In this work, we present some new results for compressed sensing and phase retrieval. For compressed sensing, it is shown that if the unknown $n$-dimensional vector can be expressed as a linear combination of $s$ unknown Vandermonde vectors (with Fourier vectors as a special case) and the measurement matrix is a Vandermonde matrix, exact recovery of the vector with $2s$ measurements and $O(\mathrm{poly}(s))$ complexity is possible when $n \geq 2s$. From these results, a measurement matrix is constructed from which it is possible to recover $s$-sparse $n$-dimensional vectors for $n \geq 2s$ with as few as $2s$ measurements and with a recovery algorithm of $O(\mathrm{poly}(s))$ complexity. In the second part of the work, these results are extended to the challenging problem of phase retrieval. The most significant discovery in this direction is that if the unknown $n$-dimensional vector is composed of $s$ frequencies with at least one being non-harmonic, $n \geq 4s - 1$ and we take at least $8s-3$ Fourier measurements, there are, remarkably, only two possible vectors producing the observed measurement values and they are easily obtainable from each other. The two vectors can be found by an algorithm with only $O(\mathrm{poly}(s))$ complexity. An immediate application of the new result is construction of a measurement matrix from which it is possible to recover almost all $s$-sparse $n$-dimensional signals (up to a global phase) from $O(s)$ magnitude-only measurements and $O(\mathrm{poly}(s))$ recovery complexity when $n \geq 4s - 1$.

翻译：本文提出了压缩感知与相位恢复领域的一些新结果。针对压缩感知问题，研究表明：若未知$n$维向量可表示为$s$个未知范德蒙德向量（傅里叶向量为特例）的线性组合，且测量矩阵为范德蒙德矩阵，则当$n \geq 2s$时，可通过$2s$次测量和$O(\mathrm{poly}(s))$复杂度的算法实现精确恢复。基于此结果，我们构造了一种测量矩阵，当$n \geq 2s$时，仅需$2s$次测量和$O(\mathrm{poly}(s))$复杂度的恢复算法即可实现$s$稀疏$n$维向量的恢复。在论文第二部分，我们将上述结果扩展至更具挑战性的相位恢复问题。该方向最重要的发现是：若未知$n$维向量由$s$个频率成分组成（其中至少一个为非谐波频率），且满足$n \geq 4s - 1$，则只需采集至少$8s-3$次傅里叶测量，即可惊人地发现仅有两种可能的向量能产生观测测量值，且两者之间极易相互转换。通过仅需$O(\mathrm{poly}(s))$复杂度的算法即可求得这两个向量。该新结果的直接应用是构造一种测量矩阵，当$n \geq 4s - 1$时，可从$O(s)$次幅度测量中以$O(\mathrm{poly}(s))$的恢复复杂度恢复几乎所有$s$稀疏$n$维信号（全局相位不确定性除外）。