Comparing Subspace Clusterings
IEEE Transactions on Knowledge and Data Engineering
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The cosines of the principal angles between the column spaces of full column rank matrices $X\in{\hbox{{\bf R}$^{m\times p}$}}$ and $Y\in{\hbox{{\bf R}$^{m\times q}$}}$ are efficiently computed, using the Björck--Golub algorithm, as the singular values of $Q_x^{T}Q_y,$ where Qx and Qy are orthonormal matrices computed by the QR factorizations of X and Y, respectively. This paper shows that the Björck--Golub algorithm is mixed stable in the following sense: the computed singular values approximate with small relative error the exact cosines of the principal angles between the column spaces of $X+\Delta X$ and $Y+\Delta Y,$ where $\Delta X,$ $\Delta Y$ are small backward errors. Further, theoretical analysis and numerical evidence show that the algorithm becomes more robust if the QR factorizations are computed with the complete pivoting scheme of Powell and Reid. Moreover, it is shown that Gaussian elimination with complete pivoting can be used as an efficient preconditioner in computation and as a useful tool in analysis of the sensitivity of the QR factorization.