A PTAS for $\ell_p$-Low Rank Approximation

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 $\|A-B\|_0=\sum_{i,j}[A_{i,j}\neq B_{i,j}]$ 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 $2^{k^{\delta}}$.