We show that, under certain circumstances, it is possible to automatically compute Jacobian-inverse-vector and Jacobian-inverse-transpose-vector products about as efficiently as Jacobian-vector and Jacobian-transpose-vector products. The key insight is to notice that the Jacobian corresponding to the use of one basis function is of a form whose sparsity is invariant to inversion. The main restriction of the method is a constraint on the number of active variables, which suggests a variety of techniques or generalization to allow the constraint to be enforced or relaxed. This technique has the potential to allow the efficient direct calculation of Newton steps as well as other numeric calculations of interest.

翻译：我们证明，在特定条件下，自动计算雅可比逆向量与雅可比逆转置向量乘积的效率，可与计算雅可比向量及雅可比转置向量乘积的效率相当。关键发现在于，对应于单一基函数使用的雅可比矩阵具有稀疏性在求逆过程中保持不变的形式。该方法的主要限制在于对活跃变量数量的约束，这提示了多种技术手段或泛化方案以实施或放宽该约束。此技术有望实现牛顿步长的高效直接计算，以及其他具有价值的数值计算。