In the study of large random systems, researchers often need to simulate dynamics in the form of iterated matrix-vector multiplications interspersed with nonlinear operations. Examples include message passing algorithms, gradient descent, and matrix iterative methods for extremal eigenvalue calculations. This paper proposes a new algorithm, named Householder Dice (HD), for simulating such dynamics on several random matrix ensembles with translation-invariant properties. Examples include the Gaussian ensemble, the Haar-distributed random orthogonal ensemble, and their complex-valued counterparts. A "direct" approach to the simulation, where one first generates a dense $n \times n$ matrix from the ensemble, requires at least $\mathcal{O}(n^2)$ resource in space and time. The HD algorithm overcomes this $\mathcal{O}(n^2)$ bottleneck by using the principle of deferred decisions: rather than fixing the entire random matrix in advance, it lets the randomness unfold with the dynamics. Key to this matrix-free construction is an adaptive and recursive construction of (random) Householder reflectors. These orthogonal transformations exploit the group symmetry of the matrix ensembles, while simultaneously maintaining the statistical correlations induced by the dynamics. The memory and computation costs of the HD algorithm are $\mathcal{O}(nT)$ and $\mathcal{O}(nT^2)$, respectively, with $T$ being the number of iterations. When $T \ll n$, which is nearly always the case in practice, the HD algorithm leads to significant reductions in runtime and memory footprint. Numerical results demonstrate the promise of the new algorithm as a new computational tool in the study of high-dimensional random systems.

翻译：在大型随机系统的研究中,研究人员往往需要以非线性操作的迭代矩阵-矢量乘法的形式模拟动态。示例包括信息传递算法、梯度下移和矩阵迭代方法来计算 extremal eigenval 。 本文提出一个新的算法, 名为Descripter Dice (HD), 用于在多个随机矩阵中模拟这种动态, 包含翻译不易变的属性。 例如, 高西亚共和( Haar) 分配的随机或全方位共和( Haar), 以及他们的复合估值对应方。 模拟的“ 直接” 方法, 第一次从 entemble 生成密度 $\ time n$ 的 矩阵 。 本文提议一个新的算法, 以延迟决定原则的方式克服了这种 $\ mathal- mal commexcial commal commexplainal ral ral_ ral max max 。 max max max max max max max max max max max max max max max max max max max max max max max max max max max max maxx maxx max maxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx