Principal Components Analysis (PCA), a.k.a. subspace learning, is one of the most widely used noise removal and dimension reduction techniques. It finds applications in a large number of scientific and exploratory data analysis applications. PCA assumes that the clean/true data (``signal``) sequence lies close to a low-dimensional subspace of the ambient space, or, equivalently, that the clean dataset forms a low-rank matrix. This talk will describe our work on practically useful, provably correct and fast solutions to two problems involving subspace learning and tracking from “bad” data – (i) Low Rank Phase Retrieval (Phaseless PCA) and (ii) Robust Subspace Tracking. For the first problem, the term ``bad`` means that the observed data is a phaseless (magnitude-only) function of a linear transformation of the true data; while for the second one it means that the data is incomplete and corrupted by outliers.
Low Rank Phase Retrieval involves recovering an n x q low-rank matrix from a set of m mutually independent magnitude-only linear projections of each of its q columns. An efficient solution to Low Rank PR can enable fast and low-cost imaging of dynamic (time-varying) scenes in a wide variety of phaseless imaging applications. Some examples include astronomical imaging of the sun’s surface properties which gradually change over time, X-ray crystallography, or Fourier ptychographic imaging of live biological specimens. Robust Subspace Tracking is simply understood as the time-varying subspace extension of Robust PCA. It involves tracking sequentially arriving data vectors, that lie in a fixed or slowly changing low-dimensional subspace, while being robust to missing entries and corruption by additive sparse outliers. It occurs in a wide variety of data analytics applications ranging from video foreground-background separation in real-time to recommendation system design.