Stability in phase retrieval: Characterizing condition numbers and the optimal vector set
Authors: Xia Yu , Xu Zhiqiang , Xu Zili 
Published: October 2024

In this paper, we primarily focus on analyzing the stability property of phase retrieval by examining the bi-Lipschitz property of the map
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi Subscript bold-italic upper A Baseline left-parenthesis bold-italic x right-parenthesis equals StartAbsoluteValue bold-italic upper A bold-italic x EndAbsoluteValue element-of double-struck upper R Subscript plus Superscript m">

Φ

A

(
x
)
=

|

A
x

|

∈

R

+
m

\Phi _{\boldsymbol {A}}(\boldsymbol {x})=|\boldsymbol {A}\boldsymbol {x}|\in \mathbb {R}_+^m

, where

x
∈

H

d

\boldsymbol {x}\in \mathbb {H}^d

and

A
∈

H

m
×
d

\boldsymbol {A}\in \mathbb {H}^{m\times d}

is the measurement matrix for

H

∈
{

R

,

C

}

\mathbb {H}\in \{\mathbb {R},\mathbb {C}\}

. We define the condition number

β

A

=

U

A

L

A

\beta _{\boldsymbol {A}}=\frac {U_{\boldsymbol {A}}}{L_{\boldsymbol {A}}}

, where

L

A

L_{\boldsymbol {A}}

and

U

A

U_{\boldsymbol {A}}

represent the optimal lower and upper Lipschitz constants, respectively. We establish the universal lower bound on

β

A

\beta _{\boldsymbol {A}}

by demonstrating that for any

A

∈

H

m
×
d

{\boldsymbol {A}}\in \mathbb {H}^{m\times d}

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta Subscript bold-italic upper A Baseline greater-than-or-equal-to beta 0 Superscript double-struck upper H Baseline equals StartLayout Enlarged left-brace 1st Row 1st Column StartRoot StartFraction pi Over pi minus 2 EndFraction EndRoot almost-equals 1.659 2nd Column a m p semicolon if double-struck upper H equals double-struck upper R comma 2nd Row 1st Column StartRoot StartFraction 4 Over 4 minus pi EndFraction EndRoot almost-equals 2.159 2nd Column a m p semicolon if double-struck upper H equals double-struck upper C period EndLayout">

β

A

≥

β
0

H

=

{

π

π
−
2

≈
1.659

a
m
p
;

if 

H

=

R

,

4

4
−
π

≈
2.159

a
m
p
;

if 

H

=

C

.

\begin{equation*} \beta _{\boldsymbol {A}}\geq \beta _0^{\mathbb {H}}= \begin {cases} \sqrt {\frac {\pi }{\pi -2}}\approx 1.659 & \text {if $\mathbb {H}=\mathbb {R}$,}\\ \sqrt {\frac {4}{4-\pi }}\approx 2.159 & \text {if $\mathbb {H}=\mathbb {C}$.} \end{cases} \end{equation*}

We prove that the condition number of a standard Gaussian matrix in

H

m
×
d

\mathbb {H}^{m\times d}

asymptotically matches the lower bound

β
0

H

\beta _0^{\mathbb {H}}

for both real and complex cases. This result indicates that the constant lower bound

β
0

H

\beta _0^{\mathbb {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

m\geq 3

, the harmonic frame

A
∈

R

m
×
2

\boldsymbol {A}\in \mathbb {R}^{m\times 2}

possesses the minimum condition number among all

A
∈

R

m
×
2

\boldsymbol {A}\in \mathbb {R}^{m\times 2}

.

To the best of our knowledge, our findings provide the first universal lower bound for the condition number in phase retrieval. Additionally, we have identified the first optimal vector set in

R

2

\mathbb {R}^2

for phase retrieval. We are confident that these findings carry substantial implications for enhancing our understanding of phase retrieval.