通过凸规划从二次测量中恢复稀疏信号
Sparse Signal Recovery from Quadratic Measurements via Convex Programming
AI总结：
本文研究从二次测量中恢复稀疏信号的问题，提出通过凸优化程序以高概率恢复信号（至多相差一个常数因子），并给出稀疏度的充分和必要条件。
摘要：
本文考虑二次方程组 |<z_j, x>|^2 = b_j, j = 1, ..., m，其中 x ∈ R^n 未知，而正态随机向量 z_j ∈ R^n 和二次测量值 b_j ∈ R 已知。假设系统是欠定的，即 m < n。我们证明，如果存在稀疏解 x（即 x 最多有 k 个非零分量），则通过求解凸优化程序，可以以高概率恢复 x（至多相差一个常数因子），只要 k ≤ O((m/log n)^(1/2))。另一方面，我们证明对于一类朴素的凸松弛方法，k ≤ O(log n (m)^(1/2)) 是精确恢复的必要条件。
英文摘要：
In this paper we consider a system of quadratic equations |<z_j, x>|^2 = b_j, j = 1, ..., m, where x in R^n is unknown while normal random vectors z_j in R_n and quadratic measurements b_j in R are known. The system is assumed to be underdetermined, i.e., m < n. We prove that if there exists a sparse solution x, i.e., at most k components of x are non-zero, then by solving a convex optimization program, we can solve for x up to a multiplicative constant with high probability, provided that k <= O((m/log n)^(1/2)). On the other hand, we prove that k <= O(log n (m)^(1/2)) is necessary for a class of naive convex relaxations to be exact.
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。
