Subspace Estimation from Incomplete Observations: A High-Dimensional Analysis

Citation:

C. Wang, Y. C. Eldar, and Y. M. Lu, “Subspace Estimation from Incomplete Observations: A High-Dimensional Analysis,” IEEE Journal of Selected Topics in Signal Processing, vol. 12, no. 6, 2018.
 subspace_estimation.pdf 721 KB

Abstract:

We present a high-dimensional analysis of three popular algorithms, namely, Oja's method, GROUSE and PETRELS, for subspace estimation from streaming and highly incomplete observations.  We show that, with proper time scaling, the time-varying principal angles between the true subspace and its estimates given by the algorithms converge weakly to deterministic processes when the ambient dimension $$n$$ tends to infinity. Moreover, the limiting processes can be exactly characterized as the unique solutions of certain ordinary differential equations (ODEs). A finite sample bound is also given, showing that the rate of convergence towards such limits is $$\mathcal{O}(1/\sqrt{n})$$. In addition to providing asymptotically exact predictions of the dynamic performance of the algorithms, our high-dimensional analysis yields several insights, including an asymptotic equivalence between Oja's method and GROUSE, and a precise scaling relationship linking the amount of missing data to the signal-to-noise ratio. By analyzing the solutions of the limiting ODEs, we also establish phase transition phenomena associated with the steady-state performance of these techniques.

arXiv:1805.06834 [cs.LG]

Last updated on 08/22/2019