Given a large random matrix, how to prove that every large submatrix whose range contains a large ball?

MathOverflow Asked on January 1, 2022

Context. Studing a problem in machine-learning, I’m led to consider the following problem in RMT…

Definition. Given positive integers $m$ and $n$ and positive real numbers $c_1$ and $c_2$, let’s say an $m$-by-$n$ real matrix $X$ is $(c_1,c_2)$-incrompressible if every submatrix of $Z$ with $k ge c_1 m$ rows and $n$ columns satisfies $c_2mathbb B^k subseteq Zmathbb B^n$, where $c_2mathbb B^k$ is the ball of radius $c_2$ in $mathbb R^k$ and $Zmathbb B^n := {Zv mid v in mathbb B^n}$.

Question. Is the definition somehow linked to the classical "restricted isometry property" ?

An answer to this the above question would allow me to directly tap into the vast RMT literature, for the purposes of attacking my subsequent questions (see below).

Special case when $c_1=1$.
In the particular case where $c_1=1$, we note that $X$ is $(1,c_2)$-incompressible iff its smallest singular value is $c_2$ or greater.

Now, fix $delta in (0, 1)$ once and for all.

Question. Do there exist universal constants $c_1 > 0$ and $c_2 > 0$ (depending only on $delta$) such that the following phenomenon holds ?

The phenomenon. Let $m$ and $n$ be positive integers with $m le delta n$ and $n$ large, and let $k ge c_1 m$. Let $X$ an $m$-by-$n$ random real matrix with iid $N(0,1)$ entries.

Question. With high probability, $X$ is $(c_1,c_2)$-incompressible!

Also, how large can this probability be as a function of $n$ and $delta$ ?

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