Previously: math for AI and AI for math
Now just kinda trying to figure out if AI can be made safe
Zygi Straznickas(Zygimantas Straznickas)
Karma: 55
Question 1 is very likely NP-hard. https://arxiv.org/abs/1203.6602 shows:
We consider the decision problem asking whether a partial rational symmetric matrix with an all-ones diagonal can be completed to a full positive semidefinite matrix of rank at most k. We show that this problem is $\NP$-hard for any fixed integer k≥2.
I’m still stretching my graph-theory muscles to see if this technically holds for (and so precisely implies the NP-hardness of Question 1.) But even without that, for applied use cases we can just fix to a very large number to see that practical algorithms are unlikely.
Nope, I’m withdrawing this answer. I looked closer into the proof and I think it’s only meaningful asymptotically, in a low rank regime. The technique doesn’t work for analyzing full-rank completions.
Thanks for the clarification! I assumed the question was using the bit computation model (biased by my own experience), in which case the complexity of SDP in the general case still seems to be unknown (https://math.stackexchange.com/questions/2619096/can-every-semidefinite-program-be-solved-in-polynomial-time)
Whether it’s a big deal depends on the person, but one objective piece of evidence is male plastic surgery statistics: looking at US plastic surgery statistics for year 2022, surgery for gynecomastia (overdevelopment of breasts) is the most popular surgery for men: 23k surgeries per year total, 50% of total male body surgeries and 30% of total male cosmetic surgeries. So it seems that not having breasts is likely quite important for a man’s body image.
(note that this ignores base rates, with more effort one could maybe compare the ratios of [prevalence vs number of surgeries] for other indications, but that’s very hard to quantify for other surgeries like tummy tuck—what’s the prevalence of needing a tummy tuck?)
I think I might have an example! (even though it’s not 100% satisfying—there’s some logical inversions needed to fit the format. But I still think it fits the spirit of the question.)
It’s in the setting of polynomial identity testing: let F be a finite field and p(x) a polynomial over that field, and assume we have access to the representation of p (i.e. not just black-box access).
Two questions we can ask about p are:
EZE (Evaluates to Zero Everywhere): does p evaluate to zero on all inputs? That is, is it the case that ∀x,p(x)=0? Unfortunately, that’s coNP-hard.
PIT (Polynomial Identity Testing) -- decide if p is equal to the zero polynomial, that is, if all of p’s coefficients are 0. (Note that these are two distinct concepts: in the two-value field F2 we have x2−x=0 for all x, even though the polynomial is not zero). Polynomial identity testing can be done efficiently in probabilistic polynomial time.
Now, make f be the inverted Polynomial Identity Testing algorithm: f(p)=0 if p is zero, and 1 if it’s not zero. Then, assume our perspective is that EZE is the “fundamental” property. In that case f(p) can be 1 for two reasons: either the polynomial p doesn’t Evaluate to Zero Everywhere, or it’s one of those rare special cases mentioned above where it Evaluates to Zero Everywhere but isn’t identically zero. These to me seem like clearly distinct mechanisms. But to distinguish between them we need to check if p Evaluates to Zero Everywhere, and that’s co-NP hard, so an efficient discriminator (likely) doesn’t exist.