Multiplicative Rank-1 Approximation using Length-Squared Sampling

TL;DR

This paper demonstrates that sampling a sufficient number of rows from a matrix according to length-squared probabilities yields a rank-1 approximation that closely matches the best possible, with a multiplicative error bound.

Contribution

The paper introduces a method for rank-1 approximation using length-squared sampling that achieves a multiplicative error bound, improving upon previous additive guarantees.

Findings

Sampling $rac{1}{ ext{epsilon}^4}$ rows suffices for a good approximation.

The method provides a multiplicative approximation for rank-1 approximation.

The approach extends length-squared sampling techniques to rank-1 approximation with strong guarantees.

Abstract

We show that the span of $\Omega(\frac{1}{\varepsilon^4})$ rows of any matrix $A \subset \mathbb{R}^{n \times d}$ sampled according to the length-squared distribution contains a rank-$1$ matrix $\tilde{A}$ such that $||A - \tilde{A}||_F^2 \leq (1 + \varepsilon) \cdot ||A - \pi_1(A)||_F^2$, where $\pi_1(A)$ denotes the best rank-$1$ approximation of $A$ under the Frobenius norm. Length-squared sampling has previously been used in the context of rank-$k$ approximation. However, the approximation obtained was additive in nature. We obtain a multiplicative approximation albeit only for rank-$1$ approximation.