Joint Low-Rank Factorizations with Shared and Unshared Components: Identifiability and Algorithms
Joint Low-Rank Factorizations with Shared and Unshared Components: Identifiability and Algorithms
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DOI：
10.23919/eusipco.2019.8903050
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发表时间：
2019-09
期刊：
影响因子：
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通讯作者：
Mikael Sørensen;N. Sidiropoulos
中科院分区：
文献类型：
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作者：
Mikael Sørensen;N. Sidiropoulos
我们研究了矩阵{X}=[\mathm{A}\\mathm{B}]\mathm{G}和$\mathm{Y}=[\mathm{A}\\mathm{C}]\mathm{H}的联合低阶分解，其中共享因子A的列对应于向量化的秩1矩阵，非共享因子B和C列满，矩阵G和H行满。首先说明了如果矩阵[A，B，C]有满列秩时，则可以从矩阵[X，Y]的零空间得到公因子矩阵A的列空间的基。这又意味着寻找共享因子矩阵A的问题归结为一个基本的典范多元分解(CPD)问题，在许多情况下，该问题可以通过特征值分解直接解决。接下来，我们解释了在计算矩阵[X，Y]的零空间时，通过考虑共享因子矩阵A的列的秩一的约束，可以得到更宽松的可辨识性条件，而不要求[A，B，C]具有满列秩.无约束零空间方法的优点是其算法简单，而一阶约束零空间方法的优点是它导致宽松的可辨识性条件。最后，简要讨论了一种由噪声观测矩阵X和Y计算共享因子矩阵A的非平衡正交Procrstes和CPD联合拟合法。
We study the joint low-rank factorization of the matrices $\mathrm{X}=[\mathrm{A}\ \mathrm{B}]\mathrm{G}$ and $\mathrm{Y}=[\mathrm{A}\ \mathrm{C}]\mathrm{H}$, in which the columns of the shared factor matrix A correspond to vectorized rank-one matrices, the unshared factors B and C have full column rank, and the matrices G and H have full row rank. The objective is to find the shared factor A, given only X and Y. We first explain that if the matrix [A B C] has full column rank, then a basis for the column space of the shared factor matrix A can be obtained from the null space of the matrix [X Y]. This in turn implies that the problem of finding the shared factor matrix A boils down to a basic Canonical Polyadic Decomposition (CPD) problem that in many cases can directly be solved by means of an eigenvalue decomposition. Next, we explain that by taking the rank-one constraint of the columns of the shared factor matrix A into account when computing the null space of the matrix [X Y], more relaxed identifiability conditions can be obtained that do not require that [A B C] has full column rank. The benefit of the unconstrained null space approach is that it leads to simple algorithms while the benefit of the rank-one constrained null space approach is that it leads to relaxed identifiability conditions. Finally, a joint unbalanced orthogonal Procrustes and CPD fitting approach for computing the shared factor matrix A from noisy observation matrices X and Y will briefly be discussed.