We introduce the first probabilistic framework tailored for sequential random projection, an approach rooted in the challenges of sequential decision-making under uncertainty. The analysis is complicated by the sequential dependence and high-dimensional nature of random variables, a byproduct of the adaptive mechanisms inherent in sequential decision processes. Our work features a novel construction of a stopped process, facilitating the analysis of a sequence of concentration events that are interconnected in a sequential manner. By employing the method of mixtures within a self-normalized process, derived from the stopped process, we achieve a desired non-asymptotic probability bound. This bound represents a non-trivial martingale extension of the Johnson-Lindenstrauss (JL) lemma, marking a pioneering contribution to the literature on random projection and sequential analysis.

翻译：我们提出了首个专用于序列随机投影的概率框架，该方法植根于不确定性下的序列决策挑战。由于序列依赖性和随机变量的高维特性（这是序列决策过程中自适应机制带来的副产品），分析过程变得复杂。我们的工作创新性地构建了一个停止过程，从而简化了对一系列相互串联的集中事件的分析。通过在此停止过程导出的自归一化过程中使用混合方法，我们获得了理想的非渐近概率界。该界是对约翰逊-林登斯特劳斯（JL）引理的非平凡鞅推广，标志着随机投影与序列分析文献中的开创性贡献。