We introduce Polytopic Matrix Factorization (PMF) as a novel data decomposition approach. In this new framework, we model input data as unknown linear transformations of some latent vectors drawn from a polytope. In this sense, the article considers a semi-structured data model, in which the input matrix is modeled as the product of a full column rank matrix and a matrix containing samples from a polytope as its column vectors. The choice of polytope reflects the presumed features of the latent components and their mutual relationships. As the factorization criterion, we propose the determinant maximization (Det-Max) for the sample autocorrelation matrix of the latent vectors. We introduce a sufficient condition for identifiability, which requires that the convex hull of the latent vectors contains the maximum volume inscribed ellipsoid of the polytope with a particular tightness constraint. Based on the Det-Max criterion and the proposed identifiability condition, we show that all polytopes that satisfy a particular symmetry restriction qualify for the PMF framework. Having infinitely many polytope choices provides a form of flexibility in characterizing latent vectors. In particular, it is possible to define latent vectors with heterogeneous features, enabling the assignment of attributes such as nonnegativity and sparsity at the subvector level. The article offers examples illustrating the connection between polytope choices and the corresponding feature representations.
翻译:我们引入了多元体矩阵系数(PMF),作为一种新型的数据分解方法。在这个新框架中,我们将输入数据作为从多层矢量中提取的某些潜在矢量的未知线性变换模型来模拟输入数据。在这个意义上,本文章认为一个半结构化的数据模型,在这个模型中,输入矩阵的模型是完整列级矩阵的产物,而一个包含从多层体矢量中提取样本的矩阵作为其柱矢量的模型。多层的选择反映了潜在组成部分及其相互关系的假定特征。作为系数化标准,我们建议对潜层矢量的样本自动调节矩阵的决定因素最大化(Det-Max)进行模型。我们引入一个充分的可识别性条件,要求潜层矢量体的矩体结构包含一个全列排名矩阵的模型的模型,其中含有以特别紧凑紧凑性限制的聚合体矩阵矩阵的模型。根据Det-MF框架的所有符合特定对等度限制要求的多元形体。具有无限多数值的多元性选择,在矢量和矢量的矢量中,使矢量的矢量的可变性具有可变的可变性特性的可塑性。