论文标题
循环矩阵倒置的张量等级范围和显式QTT表示
Tensor rank bounds and explicit QTT representations for the inverses of circulant matrices
论文作者
论文摘要
在本文中，我们关注的是循环矩阵及其量化张量训练（QTT）结构的反转。特别是，我们表明，由$ $ $（a_0，\ dots，a_ {m-1}，0，\ dots，0，a _ { - n}，\ dots，a _ { - 1}）^b by y qtt $ n等级$ qtt $ qtt $ n等级的第一列（a_0，\ dots，a_0，\ dots，a_0，\ dots，a_0，\ dots，a_0，\ dots，a_0，\ dots，a_0，\ dots，a_0，\ dots，a_0，\ dots）^qtt $ qtt $ n等级，是QTT $ n等级。根据$ a $的条目的某些假设，我们还得出了$ a^{ - 1} $的显式QTT表示。后者可以使用，例如，在数值以QTT格式的周期性边界条件求解微分方程时会产生的稳定性问题。
In this paper, we are concerned with the inversion of circulant matrices and their quantized tensor-train (QTT) structure. In particular, we show that the inverse of a complex circulant matrix $A$, generated by the first column of the form $(a_0,\dots,a_{m-1},0,\dots,0,a_{-n},\dots, a_{-1})^\top$ admits a QTT representation with the QTT ranks bounded by $(m+n)$. Under certain assumptions on the entries of $A$, we also derive an explicit QTT representation of $A^{-1}$. The latter can be used, for instance, to overcome stability issues arising when numerically solving differential equations with periodic boundary conditions in the QTT format.
