We present a unification and generalization of sequentially and hierarchically semi-separable (SSS and HSS) matrices called tree semi-separable (TSS) matrices. Our main result is to show that any dense matrix can be expressed in a TSS format. Here, the dimensions of the generators are specified by the ranks of the Hankel blocks of the matrix. TSS matrices satisfy a graph-induced rank structure (GIRS) property. It is shown that TSS matrices generalize the algebraic properties of SSS and HSS matrices under addition, products, and inversion. Subsequently, TSS matrices admit linear time matrix-vector multiply, matrix-matrix multiply, matrix-matrix addition, inversion, and solvers.

翻译：我们提出了一种统一并推广序列半可分离矩阵（SSS）与分层半可分离矩阵（HSS）的方法，称为树半可分离矩阵（TSS）。核心结论表明，任意稠密矩阵均可表示为TSS格式。其中，生成元的维度由矩阵Hankel块的秩决定。TSS矩阵满足图诱导的秩结构（GIRS）性质。研究显示，在加法、乘法及求逆运算下，TSS矩阵统一继承了SSS与HSS矩阵的代数性质。基于此，TSS矩阵支持线性时间的矩阵-向量乘法、矩阵-矩阵乘法、矩阵-矩阵加法、求逆及求解器运算。