Low-rank binary matrix approximation in column-sum norm

We consider $ell_1$-Rank-$r$ Approximation over GF(2), where for a binary $mtimes n$ matrix ${bf A}$ and a positive integer $r$, one seeks a binary matrix ${bf B}$ of rank at most $r$, minimizing the column-sum norm $||{bf A} -{bf B}||_1$. We show that for every $varepsilonin (0, 1)$, there is a randomized $(1+varepsilon)$-approximation algorithm for $ell_1$-Rank-$r$ Approximation over GF(2) of running time $m^{O(1)}n^{O(2^{4r}cdot varepsilon^{-4})}$. This is the first polynomial time approximation scheme (PTAS) for this problem.