A number of recent works have studied algorithms for entrywise $ell_p$-low rank approximation, namely, algorithms which given an $n times d$ matrix $A$ (with $n geq d$), output a rank-$k$ matrix $B$ minimizing $|A-B|_p^p=sum_{i,j}|A_{i,j}-B_{i,j}|^p$ when $p > 0$; and $|A-B|_0=sum_{i,j}[A_{i,j} eq B_{i,j}]$ for $p=0$. On the algorithmic side, for $p in (0,2)$, we give the first $(1+epsilon)$-approximation algorithm running in time $n^{text{poly}(k/epsilon)}$. Further, for $p = 0$, we give the first almost-linear time approximation scheme for what we call the Generalized Binary $ell_0$-Rank-$k$ problem. Our algorithm computes $(1+epsilon)$-approximation in time $(1/epsilon)^{2^{O(k)}/epsilon^{2}} cdot nd^{1+o(1)}$. On the hardness of approximation side, for $p in (1,2)$, assuming the Small Set Expansion Hypothesis and the Exponential Time Hypothesis (ETH), we show that there exists $delta := delta(alpha) > 0$ such that the entrywise $ell_p$-Rank-$k$ problem has no $alpha$-approximation algorithm running in time $2^{k^{delta}}$.
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