标题： 最优压缩感知对于混合随机过程 显示英文标题

标题： Optimal compressed sensing for mixing stochastic processes

摘要： 贾拉利和普尔为随机过程的压缩感知引入了一个渐近框架，证明了任何严格大于平均信息维数的速率都是用于（通用）几乎无损恢复的随机线性测量数量的上界，对于以归一化$L^2$范数衡量的$\psi^*$-混合过程而言。 在本工作中，我们表明，如果随机线性测量的归一化数量严格小于平均信息维数，则对于$\psi^*$-混合过程来说，通过任何解码器序列都无法实现几乎无损恢复。 这确立了平均信息维数作为该情况下压缩感知的基本极限（实际上，是该问题的精确阈值）。 为此，我们引入了一个新的量，与几何测度论中的技术相关：相关维数率，它被证明是任意平稳随机过程压缩感知的下界。 显示英文摘要

摘要： Jalali and Poor introduced an asymptotic framework for compressed sensing of stochastic processes, demonstrating that any rate strictly greater than the mean information dimension serves as an upper bound on the number of random linear measurements required for (universal) almost lossless recovery of $\psi^*$-mixing processes, as measured in the normalized $L^2$ norm. In this work, we show that if the normalized number of random linear measurements is strictly less than the mean information dimension, then almost lossless recovery of a $\psi^*$-mixing process is impossible by any sequence of decompressors. This establishes the mean information dimension as the fundamental limit for compressed sensing in this setting (and, in fact, the precise threshold for the problem). To this end, we introduce a new quantity, related to techniques from geometric measure theory: the correlation dimension rate, which is shown to be a lower bound for compressed sensing of arbitrary stationary stochastic processes.