# Optimal Analysis of Subset-Selection Based L_p Low Rank Approximation

arXiv: https://arxiv.org/abs/1910.13618 · HTML: https://mlsanity.com/paper/1910.13618/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Oct 2019.

## Built on by (most cited first, 4 of 4)

- [Ridge Leverage Score Sampling for ℓp Subspace Approximation](https://mlsanity.com/paper/2407.03262/) (2024)
- [One-pass additive-error subset selection for ℓp subspace approximation](https://mlsanity.com/paper/2204.12073/) (2022)
- [On Subspace Approximation and Subset Selection in Fewer Passes by MCMC Sampling](https://mlsanity.com/paper/2103.11107/) (2021)
- [Performance of ℓ1 Regularization for Sparse Convex Optimization](https://mlsanity.com/paper/2307.07405/) (2023)

## Abstract (arXiv)

We study the low rank approximation problem of any given matrix A over ℝn× m and mathbbCn× m in entry-wise ℓp loss, that is, finding a rank-k matrix X such that |A-X|p is minimized. Unlike the traditional ℓ2 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 citechierichetti2017algorithms. 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(klog 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.