Linear-size $\ell_1$ sparsifiers

2026-06-26

AI Summary

This work addresses the problem of efficiently sparsifying an arbitrary matrix while preserving $\ell_1$-norm approximation guarantees. By constructing a diagonal matrix with only a small number of non-zero entries and leveraging zonotope approximation theory together with probabilistic geometric techniques, the authors achieve a linear-size $\ell_1$ sparse approximation of both the original matrix and its associated zonotope. The main contribution lies in improving the upper bound on the number of required non-zero entries to $O\left(\frac{n}{\varepsilon^2} \log\left(\frac{1}{\varepsilon}\right)\right)$, which significantly refines the previous bound of $O\left(\frac{n}{\varepsilon^2} \log n\right)$. This advancement yields a notable reduction in both storage requirements and computational complexity.

Abstract

We prove that for any matrix $A \in \mathbb{R}^{m \times n}$ and any $\varepsilon \in (0, 1/2]$ there is a diagonal matrix $D \in \mathbb{R}_{\geq 0}^{m \times m}$ with at most $O(\frac{n}{\varepsilon^2} \log(\frac{1}{\varepsilon}))$ 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 $O(\frac{n}{\varepsilon^2} \log(\frac{1}{\varepsilon}))$ 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).

Problem

• sparsification
• ℓ₁ norm
• zonotope
• approximation
• diagonal scaling

Innovation

• ℓ₁ sparsification
• linear-size sparsifier
• zonotope approximation
• dimension reduction
• metric embedding