The restricted isometry property (RIP) is essential for the linear map to guarantee the successful recovery of low-rank matrices. The existing works show that the linear map generated by the measurement matrices with independent and identically distributed (i.i.d.) entries satisfies RIP with high probability. However, when dealing with non-i.i.d. measurement matrices, such as the rank-one measurements, the RIP compliance may not be guaranteed. In this paper, we show that the RIP can still be achieved with high probability, when the rank-one measurement matrix is constructed by the random unit-modulus vectors. Compared to the existing works, we first address the challenge of establishing RIP for the linear map in non-i.i.d. scenarios. As validated in the experiments, this linear map is memory-efficient, and not only satisfies the RIP but also exhibits similar recovery performance of the low-rank matrices to that of conventional i.i.d. measurement matrices.

翻译：受限等距性质是保证线性映射成功恢复低秩矩阵的关键性质。现有研究表明，由独立同分布测量矩阵生成的线性映射能以高概率满足受限等距性质。然而，当处理非独立同分布测量矩阵（如秩一测量）时，受限等距性质的成立性可能无法保证。本文证明：当采用随机单位模向量构造秩一测量矩阵时，仍能以高概率实现受限等距性质。与现有工作相比，我们首次解决了非独立同分布场景下线性映射受限等距性质的建立难题。实验验证表明，该线性映射不仅具有内存高效特性、满足受限等距性质，而且在低秩矩阵恢复性能方面与传统独立同分布测量矩阵表现相当。