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  • 33 votes cast
Apr
26
comment How to calculate $\langle v,w\rangle$ based only on $\langle v,x_i\rangle$ and $\langle w,x_i\rangle$?
I think you'll also need information on $<x_i,x_j>$.
Apr
26
comment How to calculate $\langle v,w\rangle$ based only on $\langle v,x_i\rangle$ and $\langle w,x_i\rangle$?
How do you use Gram–Schmidt if you don't have access to the explicit representation (i.e., the numerical entries) of the $x_i$? You are only given $2n$ numbers: $a_i=<v,x_i>$ and $b_i=<w,x_i>$. The goal is to compute $<v,w>$ as a function of these $2n$ numbers alone. As Boaz mentioned, you must assume that the $x_i$ are a basis. In this case, I think you can recover the $v,w$ exactly -- at least the number of variables matches the number of unknowns.
Apr
20
answered What is the order of the constant $K$ in the multidimensional Dvoretzky-Kiefer-Wolfowitz inequality($Ke^{-c z}$)?
Apr
7
revised What do you call the collection of all sets shattered by $F$?
added 66 characters in body
Apr
7
asked What do you call the collection of all sets shattered by $F$?
Apr
6
comment Sharpened Pinsker inequality for special case
Just a quick comment: the conjecture (now theorem in light of Iosif's answer) is indeed true for all $x\in[0,1]$. However, my weaker bound with $2.1\sqrt n x$ doesn't hold on all of $[0,1]$.
Apr
6
comment Sharpened Pinsker inequality for special case
Beautiful, @Iosif Pinelis!
Apr
6
accepted Sharpened Pinsker inequality for special case
Apr
5
asked Sharpened Pinsker inequality for special case
Mar
28
comment Metric analogues of bounded variation
Thanks again. I guess there's a reason why the Hardy-Krause extension to higher dimensions is so complicated.
Mar
26
comment Metric analogues of bounded variation
@WillieWong One last attempt before giving this a rest. What if we consider an $\epsilon$-net (i.e., minimal cover/maximal packing), and then take $\mathcal{P}_n$ to be the Voronoi regions induced by the net points?
Mar
22
comment Metric analogues of bounded variation
Very nice example, @WillieWong, and thanks Suvrit for the link. I was also pointed to arxiv.org/pdf/1301.6897v1.pdf . So I guess it's time to quite amateur conjecturing and start reading...
Mar
22
revised Metric analogues of bounded variation
added 625 characters in body
Mar
22
comment Metric analogues of bounded variation
Thank you for that example, @MartinHairer. Rather than risk further embarrassment with additional edits, let me try to continue in the comments. Suppose I define $\mathcal{P}_n$ to be the Voronoi partition induced by some $n$ points, and define the variation to be the supremum over all $\mathcal{P}_n$. Does this still admit pathological examples?
Mar
22
comment Metric analogues of bounded variation
You're of course right, @AugustCleaner! I tried to fix this in the edited version.
Mar
22
revised Metric analogues of bounded variation
added 431 characters in body
Mar
22
asked Metric analogues of bounded variation
Feb
8
comment measure of the distance between two joint distributions
See this paper by Gibbs and Su "ON CHOOSING AND BOUNDING PROBABILITYMETRICS" math.hmc.edu/~su/papers.dir/metrics.pdf
Feb
7
awarded  Custodian
Feb
7
reviewed Approve Weyl-type inequality for non-Hermitian matrices?