Householder Dice: A Matrix-Free Algorithm for Simulating Dynamics on Gaussian and Random Orthogonal Ensembles

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.