Restricted Isometry Property of Gaussian Random Projection for Finite Set of Subspaces
Restricted Isometry Property of Gaussian Random Projection for Finite Set of Subspaces
复制标题
有限子空间集高斯随机投影的受限等距性质
DOI：
10.1109/tsp.2017.2778685
复制
发表时间：
2018-04-01
影响因子：
5.4
通讯作者：
Gu, Yuantao
中科院分区：
文献类型：
--
作者：
Li, Gen;Gu, Yuantao
降维在降低求解大规模问题的复杂性方面起着至关重要的作用。著名的Johnson-Lindenstrauss（JL）引理和限制等距性质（RIP）允许在保持欧氏距离的情况下使用随机投影来降维，这导致了压缩感知和稀疏相关信号处理领域的繁荣。最近，稀疏模型在计算机视觉和机器学习中的成功应用越来越多地暗示，高维数据的底层结构看起来更像是子空间的并集。本文基于JL引理和压缩子空间聚类的新兴领域，首次研究了基于广义投影F-范数距离的高斯随机矩阵压缩子空间的RIP问题。我们从理论上证明，以高概率的亲和力或两个投影子空间之间的距离集中在他们的估计。当投影后的环境维数足够大时，两个子空间之间的亲和力和距离在随机投影后几乎保持不变。数值实验验证了理论工作。
Dimension reduction plays an essential role when decreasing the complexity of solving large-scale problems. The well-known Johnson-Lindenstrauss (JL) lemma and restricted isometry property (RIP) admit the use of random projection to reduce the dimension while keeping the Euclidean distance, which leads to the boom of compressed sensing and the field of sparsity related signal processing. Recently, successful applications of sparse models in computer vision and machine learning have increasingly hinted that the underlying structure of high dimensional data looks more like a union of subspaces. In this paper, motivated by JL lemma and an emerging field of compressed subspace clustering, we study for the first time the RIP of Gaussian random matrices for the compression of two subspaces based on the generalized projection F-norm distance. We theoretically prove that with high probability the affinity or distance between two projected subspaces are concentrated around their estimates. When the ambient dimension after projection is sufficiently large, the affinity and distance between two subspaces almost remain unchanged after random projection. Numerical experiments verify the theoretical work.