Name
Title | ||
|---|---|---|
Information Theoretic Limits for Phase Retrieval With Subsampled Haar Sensing Matrices |
Abstract | ||
|---|---|---|
We study information theoretic limits of recovering an unknown n dimensional, complex signal vector x
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sub>
with unit norm from m magnitude-only measurements of the form y
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub>
= |(Ax
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sub>
)
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub>
|
<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup>
, i = 1, 2 ..., m, where A is the sensing matrix. This is known as the Phase Retrieval problem and models practical imaging systems where measuring the phase of the observations is difficult. Since in a number of applications, the sensing matrix has orthogonal columns, we model the sensing matrix as a subsampled Haar matrix formed by picking n columns of a uniformly random m X m unitary matrix. We study this problem in the high dimensional asymptotic regime, where m, n → ∞, while m/n → δ with δ being a fixed number, and show that if m <; (2 - o
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub>
(1)) · n, then any estimator is asymptotically orthogonal to the true signal vector x
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sub>
. This lower bound is sharp since when m > (2 + o
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub>
(1)) · n, estimators that achieve a non trivial asymptotic correlation with the signal vector are known from previous works. |
Year | DOI | Venue |
|---|---|---|
2020 | 10.1109/TIT.2020.3015173 | IEEE Transactions on Information Theory |
Keywords | DocType | Volume |
Phase retrieval,random orthogonal matrices,subsampled Haar matrices,coded diffraction pattern,large deviation theory,random matrix theory,phase transition,weak recovery threshold | Journal | 66 |
Issue | ISSN | Citations |
12 | 0018-9448 | 1 |
PageRank | References | Authors |
0.35 | 0 | 3 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Dudeja Rishabh | 1 | 1 | 0.35 |
| Junjie Ma | 2 | 148 | 15.24 |
| Arian Maleki | 3 | 803 | 57.52 |