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：我们提出了首个专为顺序随机投影设计的概率框架，该方法根植于不确定性下顺序决策的挑战。由于随机变量的顺序依赖性和高维特性（这是顺序决策过程中自适应机制带来的副产物），分析过程变得复杂。我们的工作创新性地构建了一个停时过程，便于分析以顺序方式相互关联的一系列集中事件。通过在该停时过程导出的自正则化过程中应用混合方法，我们得到了理想的非渐近概率界。该界是Johnson-Lindenstrauss (JL) 引理的非平凡鞅推广，标志着随机投影与顺序分析文献中的开创性贡献。