# Information Theoretic Limits for Phase Retrieval with Subsampled Haar Sensing Matrices

We study information theoretic limits of recovering an unknown n dimensional, complex signal vector x_ with unit norm from m magnitude-only measurements of the form y_i = m|(Ax_)_i|^2, 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 × 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_n(1))· n, then any estimator is asymptotically orthogonal to the true signal vector x_. This lower bound is sharp since when m > (2+o_n(1)) · n, estimators that achieve a non trivial asymptotic correlation with the signal vector are known from previous works.

## Authors

• 6 publications
• 9 publications
• 25 publications
08/24/2020

### Universality of Linearized Message Passing for Phase Retrieval with Structured Sensing Matrices

In the phase retrieval problem one seeks to recover an unknown n dimensi...
08/20/2017

### Fundamental Limits of Weak Recovery with Applications to Phase Retrieval

In phase retrieval we want to recover an unknown signal x∈ C^d from n q...
10/04/2021

### Spiked Covariance Estimation from Modulo-Reduced Measurements

Consider the rank-1 spiked model: X=√(ν)ξu+ Z, where ν is the spike inte...
06/09/2020

### Phase retrieval in high dimensions: Statistical and computational phase transitions

We consider the phase retrieval problem of reconstructing a n-dimensiona...
02/24/2015

### Phase Transitions for High Dimensional Clustering and Related Problems

Consider a two-class clustering problem where we observe X_i = ℓ_i μ + Z...
03/07/2019

### Rigorous Analysis of Spectral Methods for Random Orthogonal Matrices

Phase retrieval refers to algorithmic methods for recovering a signal fr...
01/30/2019

### Support Recovery in the Phase Retrieval Model: Information-Theoretic Fundamental Limits

The support recovery problem consists of determining a sparse subset of ...
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