Stability in Phase Retrieval: Characterizing Condition Numbers and the Optimal Vector Set
CoRR(2024)
摘要
In this paper, we primarily focus on analyzing the stability property of
phase retrieval by examining the bi-Lipschitz property of the map
Φ_A(x)=|Ax|∈ℝ_+^m, where x∈ℍ^d and A∈ℍ^m× d is the measurement matrix for
ℍ∈{ℝ,ℂ}. We define the condition number
β_A:=U_A/L_A, where
L_A and U_A represent the optimal lower and
upper Lipschitz constants, respectively. We establish the first universal lower
bound on β_A by demonstrating that for any
A∈ℍ^m× d,
β_A≥β_0^ℍ:=√(π/π-2) ≈ 1.659 if
ℍ=ℝ,
√(4/4-π) ≈ 2.159 if ℍ=ℂ.
We prove that
the condition number of a standard Gaussian matrix in ℍ^m× d
asymptotically matches the lower bound β_0^ℍ for both real and
complex cases. This result indicates that the constant lower bound
β_0^ℍ is asymptotically tight, holding true for both the real
and complex scenarios. As an application of this result, we utilize it to
investigate the performance of quadratic models for phase retrieval. Lastly, we
establish that for any odd integer m≥ 3, the harmonic frame
A∈ℝ^m× 2 possesses the minimum condition
number among all A∈ℝ^m× 2. We are confident
that these findings carry substantial implications for enhancing our
understanding of phase retrieval.
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