Calculate the distance between spaces spanned by the column of two matrices. The distance is between 0 and 1. If the two spaces are the same, the distance is 0. if the two spaces are orthogonal, the distance is 1.
Distance(Z1, Z2)Output a number between 0 and 1.
Input a matrix with \(p \times q_1\).
Input another matrix with \(p \times q_2\)
Yong He, Ran Zhao, Wen-Xin Zhou.
Define $$\mathcal{D}(\bold{Q}_1,\bold{Q}_2)=\left(1-\frac{1}{\max{(q_1,q_2)}}\text{Tr}\left(\bold{Q}_1\bold{Q}_1^\top\bold{Q}_2\bold{Q}_2^\top\right)\right)^{1/2}.$$
By the definition of \(\mathcal{D}(\bold{Q}_1,\bold{Q}_2)\), we can easily see that \(0 \leq \mathcal{D}(\bold{Q}_1,\bold{Q}_2)\leq 1\), which measures the distance between the column spaces spanned by two orthogonal matrices \(\bold{Q}_1\) and \(\bold{Q}_2\), i.e., \(\text{span}(\bold{Q}_1)\) and \(\text{span}(\bold{Q}_2)\). In particular, \(\text{span}(\bold{Q}_1)\) and \(\text{span}(\bold{Q}_2)\) are the same when \(\mathcal{D}(\bold{Q}_1,\bold{Q}_2)=0\), while \(\text{span}(\bold{Q}_1)\) and \(\text{span}(\bold{Q}_2)\) are orthogonal when \(\mathcal{D}(\bold{Q}_1,\bold{Q}_2)=1\). The Gram-Schmidt orthogonalization can be used to make \(\bold{Q}_1\) and \(\bold{Q}_2\) column-orthogonal matrices.
He, Y., Zhao, R., & Zhou, W. X. (2023). Iterative Least Squares Algorithm for Large-dimensional Matrix Factor Model by Random Projection. <arXiv:2301.00360>.
set.seed(1111)
A=matrix(rnorm(10),5,2)
B=matrix(rnorm(15),5,3)
Distance(A,B)
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