On the robustness of PhaseLift to arbitrary
errors in the noiseless case
Paul Hand
Rice University
Department of Computational and Applied Mathematics,
MS-134, 6100 Main Street, Houston, TX 77005
Abstract—The phase retrieval problem is the task of
recovering an unknown n-vector from phaseless linear
measurements. Through the technique of lifting, this problem may be convexified into a semidefinite rank-one matrix
recovery problem known as PhaseLift. Under a linear number of exact Gaussian measurements, PhaseLift recovers
the unknown vector exactly with high probability. Under
noisy measurements, the solution to a variant of PhaseLift
has error proportional to the ℓ1 norm of the noise. In the
present paper, we prove that this variant of PhaseLift can
tolerate a small fraction of arbitrary errors under O(n)
real-valued measurements. The phase retrieval problem
under arbitrary errors can be viewed as a special case
of robust Principal Component Analysis (PCA) for a rankone matrix with Gaussian rank-one measurements. In this
case, the PhaseLift program is simpler than the standard
robust PCA optimization program because it has no free
parameters.
I. I NTRODUCTION
This paper studies an algorithm for recovering a vector
x0 ∈ Rn from phaseless linear measurements that may
contain arbitrary errors. That is, for fixed measurement
vectors ai ∈ Rn for i = 1 . . . m, our task is to find x0
satisfying
bi = |⟨x0 , ai ⟩|2 + εi
(1)
for known bi ∈ R, known ai , and unknown εi . This
problem is known as phase retrieval. In the present paper,
we will analyze it in the case that the error vector ε
is sparse yet arbitrary. This setup is noiseless in the
sense that most measurements are exact. Measurements
of form (1) arise in several applications, including Xray crystallography [15], [10], [1]. In such applications,
extremely large errors in some measurements may be
due to sensor failure, occlusions, or other effects. Ideally,
recovery algorithms could provably tolerate a small
number of such errors.
Recently, researchers have introduced algorithms for
the phase retrieval problem that have provable recovery
guarantees [1], [4]. The insight of these methods is
that the phase retrieval problem can be convexified by
lifting it to the space of matrices. That is, instead of
searching for the vector x0 , one can search for the
lifted matrix x0 xt0 . The quadratic measurements (1) then
become linear measurements on this lifted matrix. As
the desired matrix is semidefinite and low rank, one can
write a semidefinite rank minimization problem which
has a convex relaxation known as PhaseLift. In this
noiseless case, PhaseLift is the program
min tr(X) subject to X ≽ 0, {ati Xai = bi }i=1...m (2)
Here, tr(X) is a convex proxy for the rank of X. An
estimate for the underlying signal x0 can be computed
by the leading eigenvector of the optimizer of (2).
When the measurements ai are Gaussian, [7] and [2]
have shown that (2) can be simplified to the semidefinite
feasibility problem
find X ≽ 0 such that {ati Xai = bi }i=1...m
(3)
In [2], the authors showed that this program succeeds
at finding x0 xt0 exactly and with high probability when
m ≥ cn for a sufficiently large c. This result is quite
surprising because there are only O(n) measurements
for an O(n2 ) dimensional object. The semidefinite constraint apparently is sufficiently ‘pointy’ that the highdimensional affine space of data-consistent matrices intersects the semidefinite cone only at exactly one point.
In the noisy case where ε may be dense, [2] showed
that the PhaseLift variant
!
|ati Xai − bi | subject to X ≽ 0
(4)
min
i
successfully recovers a matrix near x0 xt0 with high
probability. Specifically, they prove that the solution X̂
to (4) satisfies ∥X̂ − x0 xt0 ∥F ≤ C0 ∥ε∥1 /m with high
probability. This error estimate is most interesting in the
case where the error is small in an ℓ1 sense.
The contribution of the present paper is to show that
the program (4) is additionally robust against a constant
fraction of arbitrary errors.
Theorem 1. There exist positive numbers fmin , c, γ such
that for any x0 ∈ Rn and any ε ∈ Rm the following
holds. If m ≥ cn and ∥ε∥0 /m ≤ fmin , then x0 xt0 is the
unique solution to (4) with probability at least 1−e−γm.
A. Relation to Robust PCA
Robust Principal Component Analysis (PCA) is the
task of matrix completion under gross measurement errors. This framework typically involves measuring some
of the entries of a low rank n×n matrix X0 and assuming
that some fraction of those measurements are arbitrarily
corrupted, giving the data matrix A. The matrix X0 can
then be recovered under certain conditions by a sparseplus-low-rank convex program:
min λ∥X∥∗ + ∥E∥1 subject to P(X + E) = P(A)
X,E
(5)
where ∥X∥∗ is the nuclear norm of X, λ is a constant,
∥E∥1 is the ℓ1 norm of the vectorization of E, and P
is the projection of a matrix onto the observed entries.
[3], [5], [11], [19], [8], [6], [13]. Results from this
formulation have been quite surprising. For example,
under an appropriate choice of λ and under an incoherence assumption, the sparse-plus-low-rank decomposition succeeds for sufficiently low rank X0 when O(n2 )
entries are measured and a small fraction of them have
arbitrary errors [3]. Subsequent results have been proved
that only require m ! rn polylog(n) measurements,
where r is the rank of X0 [6], [13]. These results are
information theoretically optimal except for the polylogarithmic factor.
The present paper can be viewed as a rank-one
semidefinite robust PCA problem under Gaussian rankone measurements. From this perspective, we would
naturally formulate the problem by
!
|ati Xai − bi | subject to X ≽ 0.
min λ tr(X) +
i
(6)
The present paper shows that direct rank penalization
through the trace term is not fundamental for exact
recovery in the presence of arbitrary errors. That is,
(6) can be simplified by taking λ = 0. The resulting
program has no free parameters that need to be explicitly tuned. As in [7], [2], the positive semidefinite
cone provides enough of a constraint to enforce lowrankness. The present paper differs from the robust PCA
literature in that the measurements are generic and are
not direct samples of the entries of the unknown matrix.
As in the matrix completion literature, this difference
in measurement structure affords proofs of information
theoretically optimal data scalings with no additional
logarithmic factors.
B. Numerical simulation
We now explore the empirical performance of (4) by
numerical simulation. Let the signal length n vary from
5 to 50, and let the number of measurements m vary
from 10 to 250. Let x0 = e1 , and let ai ∼ N (0, In ). For
each (n, m), we consider measurements such that
"
bi ∼ Uniform([0, 104 ]) if 1 ≤ i ≤ ⌈0.05m⌉,
bi = |⟨x0 , ai ⟩|2
otherwise.
We estimate x0 xt0 by solving (4) using the SDPT3 solver
[16], [17] and YALMIP [14]. For a given optimizer X̂,
define the capped relative error as
min(∥X̂ − x0 xt0 ∥F /∥x0 xt0 ∥F , 1).
Figure 1 plots the average capped relative error over
10 independent trials. It provides empirical evidence
that the problem (4) succeeds under a linear number of
measurements, even when a constant fraction of them
contain very large errors.
50
Signal length (n)
That is to say, x0 xt0 will be exactly recovered with
high probability, even in the presence of a linear number
of arbitrary errors among a linear number of measurements. In the case that there are no gross errors, this
theorem simplifies to the recovery results of [2] and the
rank-one case of [12].
Theorem 1 prompts several important questions. Is the
semidefinite program (4) robust against noise? Can the
theorem be strengthened to permit universal recovery for
any x0 and ε under a fixed realization of the measurement vectors ai ? Also, can the theorem be extended to
the complex-valued case? For a brief comment on the
possible complex extension, see Section II-C. We leave
all these questions to future research.
25
m=
(n+1)n
2
50
100
150
200
Number of measurements (m)
250
Fig. 1. Recovery error for the PhaseLift matrix recovery problem (4)
as a function of n and m, when 5% of measurements contain large
errors. Black represents an average recovery error of 100% over 10
independent trials. White represents an average recovery error of zero.
The solid curve depicts when the number of measurements equals the
number of degrees of freedom in a symmetric n × n matrix. The
number of measurements required for successful recovery appears to
be linear in n in the presence of large errors.
II. P ROOFS
→ Rm be the mapping X .→
where S n is the space of #
symmetric
real-valued n × n matrices. Note that A∗ λ = i λi ai ati .
Let ei be the ith standard basis vector. Without loss of
Let A : S
(ati Xai )i=1...m ,
n
generality, we take x0 = e1 . Let X0 = x0 xt0 . We can
write the measurements (1) as b = AX0 + ε. Similarly,
the optimization program (4) can be written
The second technical lemma establishes that the optimal solution X0 + H to (4) lies on the cone ∥HT ⊥ ∥∗ ≥
0.56∥HT ∥F with high probability.
min ∥AX − b∥1 such that X ≽ 0.
Lemma 2. Fix x0 ∈ Rn and ε ∈ Rm . There exists a
constant γ̃ such that the following holds. If m ≥ 100c0 n
and ∥ε∥0 /m ≤ 0.01, then with probability at least 1 −
e−γ̃m any feasible X0 +H such that ∥A(X0 +H)−b∥1 ≤
∥A(X0 ) − b∥1 satisfies ∥HT ⊥ ∥∗ ≥ 0.56∥HT ∥F .
We introduce the following notation. Let ∥X∥∗ be the
nuclear norm of the matrix X. When X ≽ 0, ∥X∥∗ =
tr(X). Denote the Frobenius and spectral norms of X
as ∥X∥F and ∥X∥, respectively. Let ⟨X, Y ⟩ = tr(Y t X)
be the Hilbert-Schmidt inner product. Let T be the
space of symmetric matrices supported on their first
row and column. The orthogonal complement T ⊥ is
then the space of matrices supported in the lower-right
n − 1 × n − 1 block. Let I be the identity matrix, and
let (E) be the indicator function of the event E.
A. Recovery by dual certificates
The proof of Theorem 1 will be based on dual
certificates, as in [9], [4], [7], [2]. A dual certificate is an
optimal variable for the dual problem to (4). Its existence
certifies the correctness of a minimizer to (4). The first
order optimality conditions at X0 for (4) are
Ỹ = A∗ λ̃
(7)
Ỹ ≽ 0
(9)
λ̃ ∈ ∂∥ · ∥1 (−ε)
⟨Ỹ , X0 ⟩ = 0
(8)
(10)
where ∂∥ · ∥1 (−ε) is the subdifferential of the ℓ1 norm
evaluated at −ε. Note that (9) and (10) imply ỸT = 0.
Such a Ỹ would be dual certificate for (4). Unfortunately,
constructing Ỹ is difficult. As in [9], [4], [7], [2], we seek
an inexact dual certificate, which approximately satisfies
these conditions. Specifically, we build a dual certificate
Y = A∗ λ that satisfies
YT ⊥ ≽ IT ⊥ ,
∥YT ∥F ≤ 1/2,
"
λi = −α sgn(εi )
λi ≤ α
(11)
(12)
if εi ̸= 0,
if εi = 0.
(13)
for some positive α,
To prove that existence of such a Y will guarantee
successful recovery of x0 xt0 with high probability, we
will rely on two technical lemmas. The first lemma
provides ℓ1 -isometry bounds on A proven in [4].
Lemma 1 ([4]). There exist constants c0 , γ0 such that if
m ≥ c0 n, then with probability at least 1 − e−γ0 m ,
%
$
1
1
∥X∥∗ for all X,
(14)
∥A(X)∥1 ≤ 1 +
m
16
%
$
1
1
∥X∥
∥A(X)∥1 ≥ 0.94 1 −
m
16
for all symmetric, rank-2 X
(15)
Proof: By assumption, ∥AH − ε∥1 ≤ ∥ε∥1 . Let S
be a set of cardinality ⌈0.01m⌉ containing the support
of ε. We have
∥AS c H∥1 + ∥AS H − ε∥1 ≤ ∥ε∥1 .
(16)
By the triangle inequality,
∥AS c H∥1 + ∥ε∥1 − ∥AS H∥1 ≤ ∥ε∥1 ,
(17)
which implies
∥AS c H∥1 ≤ ∥AS H∥1
(18)
Breaking (18) into its components on T and T ⊥ and
applying the triangle inequality, we have
∥AS c HT ∥1 − ∥AS c HT ⊥ ∥1 ≤ ∥AS HT ∥1 + ∥AS HT ⊥ ∥1
⇒∥AS c HT ∥1 ≤ ∥AS HT ∥1 + ∥AHT ⊥ ∥1
(19)
We now apply the ℓ1 isometry bounds from Lemma 1
on each term of (19). As |S c | ≥ c0 n, Lemma 1 gives
that with probability at least 1 − e−γ1 m
$
%
1
∥AS c HT ∥1 ≥ 0.94 1 −
|S c |∥HT ∥
(20)
16
$
% c
1 |S |
√ ∥HT ∥F ,
≥ 0.94 1 −
(21)
16
2
where the second inequality follows because HT has
rank at most 2. As |S| ≥ c0 n and |S| = ⌈0.01m⌉,
Lemma 1 gives that with probability at least 1 − e−γ2 m
$
%
1
∥AS HT ∥1 ≤ |S| 1 +
∥HT ∥∗
(22)
16
%
$
1 √
2∥HT ∥F
≤ |S| 1 +
(23)
16
As m ≥ c0 n, Lemma 1 also gives that with probability
at least 1 − e−γ0 m ,
∥AHT ⊥ ∥1 ≤ m(1 +
1
)∥HT ⊥ ∥∗
16
(24)
Combining (19)–(24), we have
&
)
'
(
1
0.94 1 − 16
|S c | √ |S|
(
√ '
∥HT ∥F ≤ ∥HT ⊥ ∥∗
− 2
1
m
m
2 1 + 16
Thus, 0.56∥HT ∥F ≤ ∥HT ⊥ ∥∗ with probability at least
1 − 3e−γ̃m .
We now prove that existence of an inexact dual
certificate (11)–(13) will guarantee successful recovery
of X0 = x0 xt0 with high probability.
Lemma 3. Fix x0 ∈ Rn . Let m ≥ 100c0n. Fix ε ∈ Rm
such that ∥ε∥0 /m ≤ 0.01. Suppose that there exists Y =
A∗ λ satisfying (11)–(13). Then, with probability at least
1 − e−γ̃m , X0 is the unique solution to (4).
Proof: Let X0 + H be feasible for (4) and such that
∥A(X0 + H) − b∥1 ≤ ∥A(X0 ) − b∥1 . That is,
∥AH − ε∥1 ≤ ∥ε∥1 .
By condition (13),
λ/α ∈ ∂∥ · ∥1 (−ε)
(25)
Hence,
∥ − ε∥1 + ⟨λ/α, AH⟩ ≤ ∥ε∥1
⇒
⟨λ, AH⟩ ≤ 0
⇒
(26)
(27)
⟨Y, H⟩ ≤ 0.
(28)
⟨YT , HT ⟩ ≤ −⟨YT ⊥ , HT ⊥ ⟩
(29)
Decomposing (28) into T and T ⊥ , we have
As YT ⊥ ≽ 0 and HT ⊥ ≽ 0, we have
⟨YT ⊥ , HT ⊥ ⟩ ≤ |⟨YT , HT ⟩|
(30)
The form of Y0 is due to [2], and the intuition
t
behind it is as follows.
$ Note that
% E(ai ai ) = In and
˜
β0
0
E(|⟨ai , e1 ⟩|2 ai ati ) =
, where β˜0 = Ez 4 for
0 In−1
#m
1
a standard normal z. The construction m
i=1 [β̃0 −
2
t
˜
|⟨ai , e1 ⟩| ]ai ai thus has expected value (β0 − 1)IT ⊥ ,
which would provide the exact dual certificate conditions
(9)–(10). As shown in [2], a satisfactory inexact dual
certificate can be built with m = O(n) and coefficients
that are truncated to be no larger than 7/m. In the present
formulation, the terms Y+ and Y− are then set to have
coefficients ∓7/m in order to satisfy (13).
Lemma 4. Let x0 = e1 and fix ε ∈ Rm . There
exists constants c, γ ∗ such that the following holds. Let
m ≥ cn. Suppose ∥ε∥0 /m ≤ 0.001. Let S + and S −
be disjoint sets of cardinality ⌈0.001m⌉ containing the
indices over which ε is positive and negative, respectively. Then the dual certificate Y from (31) satisfies
17
3
∥YT ∥F ≤ 1/4 and ∥YT ⊥ − 10
IT ⊥ ∥ ≤ 10
with probability
−γ ∗ m
at least 1 − e
.
Proof: It suffices to show that with high probability
,
,
,
,
,Y0,T ⊥ − 17 IT ⊥ , ≤ 0.15,
(32)
,
,
10
3
,
20
∥Y±,T ⊥ ∥ ≤ 0.015,
∥Y±,T ∥F ≤ 0.035.
∥Y0,T ∥F ≤
By conditions (11)–(12)
1
∥HT ⊥ ∥∗ ≤ |⟨YT ⊥ , HT ⊥ ⟩| ≤ |⟨YT , HT ⟩| ≤ ∥HT ∥F
2
By Lemma 2, ∥HT ⊥ ∥∗ ≥ 0.56∥HT ∥F with probability at
least 1−e−γ̃m . We conclude that HT = 0 and HT ⊥ = 0,
with probability at least 1−e−γ̃m . Thus, X0 is the unique
solution to (4).
B. Construction of the dual certificate
We now construct the dual certificate for x0 = e1 .
Let S + and S − be disjoint supersets of the indices over
which ε is positive or negative, respectively. Let S =
S + ∪ S − . For pedagogical purposes, S + and S − should
be thought of as exactly the indices over which ε is
positive or negative. For technical reasons, we let them
be supersets with cardinality linear in n, in order to use
standard concentration estimates.
For a fixed choice of S + and S − , let the inexact dual
certificate Y be defined by
!
1*!
−7ai ati +
7ai ati
Y =
m
i∈S +
i∈S −
+
!
2
(31)
+
[β0 − |⟨ai , e1 ⟩| (|⟨ai , e1 ⟩| ≤ 3)]ai ati
i∈S c
where β0 = Ez 4 (|z| ≤ 3) ≈ 2.6728 and z is a standard
normal random variable. We write (31) as Y = Y+ +
Y− + Y0 .
(33)
(34)
(35)
First, we establish (32)–(33). By Lemma 2.3 in [2],
there exist c̃, γ̃ such that if |S c | ≥ c̃n, then with
c
probability at least 1 − e−γ̃|S | ,
,
,
,
, m
17
,
,
(36)
, |S c | Y0,T ⊥ − 10 IT ⊥ , ≤ 1/10, and
,
,
, m
,
,
,
(37)
, |S c | Y0,T , ≤ 3/20.
Thus,
, ,
,
,
c
, ,
,
,
,Y0,T ⊥ − 17 IT ⊥ , ≤ ,Y0,T ⊥ − |S | 17 IT ⊥ ,
,
,
,
,
10
m 10
%
$
c
|S | 17
+ 1−
m
10
≤ 0.15
(38)
which establishes (32). By (37), we get (33) immediately.
Next, we establish (34). Let a′ be the vector formed
by
n − 1 components of a. Observe that
# the last
′ ′∗
a
a
is a Wishart random matrix, for which
i∈S + i i
standard estimates, such as Corollary 5.35 in [18], apply.
If |S + | ≥ c̃0 n and |S + | = ⌈0.001m⌉, with probability
at least 1 − e−γ̃1 m for some γ̃1 ,
,
,
, 1 !
,
,
,
a′i ai ′∗ − IT ⊥ , ≤ 1/2
, +
,
, |S |
+
i∈S
Hence,
,
,
, 1 !
,
,
′ ′∗ ,
ai ai , ≤ 3/2
, +
, |S |
,
+
i∈S
+
Thus, ∥Y+,T ⊥ ∥ ≤ 23 |Sm | , and we arrive at (34). The
bound for the Y−,T ⊥ term is identical.
+
Next,
we establish (35). Note that Y+ = −7 |Sm | ·
#
1
t
+
i∈S + ai ai . As per Lemma 5, if |S | ≥ c̃1 n and
|S + |
+
|S + | = ⌈0.001m⌉, then ∥Y+,T ∥F ≤ 7 |Sm | ·5 ≤ 7 ·0.001 ·
5, with probability at least 1 − e−γ̃1 m .
Thus (32)–(35) hold simultaneously with probabil∗
ity at least 1 − e−γ for some γ ∗ provided c ≥
max(1000c0, 2c̃, 1000c̃0, 1000c̃1 ).
The behavior of Y±,T relies on the following.
1 #m
t
Lemma 5. Let A = m
i=1 ai ai . There exists c̃, γ̃1
such that if m ≥ c̃n then ∥AT ∥F ≤ 5 with probability
at least 1 − e−γ̃1 m .
Proof: Let y be the (1, 1) entry of A, and let y ′ be
the rest of the
column. Then ∥AT ∥2F = y 2 + ∥y ′ ∥22 .
#first
m
1
So, y = m i=1 a2i (1). Hence, my ∼ χ2m . Standard
results on the concentration of chi-squared variables give
P(my ≥ 4m) ≤ e−γ̃m .
Hence P(y 2 ≥ 16) ≤ e−γ̃m .
Now, it remains to bound ∥y ′ ∥22 . We can write y ′ =
1 ′
′
′
′
m Z c, where Z = [a1 , . . . , am ] and ci = ai (i), where
′
Z and c are independent. Note that ∥c∥22 is a Chisquared random variable with m degrees of freedom.
Hence, with probability at least 1 − e−m ,
∥c∥22 ≤ 3m
For fixed ∥x∥2 = 1, ∥Z ′ x∥22 ∼ χ2n−1 , and hence with
probability at least 1 − e−γm ,
∥Z ′ x∥22 ≥ m/100,
m
when m ≥ c̃n. Hence, m2 ∥y ′ ∥22 ≤ 3m√· 100
with
−γm
′
probability at least 1−2e
. Thus ∥y ∥2 < 3/10 with
probability at least 1 − 2e−γm . We have ∥AT ∥2F ≤ 25,
and hence ∥AT ∥F ≤ 5.
The proof of Theorem 1 comes from the direct combination of Lemmas 3 and 4.
C. Comment on the complex-valued case
Extending Theorem 1 to the complex-valued case may
be possible. In the proofs above, the most central use
of the real-valued assumption is in the concentration
estimates of the dual certificate. The certificate is broken
into pieces where ε is positive, negative, and zero. In the
complex case, there are an infinite number of direction
that the values of ε can take. Hence, the complex case
will require a more involved analysis.
R EFERENCES
[1] E. Candès, Y. Eldar, T. Strohmer, and V. Voroninski. Phase
retrieval via matrix completion. SIAM J. Imaging Sci., 6(1):199–
225, 2013.
[2] Emmanuel J Candès and Xiaodong Li. Solving quadratic equations via phaselift when there are about as many equations as
unknowns. Found. of Comput. Math., pages 1–10, 2012.
[3] Emmanuel J Candès, Xiaodong Li, Yi Ma, and John Wright.
Robust principal component analysis? J. ACM, 58(3):11, 2011.
[4] Emmanuel J Candès, Thomas Strohmer, and Vladislav Voroninski. Phaselift: Exact and stable signal recovery from magnitude
measurements via convex programming. Comm. Pure Appl.
Math., 66(8):1241–1274, 2013.
[5] Venkat Chandrasekaran, Sujay Sanghavi, Pablo A Parrilo, and
Alan S Willsky. Rank-sparsity incoherence for matrix decomposition. SIAM Journal on Optimization, 21(2):572–596, 2011.
[6] Yudong Chen, Ali Jalali, Sujay Sanghavi, and Constantine Caramanis. Low-rank matrix recovery from errors and erasures.
Information Theory, IEEE Transactions on, 59(7):4324–4337,
2013.
[7] Laurent Demanet and Paul Hand. Stable optimizationless recovery from phaseless linear measurements. Journal of Fourier
Analysis and Applications, 20(1):199–221, 2014.
[8] Arvind Ganesh, John Wright, Xiaodong Li, Emmanuel J Candes,
and Yi Ma. Dense error correction for low-rank matrices via
principal component pursuit. In Information Theory Proceedings
(ISIT), 2010 IEEE International Symposium on, pages 1513–
1517. IEEE, 2010.
[9] David Gross. Recovering low-rank matrices from few coefficients in any basis. Information Theory, IEEE Transactions on,
57(3):1548–1566, 2011.
[10] Robert W Harrison. Phase problem in crystallography. J. Opt.
Soc. Am. A, 10(5):1046–1055, 1993.
[11] Daniel Hsu, Sham M Kakade, and Tong Zhang. Robust matrix
decomposition with sparse corruptions. Information Theory,
IEEE Transactions on, 57(11):7221–7234, 2011.
[12] Richard Kueng, Holger Rauhut, and Ulrich Terstiege. Low
rank matrix recovery from rank one measurements. CoRR,
abs/1410.6913, 2014.
[13] Xiaodong Li. Compressed sensing and matrix completion with
constant proportion of corruptions. Constructive Approximation,
37(1):73–99, 2013.
[14] J. Löfberg. Yalmip : A toolbox for modeling and optimization
in MATLAB. In Proceedings of the CACSD Conference, Taipei,
Taiwan, 2004.
[15] RP Millane. Phase retrieval in crystallography and optics. J. Opt.
Soc. Am. A, 7(3):394–411, 1990.
[16] K. C. Toh, M.J. Todd, and R. H. Tutuncu. Sdpt3 - a matlab
software package for semidefinite programming. Optimization
Methods and Software, 11:545–581, 1998.
[17] R.H. Tutuncu, K.C. Toh, and M.J. Todd. Solving semidefinitequadratic-linear programs using sdpt3. Mathematical Programming Ser. B, 95:189–217, 2003.
[18] R. Vershynin. Introduction to the non-asymptotic analysis of
random matrices. In Y.C. Eldar and G. Kutyniok, editors, Compressed Sensing: Theory and Applications. Cambridge University
Press, 2012.
[19] John Wright, Arvind Ganesh, Shankar Rao, Yigang Peng, and
Yi Ma. Robust principal component analysis: Exact recovery
of corrupted low-rank matrices via convex optimization. In
Advances in neural information processing systems, pages 2080–
2088, 2009.