论文标题
关于多面体矩阵分解的可识别多层表征
On Identifiable Polytope Characterization for Polytopic Matrix Factorization
论文作者
论文摘要
多重矩阵分解（PMF）是一种最近引入的矩阵分解方法，其中将数据向量建模为来自多层样品的线性变换。生成PMF模型中原始因素的成功恢复基于所选多层室的“可识别性”。在本文中，我们调查了确定多层室的可识别性的问题。可识别性条件要求多层列表为置换，并且只有/或符号不变。我们展示了如何使用图自动形态算法有效地解决此问题。特别是，我们表明，仅检查多层线的线性自动形态组的生成集，该组对应于边缘色完整图的自动形态组就足够了。该属性阻止检查置换组的所有元素，这需要阶乘算法复杂性。我们通过一些数值实验证明了所提出的方法的可行性。
Polytopic matrix factorization (PMF) is a recently introduced matrix decomposition method in which the data vectors are modeled as linear transformations of samples from a polytope. The successful recovery of the original factors in the generative PMF model is conditioned on the "identifiability" of the chosen polytope. In this article, we investigate the problem of determining the identifiability of a polytope. The identifiability condition requires the polytope to be permutation-and/or-sign-only invariant. We show how this problem can be efficiently solved by using a graph automorphism algorithm. In particular, we show that checking only the generating set of the linear automorphism group of a polytope, which corresponds to the automorphism group of an edge-colored complete graph, is sufficient. This property prevents checking all the elements of the permutation group, which requires factorial algorithm complexity. We demonstrate the feasibility of the proposed approach through some numerical experiments.
