This note is concerned with deterministic constructions of $m \times N$ matrices satisfying a restricted isometry property from $\ell_2$ to $\ell_1$ on $s$-sparse vectors. Similarly to the standard ($\ell_2$ to $\ell_2$) restricted isometry property, such constructions can be found in the regime $m \asymp s^2$, at least in theory. With effectiveness of implementation in mind, two simple constructions are presented in the less pleasing but still relevant regime $m \asymp s^4$. The first one, executing a Las Vegas strategy, is quasideterministic and applies in the real setting. The second one, exploiting Golomb rulers, is explicit and applies to the complex setting. As a stepping stone, an explicit isometric embedding from $\ell_2^n(\mathbb{C})$ to $\ell_4^{cn^2}(\mathbb{C})$ is presented. Finally, the extension of the problem from sparse vectors to low-rank matrices is raised as an open question.

翻译：本文关注从ℓ₂到ℓ₁在s-稀疏向量上满足限制等距性质的m×N矩阵的确定性构造。与标准（ℓ₂到ℓ₂）限制等距性质类似，此类构造可在m ≍ s²的范围内实现（至少在理论上）。考虑到实现效率，本文在不太理想但仍具相关性的m ≍ s⁴范围内提出了两种简单构造。第一种构造采用拉斯维加斯策略，属于准确定性方法，适用于实数域；第二种构造利用哥伦布尺，是显式方法，适用于复数域。作为铺垫，本文还给出了从ℓ₂ⁿ(ℂ)到ℓ₄^{cn²}(ℂ)的显式等距嵌入。最后，将问题从稀疏向量扩展到低秩矩阵作为开放问题提出。