TITLE: Inner products and Norms
QUESTION [6 upvotes]: Let $f:[n]\times [n] \rightarrow [0,1]$ be a function from pair of integers to the real interval [0.1]. I would like to find sets of complex vectors 
$X= \{x_i\}$ and $Y=\{y_j\}$ satisfying $x_i\cdot y_j=f(i,j)$, in such a way that 
the vectors in $X$ and in $Y$ are as small as possible. More precisely, 
set $m= max_{i,j} f(i,j)$ and $N=max_{i,j}[\|x_i\|,\|y_j\|]$.
1) What is the minimal $N$ such that $x_i\cdot y_j = f(i,j)$ for all $i,j\in [n]$? 
2) Is there an upper bound on $N$ purely in function of $m$, i.e., with no depenence 
on $n$? 
3) If the answer to question two is no, what is the best upper bound that we can 
give for $N$ in function of $n$ and $m$? A trivial upper bound is 
$$N \leq \max_i{\sum_{j} f(i,j) } \leq mn.$$ 
but I believe that the dependence on $n$ might be lowered.
cordially,
mateus

REPLY [10 votes]: Such questions have been dealt with. Note first that your $m$ is just the norm of the matrix $F$ (see Robin's comment) as an operator from $\ell_1^n$ to $\ell_\infty^n$. $M$ also has a name, it is the $\gamma_2$ norm of this operator (This is the minimal product of 
$$\|Y\|_{1\to 2}\|X\|_{2\to\infty}$$ over all  factorizations $F=XY$ . $\|Z\|_{p\to q}$ denotes the norm of Z as an operator from $\ell_p$ to $\ell_q$.)
It is not hard to see that $M=\gamma_2(F)\le \sqrt n \|F\|_{1\to\infty}=\sqrt n m$.
For a random $0,1$ matrix $F$ one gets that this estimate is tight, up to a universal constant.
You can look here http://www.springerlink.com/content/px2324p527n19xj2/ for details.
In particular Cor 5.2 there (it deals with random $\pm 1$ matrices but it is easy to go between those and random $0,1$ matrices).