Spherical designs as a tool for derandomization: The case
of PhaseLift
(Invited Paper)
Richard Kueng
David Gross
Felix Krahmer
Institute for Physics & FDM
University of Freiburg, and
School of Physics
The University of Sydney
richard.kueng@physik.uni-freiburg.de
Institute for Theoretical Physics
University of Cologne, and
Institute for Physics & FDM
University of Freiburg
david.gross@physik.uni-freiburg.de
Research Unit M15
Department of Mathematics
Technische Universität München
felix.krahmer@tum.de
Abstract—The problem of retrieving phase information from amplitude measurements alone has appeared in many scientific disciplines
over the last century. PhaseLift is a recently introduced algorithm for
phase recovery that is computationally tractable and numerically stable.
However, initial rigorous performance guarantees relied specifically on
Gaussian random measurement vectors. To date, it remains unclear
which properties of the measurements render the problem well-posed.
With this question in mind, we employ the concept of spherical t-designs
to achieve a partial derandomziation of PhaseLift. Spherical designs
are ensembles of vectors which reproduce the first 2t moments of the
uniform distribution on the complex unit sphere. As such, they provide
notions of “evenly distributed” sets of vectors, ranging from tight frames
(t = 1) to the full sphere, as t approaches infinity. Beyond the specific
case of PhaseLift, this result highlights the utility of spherical designs
for the derandomization of data recovery schemes.
Index Terms—phase retrieval, spherical designs, low rank matrix
recovery
I. I NTRODUCTION
A. The phase retrieval problem and PhaseLift
The problem of retrieving a complex signal x ∈ Cn from
measurements of the form
2
yi = |hai , xi|
i = 1, . . . , m,
(1)
where a1 , . . . , am ∈ Cn are measurement vectors, has long been
abundant in many areas of science. Quite recently, several new
recovery algorithms have been proposed and first rigorous performance
guarantees have been established. Examples include methods based
on polarization identities [1], alternating projections [2], or Wirtinger
flow [3]. In addition, there are reconstruction methods that are tailored
to specific measurement ensembles, such as the approach in [4], which
is based on polynomial representations.
The approach we will focus on has been called PhaseLift [5]–[7]
and relies on formulating the problem as a low-rank matrix recovery
task [8]–[10]. To this end, one notes [11] that the yi ’s in (1) can
equivalently be expressed as
2
yi = |hai , xi| = tr ((ai a∗i ) (xx∗ )) =: tr (Ai (xx∗ )) .
(2)
In other words, the measurement results yi are linear in the outer
product X := xx∗ of the signal x with itself. This slight reformulation
c
978-1-4673-7353-1/15/$31.00 2015
IEEE
“lifts” phase retrieval to a linear problem on the (non-linear) set of
n × n hermitian rank-one matrices {Z : Z = zz ∗ , z ∈ Cn } ⊂ H n :
find
Z ∈ Hn
subject to tr (ZAi ) = yi
(3)
i = 1, . . . , m,
rank Z = 1.
Throughout this article, we shall denote the n2 -dimensional real vector
space of hermitian n × n matrices by H n .
In general, solving linear equations over the set of rank-1 matrices
is computationally intractable. However, there are now many situations
for which it has been proved that the nuclear norm can be employed
as an efficiently computable proxy for rank [8]–[10]. (Recall that the
nuclear norm kXk∗ = tr(|X|) is the sum of the singular values of
X. In a sense, it is the natural “non-commutative”, basis-independent
matrix analogue of the vector `1 norm). These results are closely
related to the use of the `1 -norm as a convex relaxation of sparsity
in compressed sensing [12]. In particular, these findings suggest that
(3) can be substituted by the semi-definite program
minimize
kZk∗
n
(4)
Z∈H
subject to tr (ZAi ) = yi
i = 1, . . . , m.
This ansatz was dubbed the PhaseLift algorithm for phase retrieval
by its inventors [5]–[7].
The task is now to establish sufficient conditions under which the
above convex problem will indeed have the outer product X = xx∗
of the sought-for signal as its unique solution. First results proved that
this is the case (with high probability) if the number of measurements
roughly scales linearly in the problem dimension – i.e. m = O(n)
– and the measurement vectors ai are complex standard Gaussians
[6], [7]. Later works concentrated on the practically more important
case of “masked Fourier measurements” [13], [14]. However, the use
of Gaussian vectors obscures the specific properties of measurement
vectors that enable phase retrieval, while the masked Fourier case is
highly application-specified. Thus, the question we are interested in is:
Can one identify particular properties of measurement ensembles that
allow for phase retrieval via PhaseLift, that are sufficiently general
to encompass structured measurements (unlike Gaussians), but at the
same time are fairly general (unlike masked Fourier)? We will argue
below that the defining properties of spherical designs fall into this
category.
II. P HASE RETRIEVAL FROM SPHERICAL DESIGN MEASUREMENTS
To motivate the notion of spherical designs, we recall that lowrank recovery results [8]–[10] are usually phrased for measurement
ensembles that are isotropic, or drawn from a tight frame (analogous
statements apply to compressed sensing [12], but can be generalized –
see e.g. [15], [16]). One definition of such a structural property is as
follows:
Definition 1 (Isotropy). A weighted set {µ(α), Bα }α∈I ⊆ H n is
isotropic, if, for all Z ∈ H n ,
Z
Bα tr (Bα Z) dµ(α) = Z.
(5)
I
In the case of PhaseLift, full isotropy for the measurements Ai is
impossible to attain. The reason is that the Ai = ai a∗i are all outer
products and thus have positive Hilbert-Schmitt inner product with
the identity matrix:
trAi I = trAi = kai k22 > 0.
(6)
Fortunately, the vectors can be chosen in such a way that Ai ’s are
isotropic on the trace-free subspace of H n , i.e. the “overweighting
of the identity component” just noted is the only way in which the
Ai ’s deviate from being a tight frame. Indeed, for complex standard
Gaussian vectors the following identity follows from a simple direct
calculation:
Proposition 2 (Near-isotropy: Equation (5.1) in [6]). Let Z ∈ H n be
arbitrary and assume that b is a complex standard Gaussian vector
in Cn . Then
E [bb∗ tr (bb∗ Z)] = Z + tr(Z)I.
(7)
Note that the “identity component” tr(Ixx∗ ) = kxk22 of the signal is
nothing but its squared length, or intensity. From now on, we assume
that the intensity kxk22 is in fact known. Also, while not essential, we
have opted to carry out our analysis for measurement vectors ai with
unit length kai k2 = 1. With these conventions, the entire problem
only ever concerns vectors on the complex unit-sphere. The rotationinvariant measure on the unit sphere S n−1 ⊂ Cn is called the Haar
measure. One can sample from it e.g. by drawing complex standard
Gaussian vectors and normalizing them. The resulting analogue of
(7) reads
Z
1
ww∗ tr (ww∗ Z) dw =
(Z + tr(Z)I)
(8)
n(n
+ 1)
n−1
w∈S
for all Z ∈ H n . Near-isotropy for an ensemble {µ(α), bα }α∈I can
easily be seen [17, Lemma 1] to be equivalent to demanding that the
ensemble reproduces the 4th moments of the Haar measure:
Proposition 3 (Necessary and sufficient criterion for near isotropy).
Let {µ(α), bα }α∈I ⊆ S n−1 be a weighted set of unit vectors. Then
Z
Z
bα b∗α ⊗ bα b∗α dµ(α) =
ww∗ ⊗ ww∗ dw
(9)
I
w∈S n−1
holds if and only if the re-scaled set {µ(α),
near-isotropic in the sense of (7).
p
n(n + 1)bα b∗α }α∈I is
If the Ai ’s range over all measurements in a near-isotropic set, they
essentially1 form a tight frame in matrix space and thus X can be
1 As already mentioned, (6) implies that the ray {Z ∈ H n : Z = cI, c ∈ R} ∈ H n
is “overweighted”. If the signal’s intensity is known, however, this distortion can be
readily compensated.
recovered by simple linear inversion [11]. However, such a tight frame
necessarily contains at least dim H n = O(n2 ) elements – much more
than the O(n) degrees of freedom in x. Still, one could hope that
PhaseLift could be proved to succeed for linearly many ai ’s sampled
from such a set. We will prove below that, unfortunately, this is too
optimistic. In general, near-isotropy alone is insufficient for reaching
an optimal linear scaling in the number of measurements m. However,
(9) suggests a generalization which will turn out to be sufficiently
strong for achieving such a goal.
To motivate it, it is wortwhile to point out that vectors drawn
uniformly from the sphere are proportional to a tight frame in Cn (as
opposed to H n ):
Z
1
(10)
ww∗ dw = I.
n
n−1
w∈S
Combining this with (9) yields the following two structural criteria
for a weighted set {µ(α), bα }α∈I of unit vectors:
Z
Z
∗
bα bα dµ(α) =
ww∗ dw ⇒ tight frame,
(11)
n−1
I
S
Z
Z
⊗2
⊗2
(bα b∗α ) dµ(α) =
(ww∗ ) dw ⇒ near-isotropy. (12)
S n−1
I
Generalizing these equalities to arbitrary t-th tensor powers yields
the following definition which is a the heart of our work:
Definition 4 (Spherical t-design). Let t ∈ N. We call a weighted set
{µ(α), bα }α∈I ⊆ S n−1 of unit vectors a spherical t-design, if
Z
Z
⊗k
⊗k
(bα b∗α ) dµ(α) =
(ww∗ ) dw
(13)
S n−1
I
is valid for all 1 ≤ k ≤ t. The parameter t ∈ N is called the design’s
order.
While this definition underlines the resemblance of a t-design to
vectors drawn uniformly from the complex unit sphere, the expression
on the right hand side of (13) is not very practical for actual
calculations (in particular, if t is large). Fortunately, a straightforward
application of Schur’s Lemma [18, Lemma 1] yields
Z
S n−1
⊗k
(ww∗ )
dw =
−1
n+k−1
PSymk
k
for t, n ∈ N arbitrary. Here, PSymt denotes the projector onto the
⊗k
totally symmetric subspace of (Cn ) . In turn, techniques from
multilinear algebra – in particular wiring calculus [19], [20] – allow
for carrying out calculations involving PSymt explicitly.
Analytic expressions for exact designs are notoriously difficult to
find. Designs of degree 2 are widely known [21]–[24] (see also next
section). For degree 3, both real [25] and complex [26] designs are
known. For higher t, there are numerical methods based on the notion
of the frame potential [24], [26], [27], non-constructive existence
proofs [28], and constructions in sporadic dimensions (c.f. [29] and
references thererin).
In Section II-A below, we will show that drawing the measurements
from a spherical 2-design does not allow for non-trivial performance
guarantees for PhaseLift. Conversly, in Section II-B, we provide such
guarantees for designs of order t ≥ 3.
A. Phase retrieval from spherical 2-designs
Proposition 3 together with Definition 4 establishes a one-to-one
correspondence between spherical 2-designs and weighted sets of unit
vectors which fulfill near-isotropy in the sense of (7). This in turn
assures that for any X ∈ H n , measuring all projectors onto elements
of a 2-design as well as measuring tr (IX) = tr(X) allows one to
linearly invert the measurement process and determine X exactly.
In the context of the “lifted” phase retrieval problem, Balan et. al
[11] were the first to be aware of this correspondence. In turn, they
used a particular instance of a spherical 2-design
to recover X = xx∗
2
via a deterministic choice of order O n projective measurements
onto elements of this design.
Concretely, their approach uses a maximal set of mutually unbiased
bases (MUBs). Two orthonormal bases {u1 , . . . , un } and {v1 , . . . , vn }
of Cn are called mutually unbiased, if their overlap is uniformly
minimal. Concretely, this means that
Note that this statement implies that the probability of distinguishing
the orthogonal vectors x and y by means of a random phaseless
measurement (chosen uniformly from the spherical 2-design formed
by a maximal set of MUBs) is proportional to 1/n2 . In [20] the authors
use a slightly refined version of this insight together with a stopping
time argument to establish the following rigorous counter-example to
subsampling from a particular 2-design.
Theorem 6 (Theorem 2 in [20]). Let n ≥ 2 be a prime power and
let D2 ⊂ Cn be a maximal set of MUBs. Then there exist orthogonal,
normalized vectors x, y ∈ Cd which have the following property:
Suppose that m measurement vectors a1 , . . . , am are sampled
independently and uniformly at random from D2 . Then, for any ω ≥ 0,
the number of measurements must obey
ω
m ≥ n(n + 1),
(14)
4
or the event
1
∀i, j = 1, . . . , n
|hai , xi|2 = |hai , yi|2 ∀ i ∈ {1, . . . , m}
n
−ω
must hold for all i, j = 1, . . . , n. Note that this is just a generalization will occur with probability at least e .
of the incoherence property between standard and Fourier basis. In
It is worthwhile to emphasize that this no-go result only applies
prime power dimensions, a maximal set of (n + 1) such MUBs is to specific 2-designs. Particular instances of a 2-design that exhibit
known to exist (and can be constructed) [30]. Such a set is maximal additional structural properties may well allow for subsampling. In
in the sense that it is not possible to find more than (n + 1) MUBs in fact, such a measurement ensemble – “coded diffraction patterns”, or
any Hilbert space. It is well-known that equally weighted, maximal “masked Fourier measurements” – was introduced by Candès et al.
sets of MUBs form spherical 2-designs [22], [24].
in [13] and it was proven in the same paper that a total number of
Ehler and Kunis [31] also identified near isotropy – equation (7) m = O n log4 n such measurements actually allows for establishing
– and its connection to spherical 2-designs (12) (which they call a Phaselift recovery guarantee. Since these coded-diffraction patterns
curbatures of strength 2) as a crucial ingredient for performing phase fulfill near-isotropy in the sense of (7), Proposition 3 assures that
retrieval. Combining this insight with the PhaseLift approach [5], they also form a spherical 2-design. This equivalence was pointed out
[6] they conjecture that measuring O(n) projectors onto randomly in [14], where the required sampling rate for such a recovery was
chosen elements of a spherical 2-design should be sufficient for furthermore reduced to a total of O n log2 n measurements.
establishing a recovery guarantee for PhaseLift. However, in [20] a
counter-example to this conjecture is provided: without assuming and B. Phase retrieval from higher-order designs
exploiting additional properties of the measurement ensemble, random
Since a subsampled collection of random projectors onto a spherical
subsampling is not sufficient for avoiding a scaling of m = O(n2 ).
2-design can in general not be sufficient for recovering a sought for
At the heart heart of this counter-example is the following obser- signal X = xx∗ via PhaseLift, it is natural to ask if designs of higher
vation regarding the injectivity of a random phaseless measurement order allow for establishing such a recovery guarantee.
chosen from a particular spherical 2-design.
Recall that the main problem with subsampling from a spherical 2design
was the injectivity-issue pointed out in Proposition 5. However,
Proposition 5. Suppose that a is chosen uniformly at random from a
this
situation
changes dramatically for designs of higher order.
maximal set of MUBs (which forms a spherical 2-design). Then there
|hui , vj i|2 =
exist orthogonal and normalized vectors x, y ∈ Cn such that
h
i
2
2
2
Pr |ha, xi| − |ha, yi| > 0 ≤
.
n(n + 1)
Proof: Suppose that {u1 , . . . , un } ⊂ Cn is one orthonormal basis
contained in the maximal set of MUBs and set x := u1 , as well as
y := u2 . Note that by definition these vectors are orthogonal and
normalized. Due to the particular structure of MUBs, the expression
of interest obeys
(
1 if a = u1 , or a = u2 ,
2
2
|ha, u1 i| − |ha, u2 i| =
0 otherwise.
The claim now follows from noticing that a is chosen uniformly at
random from the n(n + 1) vectors contained in a maximal set of
MUBs.
Proposition 7. Suppose that a is chosen uniformly at random from
a spherical 4-design. Then for any two distinct vectors x, y ∈ Cn
h
i
1
2
2
Pr |ha, xi| − |ha, yi| > 0 ≥
96
is true.
Proof: The claim is an immediate corollary from the auxiliary
statement [32, equation (24)] used to establish Proposition 12 there.
For a chosen uniformly at random from a spherical 4-design, this
statement assures for any Z ∈ H n
"
#
2
1 − ξ2
ξkZk
2
∗
Pr |tr (aa Z)| ≥ p
≥
∀ξ ∈ [0, 1] (15)
24
n(n + 1)
by means of the Payley-Zygmund inequality. Here, kZk2 denotes the
Frobenius norm of Z. Setting Z := xx∗ − yy ∗ (note that since x and
y are distinct, the matrix Z cannot vanish and therefore kZk2 > 0
must hold) and setting ξ = 2−1/2 allows us to conclude
h
i
2
2
Pr |ha, xi| − |ha, yi| > 0 = Pr [|tr (aa∗ Z)| > 0]
"
#
2−1/2 kZk2
(1 − 1/2)2
1
∗
≥ Pr |tr (aa Z)| ≥ p
≥
=
24
96
n(n + 1)
Consequently, O(n log n) measurements randomly chosen from a
4-design are sufficient to guarantee phaseless recovery of arbitrary
signals x ∈ Cn via the convex optimization (4). Moreover, such a
sampling rate is close to optimal. As shown in [34], it follows from
the results derived in [35] that a sample size of m ≥ (4 + o(1)) n is
in fact necessary (cf. [36]).
Finally, we want to point out that Theorem 9 is also close to optimal
by means of (15).
in
terms of the design order t required. Indeed, Theorem 6 establishes
Proposition 7 assures that choosing measurement vectors indethat
a design order of at least t = 3 is required without making
pendently from any spherical 4-design behaves strikingly different
additional
assumptions on the measurement ensemble. Theorem 9 gets
from the 2-design case. In particular, this statement guarantees that
by
with
a
design order of t = 4 and no further assumptions. Fully
injectivity issues in the sense of Proposition 5 are much less severe for
closing
the
gap by establishing an analogue of Theorem 9 which is
designs of higher order. In accordance with such a disintegration of
valid
already
for 3-designs, or tightening the required sampling rate
the injecivity problem, non-trivial recovery guarantees for PhaseLift
in
Theorem
8
does constitute an intriguing open problem. Numerical
can be established for designs of higher order, as the main result in
studies
presented
in [20] suggest that this might indeed be feasible.
[20] shows.
For
the
sake
of
completeness
we have included the results of this
Theorem 8 (Theorem 1 in [20]). Let x ∈ Cn be the unknown signal study in Figure 1.
of interest. Suppose that kxk2`2 is known and that m measurement
vectors a1 , . . . , am have been sampled independently and uniformly
at random from an equally weighted t-design obeying t ≥ 3. Then,
with probability at least 1 − e−ω , PhaseLift (the convex optimization
problem (4) above) recovers x up to a global phase, provided that
the sampling rate exceeds
m ≥ ω Ct n1+2/t log2 n.
(16)
Here ω ≥ 1 is an arbitrary parameter and C is a universal constant.
Already for 3-designs, this result establishes a recovery guarantee
from subquadratically many – namely O(n5/3 log2 (n)) – projectors
onto randomly selected 3-design elements. The statement furthermore
becomes tight – i.e. the required sampling rate scales linearly in
the dimension n of the signal x – up to polylog-factors, if the
design order t is allowed to grow logarithmically with the dimension
(t = 2 log n). Note that a similar recovery guarantee for sampling
from particular t-designs can be established, even if n sampling
vectors are correlated at a time [33]. Recently, the above Theorem
was substantially strengthened and generalized in [32].
Theorem 9 (Theorem 3 in [32]). Consider the measurement process
described in (2) where the measurement vectors a1 , . . . , am have
been sampled independently from a spherical 4-design (according
to the design’s weights). Furthermore assume that the number of
measurements m obeys
m ≥ C1 nr log n,
for 1 ≤ r ≤ n arbitrary. Then with probability at least 1 − e−C2 m
it holds that for any X ∈ Hn with rank at most r, any solution
X # of the convex optimization
Pnproblem (4) with noisy measurements
yi = tr (Ai X) + i , where i=1 2i ≤ η 2 , obeys
Fig. 1. Phase Diagram for PhaseLift from (projected) stabilizer states, which form
an equally weighted 3-design in power-of-two dimensions [26]. The x-axis indicates
the problem’s dimension, while the y-axis denotes the number of independent design
measurements performed. The frequency of a successful recovery over 30 independent
runs of the experiment appears color-coded from black (zero) to white (one). To guide
the eye, we have furthermore included a red line indicating m = 4n − 4.
III. S PHERICAL DESIGNS AS A GENERAL - PURPOSE TOOL FOR
PARTIAL DERANDOMIZATION
In section II we have introduced spherical t-designs as a generalization of natural structural properties (11), (12) which assure that the
weighted vector set forms a tight frame on Cn and the corresponding
rank-one projectors obey near-isotropy (essentially meaning that they
form a slightly distorted tight-frame on H n ).
C3 η
kX − X # k2 ≤ √ .
(17)
Equivalently, one can define spherical t-designs as (usually finite)
m
weighted distributions of vectors that approximate Haar-random
Here, C1 , C2 , C3 > 0 again denote universal positive constants.
vectors (or equivalently: the distribution of complex standard Gaussian
This is a uniform recovery guarantee for recovering arbitrary rank-r vectors renormalized to unit length) up to t-th moments.
matrices that is furthermore robust towards noise. Clearly it covers
Viewing spherical t-designs from this angle reveals that they do
phase retrieval via PhaseLift as a special case – namely the one, constitute a general purpose tool for derandomizing results that
where all matrices X of interest are guaranteed to be rank one. initially required generic – i.e. Haar-random or standard Gaussian –
vectors. This utility of the design concept has long been appreciated
for example in quantum information theory [37], [38]. It has been
compared [37] to the notion of t-wise independence, which plays a
role for example in the analysis of discrete randomized algorithms.
ACKNOWLEDGEMENTS
The work of DG and RK is supported by the Excellence Initiative
of the German Federal and State Governments (Grants ZUK 43 &
81), the ARO under contracts W911NF-14-1-0098 and W911NF-141-0133 (Quantum Characterization, Verification, and Validation), the
Freiburg Research Innovation Fund, the DFG (GRO 4334 & SPP1798),
and the State Graduate Funding Program of Baden-Württemberg. FK
acknowledges support from the German Federal Ministry of Education
and Reseach (BMBF) through the cooperative research project ZeMat.
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