样本有效的低秩相位恢复
Sample-Efficient Low Rank Phase Retrieval
摘要：
本文研究了低秩相位恢复（LRPR）问题：当每个$y_k$是包含$x^*_k$的独立无相位线性投影的m长度向量时，从$y_k = |A_k^\top x^*_k|$，$k=1, 2,..., q$恢复$n \times q$秩- $r$矩阵$X^*$.不同基质$A_k$为i.i.d.。并且每个含有i.i.d.标准高斯输入我们得到了一个改进的保证AltMinLowRaP，这是一个交替的最小化解决方案，LRPR的介绍和研究，在我们最近的工作。只要$X^*$的右奇异向量满足不相干假设，我们就可以证明，如果测量总数为$mq \gtrsim nr^2 (r + \log(1/ε))$，则AltMinLowRaP估计几何收敛于$X^*$。此外，由于我们的问题具有特定的不对称性，我们还需要$m \gtrsim max(r, \log q, \log n)$。与我们最近的工作相比，我们将AltMin迭代的样本复杂度提高了$r^2$倍，将初始化的样本复杂度提高了$r$倍。我们还将我们的结果扩展到嘈杂的情况下，我们证明了稳定的腐败小加性噪声。
英文摘要：
This work studies the Low Rank Phase Retrieval (LRPR) problem: recover an $n \times q$ rank-$r$ matrix $X^*$ from $y_k = |A_k^\top x^*_k|$, $k=1, 2,..., q$, when each $y_k$ is an m-length vector containing independent phaseless linear projections of $x^*_k$. The different matrices $A_k$ are i.i.d. and each contains i.i.d. standard Gaussian entries. We obtain an improved guarantee for AltMinLowRaP, which is an Alternating Minimization solution to LRPR that was introduced and studied in our recent work. As long as the right singular vectors of $X^*$ satisfy the incoherence assumption, we can show that the AltMinLowRaP estimate converges geometrically to $X^*$ if the total number of measurements $mq \gtrsim nr^2 (r + \log(1/ε))$. In addition, we also need $m \gtrsim max(r, \log q, \log n)$ because of the specific asymmetric nature of our problem. Compared to our recent work, we improve the sample complexity of the AltMin iterations by a factor of $r^2$, and that of the initialization by a factor of $r$. We also extend our result to the noisy case; we prove stability to corruption by small additive noise.
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。
