Inexact Alternating Optimization for Phase Retrieval In the Presence of Outliers

Phase retrieval has been mainly considered in the presence of Gaussian noise. However, the performance of the algorithms proposed under the Gaussian noise model severely degrades when grossly corrupted data, i.e., outliers, exist. This paper investigates techniques for phase retrieval in the presence of heavy-tailed noise -- which is considered a better model for situations where outliers exist. An $\ell_p$-norm ($0<p<2$) based estimator is proposed for fending against such noise, and two-block inexact alternating optimization is proposed as the algorithmic framework to tackle the resulting optimization problem. Two specific algorithms are devised by exploring different local approximations within this framework. Interestingly, the core conditional minimization steps can be interpreted as iteratively reweighted least squares and gradient descent. Convergence properties of the algorithms are discussed, and the Cram\'er-Rao bound (CRB) is derived. Simulations demonstrate that the proposed algorithms approach the CRB and outperform state-of-the-art algorithms in heavy-tailed noise.Comment: 23 pages, 16 figure

Tensor Decomposition for Signal Processing and Machine Learning

Tensors or {\em multi-way arrays} are functions of three or more indices $(i,j,k,\cdots)$ -- similar to matrices (two-way arrays), which are functions of two indices $(r,c)$ for (row,column). Tensors have a rich history, stretching over almost a century, and touching upon numerous disciplines; but they have only recently become ubiquitous in signal and data analytics at the confluence of signal processing, statistics, data mining and machine learning. This overview article aims to provide a good starting point for researchers and practitioners interested in learning about and working with tensors. As such, it focuses on fundamentals and motivation (using various application examples), aiming to strike an appropriate balance of breadth {\em and depth} that will enable someone having taken first graduate courses in matrix algebra and probability to get started doing research and/or developing tensor algorithms and software. Some background in applied optimization is useful but not strictly required. The material covered includes tensor rank and rank decomposition; basic tensor factorization models and their relationships and properties (including fairly good coverage of identifiability); broad coverage of algorithms ranging from alternating optimization to stochastic gradient; statistical performance analysis; and applications ranging from source separation to collaborative filtering, mixture and topic modeling, classification, and multilinear subspace learning.Comment: revised version, overview articl