Dimensionality reduction is in demand to reduce the complexity of solving large-scale problems with data lying in latent low-dimensional structures in machine learning and computer vision. Motivated by such need, in this work we study the Restricted Isometry Property (RIP) of Gaussian random projections for low-dimensional subspaces in ${\mathbb{R}}^N$, and rigorously prove that the projection Frobenius norm distance between any two subspaces spanned by the projected data in ${\mathbb{R}}^n$ ($n<N$) remain almost the same as the distance between the original subspaces with probability no less than $1-{\mathrm{e}}^{-\mathcal{O}(n)}$. Previously the well-known Johnson-Lindenstrauss (JL) Lemma and RIP for sparse vectors have been the foundation of sparse signal processing including Compressed Sensing. As an analogy to JL Lemma and RIP for sparse vectors, this work allows the use of random projections to reduce the ambient dimension with the theoretical guarantee that the distance between subspaces after compression is well preserved.

为了降低解决大规模问题的复杂性，需要降维，而数据位于机器学习和计算机视觉中潜在的低维结构中。出于这种需求的动机，在这项工作中，我们研究了高斯随机投影在低维子空间中的受限等距性质（RIP）。${\mathbb{[R}}^{ñ}$，并严格证明投影数据中任意两个子空间之间的投影Frobenius范数距离 ${\mathbb{[R}}^{ñ}$ （$ñ<ñ$）保持与原始子空间之间的距离几乎相同，且概率不小于 $\mathrm{1个}-{\mathrm{Ë}}^{-\mathcal{Ø}（ñ）}$。以前，用于稀疏矢量的著名Johnson-Lindenstrauss（JL）引理和RIP一直是包括压缩感知在内的稀疏信号处理的基础。类似于JL Lemma和RIP的稀疏矢量，这项工作允许使用随机投影来减小环境尺寸，并在理论上保证压缩后子空间之间的距离得到很好的保留。