Efficient Removal Lemmas for Matrices

The authors and Fischer recently proved that any hereditary property of two-dimensional matrices (where the row and column order is not ignored) over a finite alphabet is testable with a constant number of queries, by establishing the following (ordered) matrix removal lemma: For any finite alphabet $\Sigma$, any hereditary property $\mathcal{P}$ of matrices over $\Sigma$, and any $\epsilon > 0$, there exists $f_{\mathcal{P}}(\epsilon)$ such that for any matrix $M$ over $\Sigma$ that is $\epsilon$-far from satisfying $\mathcal{P}$, most of the $f_{\mathcal{P}}(\epsilon) \times f_{\mathcal{P}}(\epsilon)$ submatrices of $M$ do not satisfy $\mathcal{P}$. Here being $\epsilon$-far from $\mathcal{P}$ means that one needs to modify at least an $\epsilon$-fraction of the entries of $M$ to make it satisfy $\mathcal{P}$. However, in the above general removal lemma, $f_{\mathcal{P}}(\epsilon)$ grows very fast as a function of $\epsilon^{-1}$, even when $\mathcal{P}$ is characterized by a single forbidden submatrix. In this work we establish much more efficient removal lemmas for several special cases of the above problem. In particular, we show the following: For any fixed $s \times t$ binary matrix $A$ and any $\epsilon > 0$ there exists $\delta > 0$ polynomial in $\epsilon$, such that for any binary matrix $M$ in which less than a $\delta$-fraction of the $s \times t$ submatrices are equal to $A$, there exists a set of less than an $\epsilon$-fraction of the entries of $M$ that intersects every $A$-copy in $M$. We generalize the work of Alon, Fischer and Newman [SICOMP'07] and make progress towards proving one of their conjectures. The proofs combine their efficient conditional regularity lemma for matrices with additional combinatorial and probabilistic ideas.