Low rank matrix recovery from rank one measurements (1410.6913v1)

Abstract: We study the recovery of Hermitian low rank matrices $X \in \mathbb{C}^{n \times n}$ from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form $a_j a_j^*$ for some measurement vectors $a_1,...,a_m$, i.e., the measurements are given by $y_j = \mathrm{tr}(X a_j a_j^*)$. The case where the matrix $X=x x^*$ to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements, $y_j = |\langle x,a_j\rangle|^2$ via the PhaseLift approach, which has been introduced recently. We derive bounds for the number $m$ of measurements that guarantee successful uniform recovery of Hermitian rank $r$ matrices, either for the vectors $a_j$, $j=1,...,m$, being chosen independently at random according to a standard Gaussian distribution, or $a_j$ being sampled independently from an (approximate) complex projective $t$-design with $t=4$. In the Gaussian case, we require $m \geq C r n$ measurements, while in the case of $4$-designs we need $m \geq Cr n \log(n)$. Our results are uniform in the sense that one random choice of the measurement vectors $a_j$ guarantees recovery of all rank $r$-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate $4$-designs generalizes and improves a recent bound on phase retrieval due to Gross, Kueng and Krahmer. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii.

• The paper establishes precise measurement bounds, proving recovery with m ≥ Crn for Gaussian vectors and m ≥ Crn log(n) for 4-designs.
• The paper guarantees uniform recovery of all Hermitian rank‑r matrices with high probability, ensuring robustness in noisy environments.
• The method connects phase retrieval to quantum state tomography, underlining practical implications for signal processing and quantum information.

Overview of Low Rank Matrix Recovery from Rank One Measurements

This paper, authored by Richard Kueng, Holger Rauhut, and Ulrich Terstiege, presents a paper on the recovery of Hermitian low-rank matrices from undersampled measurements using nuclear norm minimization. Specifically, the authors investigate cases where measurements stem from Frobenius inner products with random rank-one matrices, particularly of the form $a_ja_j^*$ for measurement vectors $a_1, \ldots, a_m$. The scenario where the target matrix $X=xx^*$ is of rank one links to the phase retrieval problem, specifically the PhaseLift approach.

The key objective of the paper is to ascertain the minimum number of measurements $m$ necessary to ensure the uniform recovery of Hermitian rank-$r$ matrices with high probability. Two main scenarios are considered: measurements derived from random standard Gaussian vectors and approximate complex projective $t$-designs with $t=4$. For Gaussian vectors, it is shown that $m \geq Crn$ measurements are sufficient. Meanwhile, when using 4-designs, the bound increases slightly to $m \geq Crn \log(n)$.

Theoretical Contributions

1. Measurement Bounds: The paper establishes precise bounds on the number of required measurements. When vectors $a_j$ are Gaussian, the paper proves that the recovery process succeeds with $m \geq Crn$ measurements, and when using 4-designs, $m \geq Crn \log(n)$ is required.
2. Uniform Recovery Guarantees: The presented results guarantee the uniform recovery of all rank-$r$ matrices simultaneously with high probability. This uniformity is crucial as a single random choice of measurement vectors works universally for all matrices of the given rank.
3. Robustness Against Noise: The authors demonstrate robustness to noise in the measurement process, extending the applicability of the recovery technique to more practical, noisy environments.
4. Connection to Quantum State Tomography: The paper notes that the findings can be applied to quantum state tomography, particularly for systems with low-rank density matrices, illustrating the practical significance of the work in quantum information theory.

Methodological Advances

The authors leverage a technique known as the "bowling scheme," inspired by Mendelson and Koltchinskii, which aids in deriving the lower bounds necessary for successful recovery. This technique emphasizes a deep understanding of the intersection of geometric properties of the descent cone of the nuclear norm and random measurements.

Implications and Future Directions

The implications of this work extend into practical and theoretical domains:

1. Practical Implications: The results are particularly relevant for applications demanding efficient matrix recovery with minimal measurements, such as signal processing, computer vision, and quantum mechanics.
2. Theoretical Implications: The paper extends existing theories about low-rank matrix recovery and phase retrieval by incorporating structured randomness from 4-designs, enriching the theoretical landscape of convex optimization within signal processing.
3. Future Research Directions: Potential future explorations could involve adapting these recovery frameworks to models beyond those considered, potentially including non-Hermitian matrices or alternative measurement models.

In summary, this investigation provides a substantial enhancement in our understanding of low-rank Hermitian matrix recovery. By linking nuclear norm minimization under rank-one constraints to broader applications, it presents strong theoretical underpinnings that could fuel further research and innovation in fields reliant on efficient data acquisition and recovery.