自适应压缩感知的改进算法
Improved Algorithms for Adaptive Compressed Sensing
摘要：
在自适应压缩感测的问题中，想要从$m$线性测量$A_1 x, A_2 x,\ldots, A_m x$估计近似$k$ -稀疏向量$x\in\mathbb{R}^n$，其中可以基于先前测量的结果$A_1 x,\ldots, A_{i-1} x$来选择$A_i$。目标是输出向量$\hat{x}$，其中$$\|x-\hat{x}\|_p \le C \cdot \min_{k\text{-sparse } x'} \|x-x'\|_q\,$$的概率至少为$2/3$，其中$C > 0$是近似因子。Indyk、Price和Woodruff（FOCS'11）给出了$C = 1+ε$的$p=q=2$算法，其中$\Oh((k/ε) \loglog (n/k))$测量和$\Oh(\log^*(k) \loglog (n))$轮自适应。我们首先改善他们的界限，获得一个计划与$\Oh(k \cdot \loglog (n/k) +(k/ε) \cdot \loglog(1/ε))$测量和$\Oh(\log^*(k) \loglog (n))$轮，以及一个计划与$\Oh((k/ε) \cdot \loglog (n\log (n/k)))$测量和最佳$\Oh(\loglog (n))$轮。然后，我们提供了新的自适应压缩感知方案与$(p,p)$为每个$0 < p < 2$的改进的界限。我们表明，在自适应设置中从$O(k \log(n/k))$测量到$O(k \log \log (n/k))$测量的改进可以持续，对于$p$和$q$的其他值具有更好的$ε$依赖性。例如，当$(p,q) = (1,1)$时，我们获得$O(\frac{k}{\sqrtε} \cdot \log \log n \log^3 (\frac{1}ε))$测量值。
英文摘要：
In the problem of adaptive compressed sensing, one wants to estimate an approximately $k$-sparse vector $x\in\mathbb{R}^n$ from $m$ linear measurements $A_1 x, A_2 x,\ldots, A_m x$, where $A_i$ can be chosen based on the outcomes $A_1 x,\ldots, A_{i-1} x$ of previous measurements. The goal is to output a vector $\hat{x}$ for which $$\|x-\hat{x}\|_p \le C \cdot \min_{k\text{-sparse } x'} \|x-x'\|_q\,$$ with probability at least $2/3$, where $C > 0$ is an approximation factor. Indyk, Price and Woodruff (FOCS'11) gave an algorithm for $p=q=2$ for $C = 1+ε$ with $\Oh((k/ε) \loglog (n/k))$ measurements and $\Oh(\log^*(k) \loglog (n))$ rounds of adaptivity. We first improve their bounds, obtaining a scheme with $\Oh(k \cdot \loglog (n/k) +(k/ε) \cdot \loglog(1/ε))$ measurements and $\Oh(\log^*(k) \loglog (n))$ rounds, as well as a scheme with $\Oh((k/ε) \cdot \loglog (n\log (n/k)))$ measurements and an optimal $\Oh(\loglog (n))$ rounds. We then provide novel adaptive compressed sensing schemes with improved bounds for $(p,p)$ for every $0 < p < 2$. We show that the improvement from $O(k \log(n/k))$ measurements to $O(k \log \log (n/k))$ measurements in the adaptive setting can persist with a better $ε$-dependence for other values of $p$ and $q$. For example, when $(p,q) = (1,1)$, we obtain $O(\frac{k}{\sqrtε} \cdot \log \log n \log^3 (\frac{1}ε))$ measurements.
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