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 $n \geq 2s$. Based on this result, a new class of measurement matrices is presented 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和相位恢复领域的一些新结果。对于压缩感知，研究表明，如果未知的 $n$ 维向量可以表示为 $s$ 个未知范德蒙德向量（傅里叶向量为其特例）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 $O(\mathrm{poly}(s))$ 复杂度实现向量的精确恢复。基于此结果，本文提出了一类新的测量矩阵，使得当 $n \geq 2s$ 时，能以仅 $2s$ 次测量和 $O(\mathrm{poly}(s))$ 恢复算法复杂度，恢复 $s$ 稀疏的 $n$ 维向量。在论文第二部分，这些结果被推广到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 $s$ 个频率组成（至少包含一个非谐波频率），当 $n \geq 4s - 1$ 且至少进行 $8s-3$ 次傅里叶测量时，仅有两种可能的向量能产生观测到的测量值，且两者极易相互获取。通过复杂度仅为 $O(\mathrm{poly}(s))$ 的算法可找到这两个向量。这一新结果的直接应用是构造一种测量矩阵，使得当 $n \geq 4s - 1$ 时，能从 $O(s)$ 次幅度测量和 $O(\mathrm{poly}(s))$ 恢复复杂度下，恢复几乎所有的 $s$ 稀疏 $n$ 维信号（仅相差一个全局相位）。