114 research outputs found
Phase Transitions in Phase Retrieval
Consider a scenario in which an unknown signal is transformed by a known
linear operator, and then the pointwise absolute value of the unknown output
function is reported. This scenario appears in several applications, and the
goal is to recover the unknown signal -- this is called phase retrieval. Phase
retrieval has been a popular subject of research in the last few years, both in
determining whether complete information is available with a given linear
operator, and in finding efficient and stable phase retrieval algorithms in the
cases where complete information is available. Interestingly, there are a few
ways to measure information completeness, and each way appears to be governed
by a phase transition of sorts. This chapter will survey the state of the art
with some of these phase transitions, and identify a few open problems for
further research.Comment: Book chapter, survey of recent literature, submitted to Excursions in
Harmonic Analysis: The February Fourier Talks at the Norbert Wiener Cente
Numerically erasure-robust frames
Given a channel with additive noise and adversarial erasures, the task is to
design a frame that allows for stable signal reconstruction from transmitted
frame coefficients. To meet these specifications, we introduce numerically
erasure-robust frames. We first consider a variety of constructions, including
random frames, equiangular tight frames and group frames. Later, we show that
arbitrarily large erasure rates necessarily induce numerical instability in
signal reconstruction. We conclude with a few observations, including some
implications for maximal equiangular tight frames and sparse frames.Comment: 15 page
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