We study two fundamental optimization problems: (1) scaling a symmetric positive definite matrix by a positive diagonal matrix so that the resulting matrix has row and column sums equal to 1; and (2) minimizing a quadratic function subject to hard non-negativity constraints. Both problems lend themselves to efficient algorithms based on interior point methods (IPMs). For general instances, standard self-concordance theory places a limit on the iteration complexity of these methods at $\widetilde{O}\left(n^{1/2}\right)$, where $n$ denotes the matrix dimension. We show via an amortized analysis that, when the input matrix is an M-matrix, an IPM with adaptive step sizes solves both problems in only $\widetilde{O}\left(n^{1/3}\right)$ iterations. As a corollary, using fast Laplacian solvers, we obtain an $\ell_{2}$ flow diffusion algorithm with depth $\widetilde{O}\left(n^{1/3}\right)$ and work $\widetilde{O}$$\left(n^{1/3}\cdot\text{nnz}\right)$. This result marks a significant instance in which a standard log-barrier IPM permits provably fewer than $\Theta\left(n^{1/2}\right)$ iterations.

翻译：我们研究两个基本优化问题：(1) 通过正对角矩阵缩放对称正定矩阵，使结果矩阵的行和与列和均等于1；(2) 在严格非负约束下最小化二次函数。这两个问题都适用于基于内点法的高效算法。对于一般情况，标准的自协调理论将这些方法的迭代复杂度限制在$\widetilde{O}\left(n^{1/2}\right)$，其中$n$表示矩阵维度。我们通过摊销分析证明，当输入矩阵为M矩阵时，采用自适应步长的内点法仅需$\widetilde{O}\left(n^{1/3}\right)$次迭代即可解决这两个问题。作为推论，利用快速拉普拉斯求解器，我们得到了深度为$\widetilde{O}\left(n^{1/3}\right)$、工作量为$\widetilde{O}$$\left(n^{1/3}\cdot\text{nnz}\right)$的$\ell_{2}$流扩散算法。该结果标志着标准对数障碍内点法在可证明少于$\Theta\left(n^{1/2}\right)$次迭代的情况下取得的重要突破。