发表机构
Georgia Institute of Technology（佐治亚理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文分析有限精度下Gram-Schmidt游走的集中性，推导出修正矩生成函数界，并证明更新误差不可避免，通过消融实验验证了理论结果。
AI 中文摘要
Gram-Schmidt游走是一种随机向量平衡算法，其亚高斯保证支持在差异、实验设计和数据压缩中的应用；然而，这些理论保证是在精确算术中建立的，而实际实现必须在有限精度下近似最小二乘方向、边界更新和采样概率。这一点很重要，因为微小的数值误差可以改变哪些坐标被冻结，从而改变后续的轨迹。我们直接在有界、可能带有偏差且依赖于历史的误差下分析扰动后的Gram-Schmidt游走的集中性。对于欧几里得范数至多为1的$n$个输入向量，我们获得了一个依赖于关键误差源的修正矩生成函数界，当误差趋近于零时，该界恢复原始结果。我们还构造了一个满列秩实例，其中有界更新误差产生的偏差阶为$\min\{n^2\varepsilon,n\}$，表明在此模型下更新不可避免地累积误差。最后，我们通过消融位精度和问题规模，在各种设置中验证了我们的发现。
英文摘要
The Gram-Schmidt Walk is a randomized vector-balancing algorithm whose subgaussian guarantees support applications in discrepancy, experimental design, and data compression; however, these theoretical guarantees are established in exact arithmetic, whereas implementations must approximate least-squares directions, boundary updates, and sampling probabilities in finite precision. This is important because small numerical errors can change which coordinates freeze and thereby alter the subsequent trajectory. We analyze the concentration of the perturbed Gram-Schmidt walk directly under bounded, potentially biased and history-dependent errors. For $n$ input vectors of Euclidean norm at most one, we obtain a modified MGF bound depending on key error sources which recovers the original result as the error goes to zero. We also construct a full-column-rank instance in which bounded update errors produce bias of order $\min\{n^2\varepsilon,n\}$, showing that updates accumulate error unavoidably under this model. Finally, we validate our findings in a variety of settings by ablating on the bit precision and problem size.
