Phaseless Subspace Tracking

Problem Formulation

Phaseless Subspace Tracking can be simply understood as the dynamic or timevarying subspace extension of Low rank phase retrieval (LRPR). Thus, instead of the subspace span(P) being fixed, we assume that it can change with time, albeit slowly.

Time varying subspaces is a better model (than just fixed subspace) for long data sequences, e.g., long image sequences or videos, because if one tries to use a single low-dimensional subspace to represent the entire data sequence, the required subspace dimension may end up being quite large. This can be problematic because it means that the resulting data matrix may not be sufficiently low-rank. The most general model for time-varying subspaces is to allow the subspace to change at each time. However such a model involves too many unknowns. An r dimensional subspace in n-dimensional ambient space is fully specified by $nr$ parameters. But the signal $ell_t$ is an $n \times 1$ vector (has only $n$ unknowns). Thus, allowing the subspace to change at each time will result in an increase in the number of unknowns per time instant from $n$ to $nr$ (rather than a decrease which is the purpose of incorporating structure into the PR problem). A less general model, but one that prevents an increase in the number of unknowns, is to assume that the data subspace is piecewise constant with time. This model has been extensively used in the robust subspace tracking literature (for more information please see papers in this page) which we consider this model as well.

S.Nayer and N.Vaswani, "PhaST: Model-Free Phaseless Subspace Tracking", The 44nd IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) 2019.

S.Nayer and N.Vaswani, "Phaseless Subspace Tracking", The 6th IEEE Global Conference on Signal and Information Processing (GlobalSIP) 2018.

S.Nayer, P.Narayanamurthy, N.Vaswani, "Phaseless PCA: Low-Rank Matrix Recovery from Column-wise Phaseless Measurements", ICML 2019.

S.Nayer, P.Narayanamurthy, N.Vaswani, "Provable low rank phase retrieval", IEEE Transactions on Information Theory 2020.