压缩感知中基于总结的改进次线性恢复结构
Summary Based Structures with Improved Sublinear Recovery for Compressed Sensing
摘要：
我们引入了一类新的测量矩阵的压缩感知，使用低阶摘要在给定长度的二进制序列。我们证明恢复保证三个重建算法，使用建议的测量，包括$\ell_1$最小化和两个组合的方法。具体地，算法之一在次线性时间$\text{poly}(k\log{N})$中恢复$k$ -长度为$N$的稀疏向量，并且最多需要$Ω(k\log{N}\log\log{N})$测量。该算法的经验过采样常数明显优于现有的次线性恢复算法，如Chaining Pursuit和Sudocodes。特别地，对于$10^3\leq N\leq 10^8$和$k=100$，过采样因子在3到8之间。我们提供了初步的见解，如何建议的建设，和快速恢复计划可以用于一些实际应用，如市场篮子分析，和实时的压缩感知实现。
英文摘要：
We introduce a new class of measurement matrices for compressed sensing, using low order summaries over binary sequences of a given length. We prove recovery guarantees for three reconstruction algorithms using the proposed measurements, including $\ell_1$ minimization and two combinatorial methods. In particular, one of the algorithms recovers $k$-sparse vectors of length $N$ in sublinear time $\text{poly}(k\log{N})$, and requires at most $Ω(k\log{N}\log\log{N})$ measurements. The empirical oversampling constant of the algorithm is significantly better than existing sublinear recovery algorithms such as Chaining Pursuit and Sudocodes. In particular, for $10^3\leq N\leq 10^8$ and $k=100$, the oversampling factor is between 3 to 8. We provide preliminary insight into how the proposed constructions, and the fast recovery scheme can be used in a number of practical applications such as market basket analysis, and real time compressed sensing implementation.
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