[论文解读] Low rank matrix recovery from Clifford orbits

Richard Kueng, Huangjun Zhu+1 more|arXiv (Cornell University)|Oct 25, 2016

Advanced X-ray Imaging Techniques参考文献 71被引用 14

一句话总结

该论文证明，通过从 Clifford 群轨道中采样测量——特别是稳定子态——可以高效地恢复低秩矩阵，包括量子纯态，使用凸优化方法。它建立了当测量数满足 $ m \geq C\kappa_r r d \log d $ 时，即使在噪声下也能实现稳定且近乎最优的恢复，对正定矩阵通过最小二乘法实现重建误差和恢复保证的显式界。

ABSTRACT

We prove that low-rank matrices can be recovered efficiently from a small number of measurements that are sampled from orbits of a certain matrix group. As a special case, our theory makes statements about the phase retrieval problem. Here, the task is to recover a vector given only the amplitudes of its inner product with a small number of vectors from an orbit. Variants of the group in question have appeared under different names in many areas of mathematics. In coding theory and quantum information, it is the complex Clifford group; in time-frequency analysis the oscillator group; and in mathematical physics the metaplectic group. It affords one particularly small and highly structured orbit that includes and generalizes the discrete Fourier basis: While the Fourier vectors have coefficients of constant modulus and phases that depend linearly on their index, the vectors in said orbit have phases with a quadratic dependence. In quantum information, the orbit is used extensively and is known as the set of stabilizer states. We argue that due to their rich geometric structure and their near-optimal recovery properties, stabilizer states form an ideal model for structured measurements for phase retrieval. Our results hold for $m\geq C κ_r r d \log(d)$ measurements, where the oversampling factor k varies between $κ_r=1$ and $κ_r = r^2$ depending on the orbit. The reconstruction is stable towards both additive noise and deviations from the assumption of low rank. If the matrices of interest are in addition positive semidefinite, reconstruction may be performed by a simple constrained least squares regression. Our proof methods could be adapted to cover orbits of other groups.

研究动机与目标

• 建立从 Clifford 群轨道中抽取的测量中低秩矩阵重构的理论恢复保证。
• 表明稳定子态——具有结构化、近乎最优的测量向量——可实现高效且稳定的相位恢复。
• 提供在噪声和低秩假设下实现稳定恢复所需测量数目的显式界。
• 证明可通过这些测量使用约束最小二乘回归实现正定矩阵的重构。
• 通过表示论技术为将该方法推广至其他群轨道奠定基础。

提出的方法

• 使用 PhaseLift 框架将相位恢复问题转化为矩阵上的凸优化问题。
• 利用 Clifford 群的表示理论分析其轨道（特别是稳定子态）的几何与代数结构。
• 通过分析从 Clifford 轨道中采样测量算符的限制等距性（RIP）建立恢复保证。
• 利用核范数最小化与对偶性论证，结合 $ \ell_q $-范数项处理噪声，建立重建误差界。
• 引入一个依赖于轨道和秩 $ r $ 的过采样因子 $ \kappa_r \in [1, r^2] $，影响所需测量数。
• 应用测度集中与测量集合的矩界，证明在加性噪声和低秩偏差下的稳定性。

实验结果

研究问题

• RQ1 能否从少量从 Clifford 群轨道中抽取的测量中稳定恢复低秩矩阵？
• RQ2 稳定子态——在量子信息中已知——是否构成相位恢复的近似最优测量集合？
• RQ3 稳定恢复所需的最小测量数是多少？其与秩和维度的关系如何？
• RQ4 能否利用此测量模型高效重构正定矩阵？
• RQ5 该证明框架在多大程度上可推广至 Clifford 群以外的其他群轨道？

主要发现

• 在从 Clifford 轨道中采样 $ m \geq C\kappa_r r d \log d $ 个测量下，可实现低秩矩阵恢复，其中 $ \kappa_r \in [1, r^2] $。
• 重建误差被限制为 $ \|Z^\sharp - X\|_2 \leq \frac{\hat{C}_2}{\sqrt{r}} \sigma_r(X) + \hat{C}_3 \sqrt{\kappa(z,r)} \, d m^{-1/q} \|\epsilon\|_{\ell_q} $，表明在噪声下具有稳定性。
• 对于正定矩阵，可通过约束最小二乘回归实现稳定恢复。
• 过采样因子 $ \kappa_r $ 随轨道和秩变化，最优情况下 $ \kappa_r = 1 $，最坏情况下 $ \kappa_r = r^2 $。
• 由于其丰富的几何与代数性质，稳定子态构成相位恢复的高结构性、近乎最优的测量集合。
• 启发式证据表明，$ O(n^2) $ 个无噪声的稳定子测量可能足以重构一个 $ n $-量子比特的稳定子态，尽管严格证明仍待解决。