# Multiplicative Rank-1 Approximation using Length-Squared Sampling

We show that the span of Ω(1/ε^4) rows of any matrix A ⊂R^n × d sampled according to the length-squared distribution contains a rank-1 matrix Ã such that ||A - Ã||_F^2 ≤ (1 + ε) · ||A - π_1(A)||_F^2, where π_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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