论文标题
样品有效的低等级期检索
Sample-Efficient Low Rank Phase Retrieval
论文作者
论文摘要
这项工作研究了低排名阶段检索（LRPR）问题：恢复$ n \ times q $ strix- $ r $ r $ x^*$来自$ y_k = | y_k = | a_k^\ top x^*_ k | $，$ k = 1，2，2，...，...，q $，每个$ y_k $ co $ y_k $ co $ c $ co $不同的矩阵$ a_k $是i.i.d.每个都包含I.I.D.标准高斯条目。我们为Altminlowrap获得了改进的保证，这是对LRPR的交替最小化解决方案，在我们最近的工作中引入和研究了。只要$ x^*$的正确单数向量满足不一致的假设，我们就可以证明，如果测量总数$ MQ \ gtrsim nr^2（r + \ log（r + \ log（1/ε）））$。此外，由于我们问题的特定非对称性质，我们还需要$ m \ gtrsim max（r，\ log q，\ log n）$。与我们最近的工作相比，我们将Altmin迭代的样本复杂性提高了$ r^2 $，而初始化的样本复杂性则提高了$ r $。我们还将结果扩展到嘈杂的情况；我们证明了通过小型添加噪声稳定腐败。
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.
