In probabilistic state inference, we seek to estimate the state of an (autonomous) agent from noisy observations. It can be shown that, under certain assumptions, finding the estimate is equivalent to solving a linear least squares problem. Solving such a problem is done by calculating the upper triangular matrix $\boldsymbol R$ from the coefficient matrix $\boldsymbol A$, using the QR or Cholesky factorizations; this matrix is commonly referred to as the “square root matrix”. In sequential estimation problems, we are often interested in periodic optimization of the state variable order, e.g., to reduce fill-in, or to apply a predictive variable ordering tactic; however, changing the variable order implies expensive re-factorization of the system. Thus, we address the problem of modifying an existing square root matrix $\boldsymbol R$, to convey reordering of the variables. To this end, we identify several conclusions regarding the effect of column permutation on the factorization, to allow efficient modification of $\boldsymbol R$, without accessing $\boldsymbol A$ at all, or with minimal re-factorization. The proposed parallelizable algorithm achieves a significant improvement in performance over the state-of-the-art incremental Smoothing And Mapping (iSAM2) algorithm, which utilizes incremental factorization to update $\boldsymbol R$.

中文翻译：

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上三角平方根矩阵对变量重排序的有效修改

在概率状态推断中，我们试图从嘈杂的观察中估计（自主）代理的状态。可以证明，在某些假设下，找到估计等效于解决线性最小二乘问题。解决这样的问题是通过计算上三角矩阵来完成的$\粗体符号 R$ 从系数矩阵 $\boldsymbol A$，使用 QR 或 Cholesky 分解；这个矩阵通常被称为“平方根矩阵”。在顺序估计问题中，我们经常对状态变量顺序的周期性优化感兴趣，例如，减少填充，或应用预测变量排序策略；然而，改变变量顺序意味着对系统进行昂贵的重构。因此，我们解决了修改现有平方根矩阵的问题$\粗体符号 R$, 传达变量的重新排序。为此，我们确定了关于列置换对分解的影响的几个结论，以允许有效修改$\粗体符号 R$, 无需访问 $\boldsymbol A$完全或最小的重构。与最先进的增量平滑和映射 (iSAM2) 算法相比，所提出的可并行化算法在性能上取得了显着提高，该算法利用增量分解来更新$\粗体符号 R$.