Sparse Affine Sampling: Ambiguity-Free and Efficient Sparse Phase Retrieval

TL;DR

This paper introduces a novel affine measurement-based sparse phase retrieval method that guarantees ambiguity-free reconstruction, supports noisy scenarios, and offers simple, closed-form solutions with high probability guarantees.

Contribution

It proposes a new affine measurement approach for sparse phase retrieval that ensures ambiguity-free recovery and extends to noisy cases with theoretical guarantees.

Findings

Perfect support identification under mild conditions.

Exact signal recovery using simple closed-form solutions.

High probability near-optimal performance with random sensing matrices.

Abstract

Conventional sparse phase retrieval schemes can recover sparse signals from the magnitude of linear measurements only up to a global phase ambiguity. This work proposes a novel approach that instead utilizes the magnitude of affine measurements to achieve ambiguity-free signal reconstruction. The proposed method relies on two-stage approach that consists of support identification followed by the exact recovery of nonzero signal entries. In the noise-free case, perfect support identification using a simple counting rule is guaranteed subject to a mild condition on the signal sparsity, and subsequent exact recovery of the nonzero signal entries can be obtained in closed-form. The proposed approach is then extended to two noisy scenarios, namely, sparse noise (or outliers) and non-sparse bounded noise. For both cases, perfect support identification is still ensured under mild conditions on the noise model, namely, the support size for sparse outliers and the power of the bounded noise. Under perfect support identification, exact signal recovery can be achieved using a simple majority rule for the sparse noise scenario, and reconstruction up to a bounded error can be achieved using linear least-squares (LS) estimation for the non-sparse bounded noise scenario. The obtained analytic performance guarantee for the latter case also sheds light on the construction of the sensing matrix and bias vector. In fact, we show that a near optimal performance can be achieved with high probability by the random generation of the nonzero entries of the sparse sensing matrix and bias vector according to the uniform distribution over a circle. Computer simulations using both synthetic and real-world data sets are provided to demonstrate the effectiveness of the proposed scheme.