We study a class of real robust phase retrieval problems under a Gaussian assumption on the coding matrix when the received signal is sparsely corrupted by noise. The goal is to establish conditions on the sparsity under which the input vector can be exactly recovered. The recovery problem is formulated as residual minimization in the ℓ1-norm. The main contribution is a robust phase retrieval counterpart to the seminal paper by Candes and Tao on compressed sensing (ℓ1regression) [Decoding by linear programming. IEEE Transactions on Information Theory, 51(12):4203-4215, 2005]. The analysis depends on a key new property of the coding matrix called the Absolute Range Property (ARP) which is the analogue to the Null Space Property (NSP) in compressed sensing. When the residuals are computed using squared magnitudes, we show that ARP follows from a standard Restricted Isometry Property (RIP). However, when the residuals are computed using absolute magnitudes, a different kind of RIP or growth property is required. We conclude by showing that the robust phase retrieval objectives are sharp with respect to their minimizers with high probability.

Reconstruction of a function from noisy data is key in machine learning and is often formulated as a regularized optimization problem over an infinite-dimensional reproducing kernel Hilbert space (RKHS). The solution suitably balances adherence to the observed data and the corresponding RKHS norm. When the data fit is measured using a quadratic loss, this estimator has a known statistical interpretation. Given the noisy measurements, the RKHS estimate represents the posterior mean (minimum variance estimate) of a Gaussian random field with covariance proportional to the kernel associated with the RKHS. In this brief, we provide a statistical interpretation when more general losses are used, such as absolute value, Vapnik or Huber. Specifically, for any finite set of sampling locations (that includes where the data were collected), the maximum a posteriori estimate for the signal samples is given by the RKHS estimate evaluated at the sampling locations. This connection establishes a firm statistical foundation for several stochastic approaches used to estimate unknown regularization parameters. To illustrate this, we develop a numerical scheme that implements a Bayesian estimator with an absolute value loss. This estimator is used to learn a function from measurements contaminated by outliers.