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：限制等距性（RIP）是保证线性映射能够成功恢复低秩矩阵的关键。现有研究表明，由独立同分布（i.i.d.）测量矩阵生成的线性映射能以高概率满足RIP。然而，当处理非独立同分布测量矩阵（如秩一测量）时，RIP的满足性可能无法得到保证。本文证明，当秩一测量矩阵由随机单位模向量构建时，RIP仍能以高概率实现。与现有工作相比，我们首次解决了非独立同分布场景中线性映射RIP建立的挑战。实验验证表明，该线性映射具有内存高效性，不仅满足RIP，而且在低秩矩阵的恢复性能上与传统的独立同分布测量矩阵表现相近。