whose components is orthogonal

This rule bans the use of any existing function which takes in some input and outputs a matrix which is guaranteed to be orthogonal.

please suggest improvements!

So no identity matrix functions? :P

haha

you have to make the identity all by yourself :)

Also the number 1 would be forbidden in Matlab.

the empty matrix also cannot be produced

Since scalars there are actually 1x1 matrices.

hmmm

you could allow built-ins that take no inputs, or only n

This rule bans the use of any existing function which takes in some input and outputs a matrix of size at least n by n which is guaranteed to be orthogonal. As an extreme example, if you want the n by n identity matrix, you will have to create it yourself.

better?

i prefer the less restrictive one, but that's fine too

it at least deals with feersum's example

so literals like [[1]] are fine

yes

oh, but numpy.matrix([[1]]) cannot be done

why not?

remember n > 1

yes, but i'm considering a recursive/iterative strategy where i build up to n

and it needs a base case

right but matrices smaller than n are ok

so [[1]] is ok

oh

well, that makes my strategy much better: i can just start with n-1 and not bother building up to there

that seems like an exploit

can you keep it orthogonal and uniform?

Well, dimension > 1 would cover the cases so far...

@Lembik i'm trying to

I think that's fine

I am tired :)

goodnight

good night :-)

i'll be sure to think as many ways as possible to exploit the rules while you're asleep and can't change anything :)

sounds good :)

which language are you coding in?

python with scipy

cool

good luck!

hmm, I think should be doable with plain Python.

probably, at least by gram-schmidt on a random gaussian matrix

yes

gauss is the shortest distribution in random :)

@LuisMendo negating is fine: the gaussians are independent and symmetric, so each negated gaussian is still a gaussian and is still independent

taking the absolute value of the diagonal entries would also work

@Geobits dammit

\o/ gold golf good

@xnor Thanks! Anyway, without QR this is harder than I want to accomplish...

@ConorO'Brien Congrats! Welcome to the club :-P