Permutation Matrix Sample Clauses

Permutation Matrix. ‌ On its own, the fact that a permutation matrix Q exists which reorders U1 and V1 so that their columns align with the truncation method is not particularly helpful for those who wish to implement the actual permutation. Here, we detail the construc- tion of this permutation matrix and discuss efficient implementation. We derive the permutation using U1 as an example, but the same arguments apply to V1 and Σ1. For those seeking only an explicit formula, see Equation (6.4). The goal of the permutation matrix Q, when applied on the right-hand side of U1, is to move the columns corresponding to a truncation of the matrices that form U1 to the leftmost columns, stably, and to move all of the displaced columns to the right of those, also stably. (A stable sorting retains the original ordering of equivalent values; in this case, there are three equivalent bins: columns included in the truncation, columns displaced by the reordering of the first bin, and columns which are unmoved in the permutation.) Consider an example in which UA, UB ∈ R10×10 where we want to truncate to the first 4 columns of UA and the first 3 columns of UB (that is, A = 4 and m = 3). If we form the full Kronecker product U1 = UA ⊗ UB, the columns that we want to keep are not adjacently located, as shown in Figure 6.1.
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