Richard Kueng

A Partial Derandomization of PhaseLift Using Spherical Designs

The problem of retrieving phase information from amplitude measurements alone has appeared in many scientific disciplines over the last century. PhaseLift is a recently introduced algorithm for phase recovery that is computationally tractable, numerically stable, and comes with rigorous performance guarantees. PhaseLift is optimal in the sense that the number of amplitude measurements required for phase reconstruction scales linearly with the dimension of the signal. However, it specifically demands Gaussian random measurement vectors—a limitation that restricts practical utility and obscures the specific properties of measurement ensembles that enable phase retrieval. Here we present a partial derandomization of PhaseLift that only requires sampling from certain polynomial size vector configurations, called

STABLE LOW-RANK MATRIX RECOVERY VIA NULL SPACE PROPERTIES

t

Comparing Experiments to the Fault-Tolerance Threshold

t-designs. Such configurations have been studied in algebraic combinatorics, coding theory, and quantum information. We prove reconstruction guarantees for a number of measurements that depends on the degree

Improving Compressed Sensing With the Diamond Norm

t

Near-optimal quantum tomography: estimators and bounds

t of the design. If the degree is allowed to grow logarithmically with the dimension, the bounds become tight up to polylog-factors. Beyond the specific case of PhaseLift, this work highlights the utility of spherical designs for the derandomization of data recovery schemes.

Note on the saturation of the norm inequalities between diamond and nuclear norm.

In this work we analyze the problem of phase retrieval from Fourier measurements with random diffraction patterns. To this end, we consider the recently introduced PhaseLift algorithm, which expresses the problem in the language of convex optimization. We provide recovery guarantees which require O(log^2 d) different diffraction patterns, thus improving on recent results by Candes et al. [arXiv:1310.3240], which require O(log^4 d) different patterns.