A PTAS for $\ell_p$Low Rank Approximation
Abstract
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 $\AB\_p^p=\sum_{i,j}A_{i,j}B_{i,j}^p$ when $p > 0$; and $\AB\_0=\sum_{i,j}[A_{i,j}\neq B_{i,j}]$ 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 almostlinear 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 $2^{k^{\delta}}$.
 Publication:

arXiv eprints
 Pub Date:
 July 2018
 arXiv:
 arXiv:1807.06101
 Bibcode:
 2018arXiv180706101B
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Computational Complexity;
 Computer Science  Machine Learning
 EPrint:
 Accepted at SODA'19, 61 pages