Exact and Stable Covariance Estimation from Quadratic Sampling via Convex Programming (1310.0807v5)

Abstract: Statistical inference and information processing of high-dimensional data often require efficient and accurate estimation of their second-order statistics. With rapidly changing data, limited processing power and storage at the acquisition devices, it is desirable to extract the covariance structure from a single pass over the data and a small number of stored measurements. In this paper, we explore a quadratic (or rank-one) measurement model which imposes minimal memory requirements and low computational complexity during the sampling process, and is shown to be optimal in preserving various low-dimensional covariance structures. Specifically, four popular structural assumptions of covariance matrices, namely low rank, Toeplitz low rank, sparsity, jointly rank-one and sparse structure, are investigated, while recovery is achieved via convex relaxation paradigms for the respective structure. The proposed quadratic sampling framework has a variety of potential applications including streaming data processing, high-frequency wireless communication, phase space tomography and phase retrieval in optics, and non-coherent subspace detection. Our method admits universally accurate covariance estimation in the absence of noise, as soon as the number of measurements exceeds the information theoretic limits. We also demonstrate the robustness of this approach against noise and imperfect structural assumptions. Our analysis is established upon a novel notion called the mixed-norm restricted isometry property (RIP-$\ell_{2}/\ell_{1}$), as well as the conventional RIP-$\ell_{2}/\ell_{2}$ for near-isotropic and bounded measurements. In addition, our results improve upon the best-known phase retrieval (including both dense and sparse signals) guarantees using PhaseLift with a significantly simpler approach.

• The paper introduces a covariance sketching framework that reduces memory and computational complexity while accurately recovering structured covariance matrices from quadratic measurements.
• It leverages convex programming and a novel mixed-norm RIP (ℓ2/ℓ1) to achieve optimal covariance estimation even in challenging, low-measurement regimes.
• Strong numerical experiments validate the method’s effectiveness, highlighting its potential in streaming data processing, high-frequency communications, and phase retrieval applications.

Overview of "Exact and Stable Covariance Estimation from Quadratic Sampling via Convex Programming"

The paper, "Exact and Stable Covariance Estimation from Quadratic Sampling via Convex Programming," by Yuxin Chen, Yuejie Chi, and Andrea J. Goldsmith, provides a comprehensive paper on covariance estimation of high-dimensional data from quadratic measurements. The authors propose a novel approach using convex programming to exploit the structural assumptions of covariance matrices, such as low-rank, Toeplitz low-rank, sparsity, and jointly sparse and rank-one structures. This work is crucial in scenarios where rapid changes in data, limited device capability, or constrained memory capacity necessitate efficient data processing techniques.

The paper considers a quadratic measurement model that reduces complexity and memory requirements during sampling. It is demonstrated that this model preserves several low-dimensional covariance structures optimally. Leveraging convex relaxation paradigms results in accurate recovery of covariance matrices under various structural assumptions.

Key Contributions

1. Covariance Sketching Framework: The authors present a framework for extracting covariance matrices by leveraging minimal memory and reducing computational complexity. Applications span streaming data processing, high-frequency communications, and phase retrieval in fields like optics and tomography.
2. Optimal Covariance Estimation: By using quadratic measurements, the paper guarantees universally accurate covariance estimation without noise when the number of measurements exceeds information-theoretic limits. The paper also explores robust estimation methods in noisy environments and imperfect structural assumptions.
3. Mixed-Norm Restricted Isometry Property (RIP-ℓ2/ℓ1): The paper introduces RIP-ℓ2/ℓ1, a new notion that allows simpler yet effective analysis of the measurement model. This property is pivotal in understanding the relation between convex programming and the types of measurements needed for accurate recovery.
4. Applications and Extensions: The results are significant for real-world high-dimensional signal processing tasks, offering computational and storage advantages compared to other measurement schemes. The paper's framework extends beyond covariance matrices to general symmetric matrices and phase retrieval problems.
5. Strong Numerical and Theoretical Results: The paper demonstrates solid numerical experiments aligning with theoretical predictions, showcasing close optimality in practical performance and pushing the boundaries of existing theoretical guarantees such as those found in phase retrieval.

Implications and Future Directions

The implications of this research extend into practical applications in data science, particularly where high-dimensional data streams are involved. The reliability and efficiency of the proposed techniques make them suitable for modern scalable systems where data is constantly evolving. The transferability of RIP-ℓ2/ℓ1 to other domains poses intriguing questions, and potential further exploration in areas like sparse inverse covariance estimation can be envisioned. Additionally, this foundational work invites further studies on RIP-less theories for other measurement models, enhancing its applicability to a wider array of signal processing challenges.

In summary, this paper makes significant strides in statistical inference from quadratic sampling, presenting a useful methodology for efficient data-driven solutions and opening the path for ongoing research in the broader context of structured signal estimation.