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One-bit Quantization for Phase Retrieval

Dylan Domel-White

Vanderbilt University, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak8625c9dcdd62a15c6e9738cf55dfb574').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy8625c9dcdd62a15c6e9738cf55dfb574 = 'dyl&#97;n.d&#111;m&#101;l-wh&#105;t&#101;' + '&#64;'; addy8625c9dcdd62a15c6e9738cf55dfb574 = addy8625c9dcdd62a15c6e9738cf55dfb574 + 'v&#97;nd&#101;rb&#105;lt' + '&#46;' + '&#101;d&#117;'; var addy_text8625c9dcdd62a15c6e9738cf55dfb574 = 'dyl&#97;n.d&#111;m&#101;l-wh&#105;t&#101;' + '&#64;' + 'v&#97;nd&#101;rb&#105;lt' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak8625c9dcdd62a15c6e9738cf55dfb574').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy8625c9dcdd62a15c6e9738cf55dfb574 + '\'>'+addy_text8625c9dcdd62a15c6e9738cf55dfb574+'<\/a>';

Abstractly, phase retrieval is the problem of recovering a vector in a real or complex Hilbert space (up to a global phase factor) from magnitudes of linear functionals applied to the vector. In other words, rather than having access to linear functional measurements of the form $x \mapsto \left\langle x,v \right\rangle$ as in many sensing applications, we have nonlinear measurements $x \mapsto \left|\left\langle x,v \right\rangle\right|$. If $v$ is a unit vector, then $\left|\left\langle x,v \right\rangle\right|$ is the norm of the projection of $x$ onto the span of $v$, and so norms of projections onto higher dimensional subspaces are a natural generalization of typical phase retrieval measurements as described above. We present a measurement and recovery algorithm along with theoretical performance guarantees for \textit{one-bit} phase retrieval, where only one bit of information is recorded from each projection norm. For our algorithm, we obtain one-bit information by thresholding the norms of projections onto random subspaces of dimension equal to half that of the Hilbert space, and recover by finding the principal component of an auxiliary matrix constructed using the one-bit measurements.

Joint work with Bernhard Bodmann (University of Houston, TX, United States).