We prove that every element of the special linear group can be represented as the product of at most six block unitriangular matrices, and that there exist matrices for which six products are necessary, independent of indexing. We present an analogous result for the general linear group. These results serve as general statements regarding the representational power of alternating linear updates. The factorizations and lower bounds of this work immediately imply tight estimates on the expressive power of linear affine coupling blocks in machine learning.

翻译：我们证明了特殊线性群中的每个元素都可以表示为最多六个分块单位三角矩阵的乘积，并且存在一些矩阵，其六个乘积是必要的，这一结果与指标选取无关。我们给出了关于一般线性群的类似结论。这些结果可作为交替线性更新表示能力的通用性论断。本文中的分解与下界直接推演出机器学习中线性仿射耦合块表达能力的紧致估计。