TITLE: Distance of low-rank matrices to the identity for the $\infty$-norm
QUESTION [5 upvotes]: I am trying to get a lower bound (or even the exact value) of
$$
\min_{X \in \mathbb{R}^{n\times n}} \|X - I_n\|_{\infty} \enspace \text{s.t.} \enspace \mbox{Rank}(X) = m
$$
where $m \leq n$, and the infinity norm is
$$ \| X \|_{\infty} := \max_{ij}|X_{ij}| $$
I have a very simple lower bound, obtained with norm equivalence:
$$\|X - I_n\|_{\infty} \geq \frac{1}{n} \|X - I_n\|_F\geq \frac{\sqrt{n - m}}{n}$$
but this is obviously not tight, and experiments suggest that the scaling is wrong.
Thanks a lot ! :)

REPLY [4 votes]: Theorem 1.1 from Alon's paper "Perturbed identity matrices have high rank: proof and applications" says that if $\|X-I_n\|_\infty<c$ with $1/(2\sqrt n)<c<1/4$, then $m\gg \log n/(c^2\log(1/c))$ with an absolute implicit constant. This is exactly what you need: if $X$ is close to the identity matrix in the infinity norm, then the rank of $X$ is large. Incidentally, ALon's bound is sharp up to a logarithmic factor; therefore, you cannot expect to get much better estimates.

REPLY [2 votes]: Your simple lower bound is not so bad, in particular when $m$ is of order $cn$ for $0<c<1$.
Indeed, it follows from the answers to this question that there are unit vectors $u_1,...,u_n$ in $\mathbf{R}^m$ such that $|\langle u_i,u_j\rangle| \leq C \sqrt{\frac{\log n}{m}}$ for every $i \neq j$. Taking $X = (\langle u_i,u_j\rangle)_{i,j}$ yields that the minimum is at most $C \sqrt{\frac{\log n}{m}}$.