Zero-One Rounding of Singular Vectors

Abstract

We propose a generic and simple technique called dyadic rounding for rounding real vectors to zero-one vectors, and show its several applications in approximating singular vectors of matrices by zero-one vectors, cut decompositions of matrices, and norm optimization problems. Our rounding technique leads to the following consequences.

• 1 Given any A ∈ ℝ^{m ×n}, there exists z ∈ {0, 1}ⁿ such that

  $$ \frac{\left\|Az\right\|_{q}}{\left\|z\right\|_{p}} \geq \Omega\left(p^{1 - \frac{1}{p}} (\log n)^{\frac{1}{p} - 1}\right) \left\|A\right\|_{p \mapsto q}, $$

  where $\left\|A\right\|_{p \mapsto q} = \max_{x \neq 0} \left\|Ax\right\|_{q} / \left\|x\right\|_{p}$. Moreover, given any vector v ∈ ℝⁿ we can round it to a vector z ∈ {0, 1}ⁿ with the same approximation guarantee as above, but now the guarantee is with respect to $\left\|Av\right\|_{q}/\left\|Av\right\|_{p}$ instead of $\left\|A\right\|_{p \mapsto q}$. Although stated for p ↦q norm, this generalizes to the case when $\left\|Az\right\|_{q}$ is replaced by any norm of z.

• 2 Given any A ∈ ℝ^{m ×n}, we can efficiently find z ∈ {0, 1}ⁿ such that

  $$ \frac{\left\|Az\right\|}{\left\|z\right\|} \geq \frac{\sigma_{1}(A)}{2 \sqrt{2 \log n}}, $$

  where σ ₁(A) is the top singular value of A. Extending this, we can efficiently find orthogonal z ₁, z ₂, …, z _k  ∈ {0, 1}ⁿ such that

  $$ \frac{\left\|Az_{i}\right\|}{\left\|z_{i}\right\|} \geq \Omega\left(\frac{\sigma_{k}(A)}{\sqrt{k \log n}}\right), \quad \text{for all $i \in[k]$}. $$

  We complement these results by showing that they are almost tight.

• 3 Given any A ∈ ℝ^{m ×n} of rank r, we can approximate it (under the Frobenius norm) by a sum of O(r log² m log² n) cut-matrices, within an error of at most $\left\|A\right\|_{F}/\text{poly}(m, n)$. In comparison, the Singular Value Decomposition uses r rank-1 terms in the sum (but not necessarily cut matrices) and has zero error, whereas the cut decomposition lemma by Frieze and Kannan in their algorithmic version of Szemerédi’s regularity partition [9,10] uses only O(1/ε ²) cut matrices but has a large ${\epsilon} \sqrt{mn} \left\|A\right\|_{F}$ error (under the cut norm). Our algorithm is deterministic and more efficient for the corresponding error range.