## Abstract

We study the low rank approximation problem of any given matrix A over R^{n}×m and C^{n}×^{m} in entry-wise l_{p} loss, that is, finding a rank-k matrix X such that kA - Xk_{p} is minimized. Unlike the traditional l_{2} setting, this particular variant is NP-Hard. We show that the algorithm of column subset selection, which was an algorithmic foundation of many existing algorithms, enjoys approximation ratio (k + 1)^{1/p} for 1 = p = 2 and (k + 1)^{1-1/p} for p = 2. This improves upon the previous O(k + 1) bound for p = 1 [1]. We complement our analysis with lower bounds; these bounds match our upper bounds up to constant 1 when p = 2. At the core of our techniques is an application of Riesz-Thorin interpolation theorem from harmonic analysis, which might be of independent interest to other algorithmic designs and analysis more broadly. As a consequence of our analysis, we provide better approximation guarantees for several other algorithms with various time complexity. For example, to make the algorithm of column subset selection computationally efficient, we analyze a polynomial time bi-criteria algorithm which selects O(k log m) columns. We show that this algorithm has an approximation ratio of O((k + 1)^{1/p}) for 1 = p = 2 and O((k + 1)^{1-1/p}) for p = 2. This improves over the best-known bound with an O(k + 1) approximation ratio. Our bi-criteria algorithm also implies an exact-rank method in polynomial time with a slightly larger approximation ratio.

Original language | English (US) |
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Journal | Advances in Neural Information Processing Systems |

Volume | 32 |

State | Published - 2019 |

Event | 33rd Annual Conference on Neural Information Processing Systems, NeurIPS 2019 - Vancouver, Canada Duration: Dec 8 2019 → Dec 14 2019 |

## ASJC Scopus subject areas

- Computer Networks and Communications
- Information Systems
- Signal Processing

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