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We consider the high-dimensional linear regression model and assume that a fraction of the measurements are altered by an adversary with complete knowledge of the data and the underlying distribution. We are interested in a scenario where dense additive noise is heavy-tailed while the measurement vectors follow a sub-Gaussian distribution. Within this framework, we establish minimax lower bounds for the performance of an arbitrary estimator that depend on the the fraction of corrupted observations as well as the tail behavior of the additive noise. Moreover, we design a modification of the so-called Square-Root Slope estimator with several desirable features: (a) it is provably robust to adversarial contamination, and satisfies performance guarantees in the form of sub-Gaussian deviation inequalities that match the lower error bounds, up to logarithmic factors; (b) it is fully adaptive with respect to the unknown sparsity level and the variance of the additive noise, and (c) it is computationally tractable as a solution of a convex optimization problem. To analyze performance of the proposed estimator, we prove several properties of matrices with sub-Gaussian rows that may be of independent interest. 

Third harmonic generation (THG) provides a valuable, label-free approach to imaging biological systems. To date, THG microscopy has been performed using point-scanning methods that rely on intensity measurements lacking phase information of the complex field. We report the first demonstration, to the best of our knowledge, of THG holographic microscopy and the reconstruction of the complex THG signal field with spatial synthetic aperture imaging. Phase distortions arising from measurement-to-measurement fluctuations and imaging components cause optical aberrations in the reconstructed THG field. We have developed an aberration-correction algorithm that estimates and corrects these phase distortions to reconstruct the spatial synthetic aperture THG field without optical aberrations. 

Second harmonic generation (SHG) microscopy is a valuable tool for optical microscopy. SHG microscopy is normally performed as a point scanning imaging method, which lacks phase information and is limited in spatial resolution by the spatial frequency support of the illumination optics. In addition, aberrations in the illumination are difficult to remove. We propose and demonstrate SHG holographic synthetic aperture holographic imaging in both the forward (transmission) and backward (epi) imaging geometries. By taking a set of holograms with varying incident angle plane wave illumination, the spatial frequency support is increased and the input and output pupil phase aberrations are estimated and corrected – producing diffraction limited SHG imaging that combines the spatial frequency support of the input and output optics. The phase correction algorithm is computationally efficient and robust and can be applied to any set of measured field imaging data.