关于按元素幂矩阵分解的复杂性
On the Complexity of Low-Rank Matrix Signing and Entrywise Power Matrix Factorization
AI总结：
研究非负矩阵的按元素幂矩阵分解，分析其精确决策和弗罗贝尼乌斯范数近似问题的计算复杂性，建立完整复杂性图景，明确精确情形与符号问题等价及相关复杂度结果，近似情形下也证明了NP难。
AI中文摘要：
给定非负矩阵\(X\)、分解秩\(r\)和实参数\(p\)，按元素幂矩阵分解（EPMF）寻找低秩矩阵\(X_r\)使\(X = |X_r|^{\circ p}\)（精确情形）或\(X \approx |X_r|^{\circ p}\)（近似情形）。分析了精确决策和弗罗贝尼乌斯范数近似问题的计算复杂性，建立了完整的复杂性图景。
英文摘要：
Given a nonnegative matrix $X$, a factorization rank $r$ and {a positive integer $p$}, entrywise power matrix factorization (EPMF) looks for a low-rank matrix $X_r$ such that $X = |X_r|^{\circ p}$ (exact case) or $X \approx |X_r|^{\circ p}$ (approximate case), where $(\cdot)^{\circ p}$ denotes the componentwise exponent. EPMF includes the modulus model ($p=1$) and componentwise square factorization ($p=2$) as special cases, the latter being closely related to the square root rank. We analyze the computational complexity of the exact decision problem and the Frobenius-norm approximation problem, and establish a complete complexity landscape. In the exact case, we show that EPMF is equivalent to the combinatorial problem of flipping the signs of the entries of a given matrix $X$ to obtain a rank-$r$ matrix, which we refer to as the low-rank matrix signing (LRMS) problem. We first show that LRMS, and hence exact EPMF, is strongly NP-hard, improving a weak NP-hardness result for the square-root-rank (Math. Prog., 2015). We then show that LRMS can be solved in polynomial time when $r$ is fixed. Moreover, when the rank $r$ is part of the input, we show that for generic matrices the algorithm is fixed-parameter tractable (FPT) in the parameter $r$; in fact, the running time is fixed-parameter linear in the number of entries of the input matrix. In the approximate case using the Frobenius norm as an error measure, we show that EPMF is NP-hard, already when $r=2$, the smallest nontrivial case.
