Linear-size $\ell_1$ sparsifiers

Victor Reis; Thomas Rothvoss

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arXiv · 2606.28147

Linear-size $\ell_1$ sparsifiers

Victor Reis·Thomas Rothvoss

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Abstract

We prove that for any matrix $A \in \mathbb{R}^{m \times n}$ and any $\varepsilon > 0$ there is a diagonal matrix $D \in \mathbb{R}_{\geq 0}^{m \times m}$ with at most $O(\frac{n}{\varepsilon^2})$ nonzero entries so that \[ (1-\varepsilon) \|Ax\|_1 \leq \|DAx\|_1 \leq (1+\varepsilon)\|Ax\|_1 \quad \forall x \in \mathbb{R}^n. \] In particular, for any zonotope $Z \subseteq \mathbb{R}^{n}$ there exists a zonotope $Z' \subseteq \mathbb{R}^{n}$ generated by at most $O(\frac{n}{\varepsilon^2})$ segments so that $(1-\varepsilon) Z \subseteq Z' \subseteq (1+\varepsilon) Z$. Previously, the best known bound was $O(\frac{n}{\varepsilon^2} \log n)$ due to Talagrand (1990).

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Victor Reis, Thomas Rothvoss. 2026-06-26. Linear-size $\ell_1$ sparsifiers. https://arxiv.org/abs/2606.28147

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