Asymmetry Helps: Eigenvalue and Eigenvector Analyses of Asymmetrically Perturbed Low-Rank Matrices

This paper is concerned with the interplay between statistical asymmetry and spectral methods. Suppose we are interested in estimating a rank-1 and symmetric matrix $\mathbf{M}^{\star}\in \mathbb{R}^{n\times n}$, yet only a randomly perturbed version $\mathbf{M}$ is observed. The noise matrix $\mathbf{M}-\mathbf{M}^{\star}$ is composed of zero-mean independent (but not necessarily homoscedastic) entries and is, therefore, not symmetric in general. This might arise, for example, when we have two independent samples for each entry of $\mathbf{M}^{\star}$ and arrange them into an {\em asymmetric} data matrix $\mathbf{M}$. The aim is to estimate the leading eigenvalue and eigenvector of $\mathbf{M}^{\star}$. We demonstrate that the leading eigenvalue of the data matrix $\mathbf{M}$ can be $O(\sqrt{n})$ times more accurate --- up to some log factor --- than its (unadjusted) leading singular value in eigenvalue estimation. Further, the perturbation of any linear form of the leading eigenvector of $\mathbf{M}$ --- say, entrywise eigenvector perturbation --- is provably well-controlled. This eigen-decomposition approach is fully adaptive to heteroscedasticity of noise without the need of careful bias correction or any prior knowledge about the noise variance. We also provide partial theory for the more general rank-$r$ case. The takeaway message is this: arranging the data samples in an asymmetric manner and performing eigen-decomposition could sometimes be beneficial.Comment: accepted to Annals of Statistics, 2020. 37 page

Reconstruction of sparse wavelet signals from partial Fourier measurements

In this paper, we show that high-dimensional sparse wavelet signals of finite levels can be constructed from their partial Fourier measurements on a deterministic sampling set with cardinality about a multiple of signal sparsity

Energy-Throughput Tradeoff in Sustainable Cloud-RAN with Energy Harvesting

In this paper, we investigate joint beamforming for energy-throughput tradeoff in a sustainable cloud radio access network system, where multiple base stations (BSs) powered by independent renewable energy sources will collaboratively transmit wireless information and energy to the data receiver and the energy receiver simultaneously. In order to obtain the optimal joint beamforming design over a finite time horizon, we formulate an optimization problem to maximize the throughput of the data receiver while guaranteeing sufficient RF charged energy of the energy receiver. Although such problem is non-convex, it can be relaxed into a convex form and upper bounded by the optimal value of the relaxed problem. We further prove tightness of the upper bound by showing the optimal solution to the relaxed problem is rank one. Motivated by the optimal solution, an efficient online algorithm is also proposed for practical implementation. Finally, extensive simulations are performed to verify the superiority of the proposed joint beamforming strategy to other beamforming designs.Comment: Accepted by ICC 201