February Fourier Talks 2013

Matthew Fickus


Group-theoretic constructions of erasure-robust frames


A recently proposed method for phase retrieval requires frames which are robust against the removal of any fixed proportion of the frame elements. However, such numerically erasure robust frames (NERFs) are difficult to construct explicitly since, like Restricted Isometry Property (RIP) matrices, they must satisfy a combinatorially large number of conditions. We discuss a new method for constructing such frames. We begin by focusing on a subtle difference between the definition of a NERF and that of an RIP matrix, one that allows us to introduce a new computational trick for quickly estimating NERF bounds. In short, we estimate these bounds by evaluating the frame analysis operator at every point of an epsilon-net for the unit sphere. We then borrow ideas from the theory of group frames to construct explicit frames and epsilon-nets with such high degrees of symmetry that the requisite number of operator evaluations is greatly reduced. We conclude with numerical results, using these new ideas to quickly produce decent estimates of NERF bounds which would otherwise take an eternity.