次线性时间、测量最优、稀疏恢复
Sublinear Time, Measurement-Optimal, Sparse Recovery For All
摘要：
在ell_1范数下的近似稀疏恢复系统形式上由参数N、k、m × N测量矩阵Phi和解码算法D组成。给定一个向量x，其中x_k表示x的最佳k项逼近，系统通过hat_x = D（φ x）逼近x，它必须满足||hat_x - x||_1 <=（1+ 1）||x-x_k||_1。设计这样的系统的目标之一是最小化m和D的运行时间。我们考虑“forall”模型，其中单个矩阵Phi用于所有信号x。所有使用最佳测量数m=O（k log（N/k））的先前算法需要超线性时间Omega（N log（N/k））。在本文中，我们给出了这个问题的第一个算法，使用的最佳数量的测量（到一个常数因子）和运行在次线性时间为O（N）时，k=O（N），假设访问一个数据结构需要空间和预处理O（N）。
英文摘要：
An approximate sparse recovery system in ell_1 norm formally consists of parameters N, k, epsilon an m-by-N measurement matrix, Phi, and a decoding algorithm, D. Given a vector, x, where x_k denotes the optimal k-term approximation to x, the system approximates x by hat_x = D(Phi.x), which must satisfy ||hat_x - x||_1 <= (1+epsilon)||x - x_k||_1. Among the goals in designing such systems are minimizing m and the runtime of D. We consider the "forall" model, in which a single matrix Phi is used for all signals x. All previous algorithms that use the optimal number m=O(k log(N/k)) of measurements require superlinear time Omega(N log(N/k)). In this paper, we give the first algorithm for this problem that uses the optimum number of measurements (up to a constant factor) and runs in sublinear time o(N) when k=o(N), assuming access to a data structure requiring space and preprocessing O(N).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
