We study lower bounds on adaptive sensing algorithms for recovering low rank matrices using linear measurements. Given an $n \times n$ matrix $A$, a general linear measurement $S(A)$, for an $n \times n$ matrix $S$, is just the inner product of $S$ and $A$, each treated as $n^2$-dimensional vectors. By performing as few linear measurements as possible on a rank-$r$ matrix $A$, we hope to construct a matrix $\hat{A}$ that satisfies $\|A - \hat{A}\|_F^2 \le c\|A\|_F^2$, for a small constant $c$. It is commonly assumed that when measuring $A$ with $S$, the response is corrupted with an independent Gaussian random variable of mean $0$ and variance $\sigma^2$. Cand\'es and Plan study non-adaptive algorithms for low rank matrix recovery using random linear measurements. At a certain noise level, it is known that their non-adaptive algorithms need to perform $\Omega(n^2)$ measurements, which amounts to reading the entire matrix. An important question is whether adaptivity helps in decreasing the overall number of measurements. We show that any adaptive algorithm that uses $k$ linear measurements in each round and outputs an approximation to the underlying matrix with probability $\ge 9/10$ must run for $t = \Omega(\log(n^2/k)/\log\log n)$ rounds showing that any adaptive algorithm which uses $n^{2-\beta}$ linear measurements in each round must run for $\Omega(\log n/\log\log n)$ rounds to compute a reconstruction with probability $\ge 9/10$. Hence any adaptive algorithm that has $o(\log n/\log\log n)$ rounds must use an overall $\Omega(n^2)$ linear measurements. Our techniques also readily extend to obtain lower bounds on adaptive algorithms for tensor recovery and obtain measurement-vs-rounds trade-off for many sensing problems in numerical linear algebra, such as spectral norm low rank approximation, Frobenius norm low rank approximation, singular vector approximation, and more.

翻译：我们研究了使用线性测量恢复低秩矩阵的自适应感知算法的下界。给定一个$n \times n$矩阵$A$，一般线性测量$S(A)$（其中$S$为$n \times n$矩阵）即是将$S$和$A$分别视为$n^2$维向量后的内积。通过尽可能少地对秩为$r$的矩阵$A$进行线性测量，我们希望构造一个矩阵$\hat{A}$，使得$\|A - \hat{A}\|_F^2 \le c\|A\|_F^2$，其中$c$为小常数。通常假设用$S$测量$A$时，响应受到均值为$0$、方差为$\sigma^2$的独立高斯随机变量干扰。Candès和Plan研究了使用随机线性测量恢复低秩矩阵的非自适应算法。在特定噪声水平下，已知其非自适应算法需要执行$\Omega(n^2)$次测量，相当于读取整个矩阵。一个重要问题是自适应性是否有助于减少总测量次数。我们证明，每轮使用$k$次线性测量并以概率$\ge 9/10$输出底层矩阵近似值的任意自适应算法，必须运行$t = \Omega(\log(n^2/k)/\log\log n)$轮。这表明，每轮使用$n^{2-\beta}$次线性测量并以概率$\ge 9/10$输出重构结果的任意自适应算法，必须运行$\Omega(\log n/\log\log n)$轮。因此，任何少于$o(\log n/\log\log n)$轮的自适应算法必须使用总计$\Omega(n^2)$次线性测量。我们的技术还可直接扩展至张量恢复自适应算法的下界，并为数值线性代数中的多种感知问题（如谱范数低秩近似、弗罗贝尼乌斯范数低秩近似、奇异向量近似等）建立测量-轮次权衡。