通过确定性降维映射将稀疏向量从 $\ell_2$ 线性嵌入到 $\ell_1$
Linearly Embedding Sparse Vectors from $\ell_2$ to $\ell_1$ via Deterministic Dimension-Reducing Maps
AI总结：
本文研究从 $\ell_2$ 到 $\ell_1$ 的稀疏向量限制等距嵌入的确定性构造，在 $m \asymp s^4$ 范围内提出实数情形的准确定性拉斯维加斯构造和复数情形的显式戈洛姆尺构造，并将低秩矩阵扩展留作开放问题。
摘要：
本文关注的是在 $s$-稀疏向量上满足从 $\ell_2$ 到 $\ell_1$ 的限制等距性质的 $m \times N$ 矩阵的确定性构造。与标准的（$\ell_2$ 到 $\ell_2$）限制等距性质类似，至少在理论上，此类构造可以在 $m \asymp s^2$ 的范围内找到。考虑到实现的有效性，本文在不太理想但仍然相关的 $m \asymp s^4$ 范围内给出了两种简单构造。第一种构造采用拉斯维加斯策略，是准确定性的，适用于实数情形。第二种构造利用戈洛姆尺，是显式的，适用于复数情形。作为铺垫，本文给出了一个从 $\ell_2^n(\mathbb{C})$ 到 $\ell_4^{cn^2}(\mathbb{C})$ 的显式等距嵌入。最后，本文将问题从稀疏向量扩展到低秩矩阵，并作为一个开放问题提出。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
