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arXiv · 1909.07515

Multiplicative Rank-1 Approximation using Length-Squared Sampling

Ragesh Jaiswal·Amit Kumar
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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.

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Ragesh Jaiswal, Amit Kumar. 2019-09-16. Multiplicative Rank-1 Approximation using Length-Squared Sampling. https://arxiv.org/abs/1909.07515

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