Authors Jop Bri{\"e}t, Oded Regev, Rishi Saket,
License CC-BY-3.0
T HEORY OF C OMPUTING, Volume 13 (15), 2017, pp. 1–24
www.theoryofcomputing.org
Tight Hardness of the
Non-Commutative Grothendieck Problem
Jop Briët∗ Oded Regev† Rishi Saket
Received February 2, 2016; Revised January 20, 2017; Published December 2, 2017
Abstract: We prove that for any ε > 0 it is NP-hard to approximate the non-commutative
Grothendieck problem to within a factor 1/2 + ε, which matches the approximation ratio
of the algorithm of Naor, Regev, and Vidick (STOC’13). Our proof uses an embedding
of `2 into the space of matrices endowed with the trace norm with the property that the
image of standard basis vectors is longer than that of unit vectors with no large coordinates.
We also observe that one can obtain a tight NP-hardness result for the commutative Lit-
tle Grothendieck problem; previously, this was only known based on the Unique Games
Conjecture (Khot and Naor, Mathematika 2009).
ACM Classification: G.1.6
AMS Classification: 68Q17, 15A60, 32A70, 03D15
Key words and phrases: hardness of approximation, semidefinite programming, Grothendieck inequality
1 Introduction
The subject of this paper, the non-commutative Grothendieck problem, has its roots in celebrated work of
Grothendieck [11], sometimes (jokingly?) referred to as “Grothendieck’s résumé.” His paper laid the
foundation for the study of the geometry of tensor products of Banach spaces, though its significance only
A conference version of this paper appeared in the Proceedings of the 56th Annual IEEE Symposium on Foundations of
Computer Science (FOCS 2015) [8].
∗ Supported by a Rubicon grant from the Netherlands Organisation for Scientific Research (NWO).
† Supported by the Simons Collaboration on Algorithms and Geometry and by the National Science Foundation (NSF) under
Grant No. CCF-1320188. Any opinions, findings, and conclusions or recommendations expressed in this material are those of
the authors and do not necessarily reflect the views of the NSF.
© 2017 Jop Briët, Oded Regev, and Rishi Saket
c b Licensed under a Creative Commons Attribution License (CC-BY) DOI: 10.4086/toc.2017.v013a015
J OP B RI ËT, O DED R EGEV, AND R ISHI S AKET
became widely recognized after it was revamped by Lindenstrauss and Pełczyński [27]. The main result
of the paper, now known as Grothendieck’s inequality, shows a close relationship between the following
two quantities. For a complex d×d matrix M let
d
OPT(M) = sup ∑ Mi j αi β j , (1.1)
αi ,β j i, j=1
where the supremum goes over scalars on the complex unit circle, and let
d
SDP(M) = sup ∑ Mi j hai , b j i ,
ai ,b j i, j=1
(1.2)
where the supremum goes over vectors on a complex Euclidean unit sphere of any dimension. Since the
circle is the sphere in dimension one, we clearly have SDP(M) ≥ OPT(M). Grothendieck’s inequality
states that there exists a universal constant KGC < ∞ such that for any positive integer d and any d×d
matrix M, we also have SDP(M) ≤ KGC OPT(M). This result found an enormous number of applications
both within and far beyond its original scope and we give some examples below (see [23, 32] for extensive
surveys). Despite this, finding the optimal value of KGC is the only one of six problems posed in [11]
that remains unsolved today; the current best upper and lower bounds are 1.4049 [14] and 1.338 [9],
respectively. The situation is similar for the real variant of the problem, where all objects involved are
over the real numbers. The constant in that case is denoted KG and is known to be between 1.6769 and
1.7823 (see [6]).
The non-commutative Grothendieck problem, to which we will refer as the NCG, is the optimization
problem in which we are asked to maximize a given bilinear form over all pairs of unitary matrices. More
explicitly, we are given a four-dimensional array of complex numbers (Ti jkl )di, j,k,l=1 and are asked to find
or approximate the value
d
OPT(T ) = sup ∑ Ti jkl Ai j Bkl , (1.3)
A,B i, j,k,l=1
where the supremum is over pairs of d×d unitary matrices. (The word “non-commutative” simply refers
to the fact that optimization is over matrices.) It is not difficult to see that the (commutative) Grothendieck
problem of computing OPT(M) as in (1.1) is the special case where T has Tii j j = Mi j and zeros elsewhere.
Seen at first, the problem might seem overly abstract, but in fact, as we will illustrate below, it captures
many natural questions as special cases. Grothendieck conjectured that his namesake inequality has an
extension that relates (1.3) and the quantity
d
SDP(T ) = sup ∑ Ti jkl ~Ai j , ~Bkl ,
~A,~B i, j,k,l=1
(1.4)
where ~A, ~B range over all d×d matrices whose entries are complex vectors of arbitrary dimension satisfy-
ing a certain “unitarity” constraint.1 Namely, he conjectured that there exists a universal constant K < ∞
1 Namely, we require that ~ A∗~A = 1 and ~A~A∗ = 1 and similarly for ~B, where the multiplication of two vector-entried matrices
is a scalar-valued matrix computed just like a normal matrix multiplication except the scalar multiplication is replaced by an
inner product, e. g., the (i, j)-coordinate of ~A~A∗ is given by ∑k h~Aik , ~A jk i.
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T IGHT H ARDNESS OF THE N ON -C OMMUTATIVE G ROTHENDIECK P ROBLEM
such that for every positive integer d and array T as above, we have OPT(T ) ≤ SDP(T ) ≤ K OPT(T ),
where the first inequality follows immediately from the definition. Over twenty-five years after being
posed, the non-trivial content of Grothendieck’s conjecture, SDP(T ) ≤ K OPT(T ), was finally settled in
the positive by Pisier [31]. This result is now known as the non-commutative Grothendieck inequality.
In contrast with the commutative case, and somewhat surprisingly, the optimal value of K is known:
Haagerup [13] lowered Pisier’s original estimate to K ≤ 2 and this was later shown to be sharp by
Haagerup and Itoh [15].
Algorithmic applications. The importance of Grothendieck’s inequality to computer science was
pointed out by Alon and Naor [1], who placed it in the context of approximation algorithms for combina-
torial optimization problems. They observed that computing SDP(M) is a semidefinite programming
(SDP) problem that can be solved efficiently (to within arbitrary precision), and they translated an upper
bound of about 1.78 on KG due to Krivine [26] to an efficient rounding scheme that turns SDP vectors
into a feasible solution for the real Grothendieck problem (1.1) achieving value at least SDP(M)/1.78.
It is known that whatever the value of KG is, there exists an efficient algorithm achieving value at
least (1/KG − ε) SDP(M) for any constant ε > 0. (This was first shown in [33], but can also be derived
from the results in [7] using a simple discretization argument combined with a brute-force search.) The
Grothendieck problem shows up in a number of different areas such as graph partition problems and
computing the Cut-Norm of a matrix [1], in statistical physics where it gives ground state energies in the
spin glass model [25], and in quantum physics where it is related to Bell inequalities [37].
In the same spirit, Naor, Regev, and Vidick [28] recently translated the non-commutative Grothendieck
inequality into an efficient SDP-based approximation algorithm for the NCG problem (1.3) that achieves
value at least SDP(T )/2. They also considered the real variant and √ a Hermitian variant, for which
they gave analogous algorithms achieving value at least SDP(T )/2 2. This in turn implies efficient
constant-factor approximation algorithms for a variety of problems, including the Procrustes problem and
two robust versions of Principal Component Analysis [28] and quantum XOR games [34]. In a related
result, Bandeira et al. [2] considered a special case of the NCG (in fact a special case even of the Little
NCG defined below) and showed how to obtain better approximation factors for it. They also show that it
is rich enough to capture some applications such as the Procrustes problem and another natural problem
called the Global Registration Problem.
Hardness of approximation. For simplicity we momentarily turn to the real setting, but similar results
hold over the complex numbers. The Grothendieck problem contains M AX C UT as a special case (in fact,
it is a special case of the “Little Grothendieck Problem,” discussed below, in which M is the positive
semidefinite Laplacian matrix of a graph [1]). It therefore follows from Håstad’s inapproximability
result [16] that it is NP-hard to approximate the value (1.1) to any factor larger than 16/17 ≈ .941. Based
on the current best-known lower bound of about 1.676 on KG , Khot and O’Donnell [24] proved that (1.1)
is Unique-Games-hard to approximate to within a factor larger than 1/1.676 ≈ .597. Moreover, despite
the fact that the exact value of KG is still unknown, Raghavendra and Steurer [33] were able to improve
this Unique Games hardness to 1/KG . (See for instance [20, 36] for background on the Unique Games
conjecture.)
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J OP B RI ËT, O DED R EGEV, AND R ISHI S AKET
Our result. Whereas the hardness situation for the commutative version of Grothendieck’s problem is
reasonably well understood (apart from the yet-unknown exact value of KG ), no tight hardness result was
previously known for the non-commutative version. In fact, we are not even aware of any hardness result
that is better than what follows from the commutative case. Here we settle this question.
Theorem 1.1. For any constant ε > 0 it is NP-hard to approximate the optimum (1.3) of the non-
commutative Grothendieck problem to within a factor greater than 1/2 + ε.
Little Grothendieck. In fact, we prove a stronger result than Theorem 1.1 that concerns a special case
of the NCG called the Little NCG. Let us start by describing the (real case of the) commutative Little
Grothendieck problem (a. k. a. the positive-semidefinite Grothendieck problem). A convenient way to
phrase it is as asking for the operator norm of a linear map F : Rn → `d1 (where `dp denotes Rd endowed
with the ` p norm), defined as kFk = supa kF(a)k`1 where the vector a ranges over the n-dimensional
Euclidean unit ball. It turns out that this is a special case of (the real version of) Equation (1.1): for any F
there exists a positive semidefinite d×d matrix M such that OPT(M) = kFk2 ; and vice versa, one can
also map any such M into a corresponding operator F (see, e. g., [32] or Section 6). We wish to highlight
that for such instances, the constant KG may be replaced by the smaller value π/2 [35] and that this value
is known to be optimal [11]. Moreover, Nesterov p made this algorithmic, namely, he showed an algorithm
that approximates kFk as above to within 2/π [29]. Finally, Khot and Naor [22], as part of a more
general result, showed
p that this is tight: the Unique-Games-hardness threshold for the Little Grothendieck
problem is exactly 2/π. As an aside, we note that other operator norms, in particular of operators in
Rn → `d4 , have played an important role recently in theoretical computer science (see, e. g., [3]).
The Little non-commutative Grothendieck problem is formulated in terms of the (normalized)√trace
norm, also known as the Schatten-1 norm, which for a d×d matrix A is given by kAkS1 = d −1 Tr A∗ A.
In other words, kAkS1 is the average of the singular values of A. The space of matrices endowed with this
norm is denoted by S1 and by S1d if we restrict to d×d matrices. The problem then asks for the operator
2
norm of a linear map F : Cn → S1d . This problem is a special case of the NCG where OPT(T √ ) = kFk (see
Section 6). In particular, it follows from [13, 28] that there is an efficient SDP-based 1/ 2-approximation
algorithm for the Little NCG. Our stronger result alluded to above shows tight hardness for the Little
NCG, which directly implies Theorem 1.1.
Theorem 1.2. For any constant ε > 0 it is NP-hard√to approximate the Little non-commutative
Grothendieck problem to within a factor greater than 1/ 2 + ε.
While this result applies to the complex case, an easy transformation shows that it directly implies
the same result for the real and Hermitian cases introduced in [28] (see Section 5.1). Finally, as we
show in the “warm-up” section of this paper (Section 4), we also get a tight NP-hardness result for the
commutative Little Grothendieck problem, strengthening the unique-games-based result of [22].
Theorem 1.3. For any constant ε > 0 it is NP-hard p to approximate the real Little commutative
Grothendieck problem to within a factor greaterpthan 2/π + ε. Similarly, the complex case is NP-hard
to approximate to within a factor greater than π/4 + ε.
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Techniques. Nearly all recent work on hardness of approximation, including for commutative Grothen-
dieck problems [33, 22], uses the machinery of Fourier analysis over the hypercube, influences, or the
majority is stablest theorem [30]. Our attempts to apply these techniques here failed. Instead, we use a
more direct approach similar to that taken in [12] and avoid the use of the hypercube altogether. The role
of dictator functions is played in our proof simply by the standard basis vectors of Cn . The dictatorship
test, which is our main technical contribution, comes in the form of a linear operator F : Cn → S1d with
the following notable property: it maps the n standard basis vectors to matrices with trace√norm 1, and
it maps any unit vector with no large coordinate to a matrix with trace norm close to 1/ 2. Roughly
speaking, one can think of F as identifying an interesting subspace of S1d (namely, the √ image of F) in
which the unit ball looks somewhat like the intersection of the Euclidean ball of radius√ 2 with the `∞
ball [−1, 1]n (since with such a unit ball, the norm of a vector a is given by max{kak2 / 2, kak∞ }).
A first attempt to construct an operator F as above might be to map each standard basis vector to a
random unitary matrix. This, however, leads to a very poor map—while standard basis vectors are mapped
to matrices of trace norm 1, vectors with no large coordinates are mapped to matrices of trace norm close
to 8/(3π) ≈ 0.848 by Wigner’s semicircle law. Another natural approach is to look at the construction
by Haagerup and Itoh [15] (see also [32, Section 11] for a self-contained description) which shows the
factor-2 lower bound in the non-commutative Grothendieck inequality, i. e., the tight integrality gap of
the SDP (1.4). Their construction relies on the so-called CAR algebra (after canonical anticommutation
relations) and provides an isometric mapping from Cn to S1 , i. e., all unit vectors are mapped to matrices
of trace norm 1. Directly modifying this construction (akin to how, e. g., Khot et al. [21] obtained tight
hardness of M AX C UT by restricting the tight integrality gap instances by Feige and Schechtman [10]
from the sphere to the hypercube) does not seem to work. Instead, our construction of F relies on a
different (yet related) algebra known as the Clifford algebra. The Clifford algebra was used before in
a celebrated result by Tsirelson [37] (to show that Grothendieck’s inequality can be interpreted as a
statement about XOR games with entanglement). His result crucially relies on the fact that the Clifford
algebra gives an isometric mapping from Rn to S1 . Notice that this is again an isometric embedding,
but now only over the reals. Our main observation here (Lemma 5.2) is that the same mapping, when
extended to Cn , exhibits intriguing cancellations when the phases in the input vectors are not aligned,
and this leads to the construction of F (Lemma 5.1). Even though the proof of this fact is simple, we
find it surprising; we are not aware of any previous application of such “complex extensions” of Clifford
algebras.
√
Open questions. For the real and Hermitian cases there is a gap of 2 between the guarantee of
the [28] algorithms and our hardness result. It also would be interesting to explore whether hardness of
approximation results can be derived to some of the applications of the NCG, including the Procrustes
problem and robust Principle Component Analysis. We believe that our embedding would be useful there
too.
Outline. The rest of the paper is organized as follows. In Section 2, we set some notational conventions,
gather basic preliminary facts about relevant Banach spaces, and give a detailed formulation of the
Smooth Label Cover problem. In Section 3, we prove hardness of approximation for the problem of
computing the norm of a general class of Banach-space-valued functions, closely following [12]. In
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J OP B RI ËT, O DED R EGEV, AND R ISHI S AKET
Section 4, as a “warm up,” we prove Theorem 1.3 using the generic result of Section 3 and straightforward
applications of real and complex versions of the Berry-Esséen Theorem. Section 5 contains our main
technical contribution, which we use there to finish the proof of our main result (Theorem 1.2).
Acknowledgements. We thank Steve Heilman and Thomas Vidick for early discussions. We also thank
Gilles Pisier and anonymous referees for helpful comments on an earlier version of this manuscript.
2 Preliminaries
Notation and relevant Banach spaces. For a positive integer n we denote [n] = {1, . . . , n}. For a
graph G and vertices v, w ∈ V (G) we write v ∼ w to denote that v and w are adjacent. We write Pre∼v [·]
for the probability with respect to a uniformly distributed random edge incident with v. For a finite set U
we denote by Eu∈U [·] the expectation with respect to the uniform distribution over U. For a complex
number c ∈ C, we denote its real and imaginary parts by ℜ(c) and ℑ(c), respectively. All Banach spaces
are assumed to be finite-dimensional (so we can equivalently talk about normed spaces). Recall that for
Banach spaces X,Y the operator norm of a linear operator F : X → Y is given by
kFk = sup kF(x)kY .
x∈X: kxkX ≤1
For a real number p ≥ 1, the p-norm of a vector a ∈ Cn is given by
!1/p
n
kak` p = ∑ |ai | p .
i=1
As usual we implicitly endow Cn with the Euclidean norm kak`2 . For a finite set U endowed with the
uniform probability measure we denote by L p (U) the space of functions f : U → C with the norm
1/p
k f kL p (U) = Eu∈U | f (u)| p
.
More generally, for a Banach space X we denote by L p (U, X) the space of functions f : U → X with the
norm h i 1/p
k f kL p (U,X) = Eu∈U k f (u)kXp .
We will write L p (X) if U is not explicitly given and k f kL p instead of k f kL p (U,X) when there is no danger
of ambiguity. Note that L2 (U, Cn ) is a Hilbert space.
Smooth Label Cover. An instance of Smooth Label Cover is given by a quadruple (G, [n], [k], Σ) that
consists of a regular connected (undirected) graph G = (V, E), a label set [n] for some positive integer n,
and a collection
Σ = (πev , πew ) : e = (v, w) ∈ E
of pairs of maps both from [n] to [k] associated with the endpoints of the edges in E. Given
an assignment
A : V → [n], we say that an edge e = (v, w) ∈ E is satisfied if πev A(v) = πew A(w) .
The following hardness result for Smooth Label Cover, given in [12],2 is a slight variant of the original
2 For convenience, we make implicit some of the parameters in the statement of the theorem.
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construction due to Khot [19]. The theorem also describes the various structural properties, including
smoothness, that are satisfied by the hard instances.
Theorem 2.1. For any positive real numbers ζ , γ there exist positive integers n = n(ζ , γ), k = k(ζ , γ),
and t = t(ζ ), and a family of Smooth Label Cover instances (G, [n], [k], Σ) as above such that
• (Hardness) It is NP-hard to distinguish between the following two cases:
– (YES Case) There is an assignment that satisfies all edges.
– (NO Case) Every assignment satisfies less than a ζ -fraction of the edges.
• (Structural properties)
– (Smoothness) For every vertex v ∈ V and distinct i, j ∈ [n], we have
Pre∼v [πev (i) = πev ( j)] ≤ γ . (2.1)
−1 (i)| ≤ t; that is, at
– For every vertex v ∈ V , edge e ∈ E incident on v, and i ∈ [k], we have |πev
most t elements in [n] are mapped to the same element in [k].
– (Weak Expansion) For any δ > 0 and vertex subset V 0 ⊆ V such that |V 0 | = δ · |V |, the
number of edges between the vertices in V 0 is at least (δ 2 /2)|E|.
3 Hardness for general Banach-space valued operators
The following proposition shows hardness of approximation for the problem of computing the norm of a
linear map from Cn to any Banach space that allows for a “dictatorship test,” namely, a linear function
that maps the standard basis vectors to long vectors, and maps “spread” unit vectors to short vectors.
As stated, the proposition assumes the underlying field to be C; we note that the proposition holds with
exactly the same proof also in the case of the real field R.
Theorem 3.1. Let (Xn )n∈N be a family of finite-dimensional Banach spaces, and η and τ be positive
numbers such that η > τ. Suppose that for each positive integer n there exists a linear operator
f : Cn → Xn with the following properties:
• For any vector a ∈ Cn , we have k f (a)kXn ≤ kak`2 .
• For each standard basis vector ei , we have k f (ei )kXn ≥ η.
• For any ε > 0, there is a δ = δ (ε) > 0 such that k f (a)kXn > (τ + ε)kak`2 implies kak`4 > δ kak`2 .
Then, for any ε 0 > 0 there exists a positive integer n such that it is NP-hard to approximate the norm of
an explicitly given linear operator F : L2 → L1 (Xn ) to within a factor greater than (τ/η) + ε 0 .
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3.1 The hardness reduction
To set up the reduction, we begin by defining a linear operator F = Fζ ,γ for any choice of positive real
numbers ζ , γ. Afterwards we show that there is a choice of these parameters giving the desired result.
For positive real numbers ζ , γ, let n, k, and t be positive integers (depending on ζ , γ) and (G, [n], [k], Σ) a
Smooth Label Cover instance as in Theorem 2.1, where G = (V, E) is a regular graph. Note that ζ controls
the “satisfiability” of the instance in the NO case, that γ controls the “smoothness,” and that t depends on ζ
only. Endow the vertex set V with the uniform probability measure. To define F we consider a special
linear subspace H of the Hilbert space L2 (V, Cn ). It will be helpful to view a vector a ∈ L2 (V, Cn ) as an
assignment a = (av )v∈V of vectors av ∈ Cn to V . Let H ⊆ L2 (V, Cn ) be the subspace of vectors a = (av )v∈V
that satisfy for every e = (v, w) ∈ E and j ∈ [k] the homogeneous linear constraint
∑ av (i) = ∑ aw (i) ,
−1 −1
(3.1)
i∈πev ( j) i∈πew ( j)
where av (i) denotes the ith coordinate of the vector av . Notice that if an assignment A : V → [n] satisfies
the edge e= (v, w), then the standard basis vectors av = eA(v) and aw = eA(w) satisfy (3.1); indeed,
if πev A(v) = πew A(w) = j0 then both sides of (3.1) equal 1 if j = j0 and equal zero otherwise.
Now let η, τ and f be as in Theorem 3.1. We associate with the Smooth Label Cover instance
(G, [n], [k], Σ) from above the linear operator F : H → L1 (V, Xn ) given by
(F(a)) (v) = f (av ) . (3.2)
The operator F thus maps a Cn -valued assignment a = (av )v∈V satisfying (3.1) to an Xn -valued assignment
given by f (av ) for each v ∈ V . Theorem 3.1 follows from the following two lemmas, which we prove in
Sections 3.2 and 3.3, respectively.
Lemma 3.2 (Completeness). Suppose that there exists an assignment A : V → [n] that satisfies all the
edges in E. Then, kFk ≥ η.
Lemma 3.3 (Soundness). For any ε > 0 there exists a choice of ζ , γ > 0 such that if kFk > τ + 4ε then
there exists an assignment that satisfies at least a ζ -fraction of the edges of G.
Proof of Theorem 3.1. Let ε > 0 be arbitrary, let ζ , γ be as in Lemma 3.3, and let n = n(ζ , γ) and k =
k(ζ , γ) be as in Theorem 2.1. We use the reduction described above, which maps a Smooth Label Cover
instance (G, [n], [k], Σ) to the linear operator F : L2 → L1 (Xn ) specified in (3.2). By Lemma 3.2, YES
instances are mapped to F satisfying kFk ≥ η, whereas by Lemma 3.3, NO instances are mapped to F
satisfying kFk < τ + 4ε. We therefore obtain hardness of approximation to within a factor (τ + 4ε)/η.
Since ε is arbitrary, we are done.
3.2 Completeness
Here we prove Lemma 3.2.
Proof of Lemma 3.2. Let A : V → [n] be an assignment that satisfies all the edges. Consider the vector
a ∈ L2 (V, Cn ) where av = eA(v) and notice that kakL2 = 1. Since A satisfies all edges, a satisfies the
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constraint (3.1) for every e ∈ E and j ∈ [n], and thus a lies in the domain H of F. Moreover, by the
second property of f given in Theorem 3.1,
kF(a)kL1 (V,Xn ) = Ev∈V k f (av )kXn ≥ η.
Hence, kFk ≥ η.
3.3 Soundness
Here we prove Lemma 3.3 and show that among the family of operators F = Fζ ,γ as in (3.2), for any ε > 0
there is a choice of ζ , γ > 0 such that if kFk > τ + 4ε, then there exists an assignment satisfying a ζ -
fraction of the edges in the Smooth Label Cover instance associated with F. To begin, assume that
kFk > τ + 4ε for some ε > 0. Let b ∈ H be a vector such that kbkL2 = 1 and
Ev∈V k f (bv )k = kF(b)kL1 ≥ τ + 4ε . (3.3)
The weak expansion property in Theorem 2.1 implies that it suffices to find a “good” assignment for a
large subset of the vertices, as any large set of vertices will induce a large set of edges. For δ = δ (ε) as
in Theorem 3.1, we will consider set of vertices
V0 = {v ∈ V | kbv k`4 > δ ε and kbv k`2 ≤ 1/ε} . (3.4)
The following lemma shows that V0 contains a significant fraction of vertices.
Lemma 3.4. For V0 ⊆ V defined as in (3.4), we have |V0 | ≥ ε 2 |V |.
Proof. Define the sets
V1 = {v ∈ V | kbv k`4 ≤ δ ε and kbv k`2 < ε} ,
V2 = {v ∈ V | kbv k`4 ≤ δ ε and kbv k`2 ≥ ε} ,
V3 = {v ∈ V | kbv k`2 > 1/ε} .
From (3.3), we have
∑ k f (bv )kX + ∑ k f (bv )kX + ∑ k f (bv )kX + ∑ k f (bv )kX ≥ (τ + 4ε)|V | .
n n n n (3.5)
v∈V0 v∈V1 v∈V2 v∈V3
We bound the four sums on the left-hand side of (3.5) individually. Since (by the first item in Theorem 3.1)
we have k f (bv )kXn ≤ kbv k`2 , and since kbv k`2 ≤ 1/ε for every v ∈ V0 , the first sum in (3.5) can be bounded
by
∑ k f (bv )kXn ≤ |V0 |/ε . (3.6)
v∈V0
Similarly, using the definition of V1 the second sum in (3.5) is at most ε|V |. Next, from the third property
of f in Theorem 3.1, for each v ∈ V2 , we have k f (bv )kXn ≤ (τ + ε)kbv k`2 . Therefore, the third sum
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in (3.5) is bounded as
∑ k f (bv )kX ≤ (τ + ε) ∑ kbv k`
n 2
v∈V2 v∈V2
1
21
≤ (τ + ε)|V2 | 2
∑ kbv k2`2 (by Cauchy-Schwartz)
v∈V2
1
12
≤ (τ + ε)|V | 2 ∑ kbv k2`2
v∈V
= (τ + ε)|V | , (3.7)
where the last inequality uses kbkL2 = 1. Finally, the fourth sum in (3.5) is bounded by
∑ k f (bv )kX ≤ ∑ kbv k`
n 2
(by the property of f )
v∈V3 v∈V3
< ∑ εkbv k2`2
v∈V3
≤ ε ∑ kbv k2`2
v∈V
= ε|V |kbk2L2 = ε|V | . (3.8)
Combining the above with Equation (3.5) yields, |V0 |/ε ≥ ε|V |, which proves Lemma 3.4.
Lemma 3.4 and the weak expansion property implies that the set E(V0 ) of edges induced by V0 has
cardinality
|E(V0 )| ≥ (ε 4 /2)|E| . (3.9)
We set out to show that there exists an assignment to the vertices in V0 that satisfies a significant
fraction of edges in E(V0 ). Roughly speaking, we do this by randomly assigning each v ∈ V0 one of the
coordinates of the vector bv at which it has large magnitude. (Assigning the largest coordinate may not
work.) The following simple proposition shows that those vectors indeed have large coordinates.
Proposition 3.5. Let β = β (ε) be given by β = δ 2 ε 3 . Then, for each v ∈ V0 , we have kbv k`∞ ≥ β .
Proof. For every v ∈ V0 , we have
δ 4 ε 4 ≤ kbv k4`4 ≤ kbv k2`∞ kbv k2`2 ≤ kbv k2`∞ /ε 2 , (3.10)
giving the claim.
For the to-be-determined value of ζ let t = t(ζ ) be as in Theorem 2.1 and for each v ∈ V0 let
n βo n βo
Av1 = i ∈ [n] |bv (i)| ≥ and Av2 = i ∈ [n] |bv (i)| ≥ .
4 4t
By Proposition 3.5 these sets are nonempty and clearly Av1 ⊆ Av2 . Moreover, since kbv k`2 ≤ 1/ε, we
have,
16 16t 2
|Av1 | ≤ 2 2 and |Av2 | ≤ 2 2 . (3.11)
ε β ε β
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Now consider a random assignment A : V0 → [n] that independently assigns each vertex v ∈ V0 a
uniformly random label from Av1 and assigns the remaining vertices in V some fixed arbitrary label. The
following lemma shows that on average, this assignment satisfies a significant fraction of edges.
Lemma 3.6. There exists a γ > 0 depending only on ε and ζ such that for some absolute constant c > 0
the expected fraction of edges in E satisfied by the random assignment A given above is at least cε 8 β 4 .
Setting γ appropriately as in the above lemma and ζ = cε 8 δ 4 then gives Lemma 3.3; indeed, notice
that then ζ , and therefore also γ, depend on ε alone.
The remainder of this section is devoted to the proof of Lemma 3.6. Let E 0 ⊆ E(V0 ) be the subset
of edges e = (v, w) whose projections πev and πew are injective on the subsets Av2 and Aw2 respectively.
Formally,
E 0 = e = (v, w) ∈ E(V0 ) |πev (Av2 )| = |Av2 |, and |πew (Aw2 )| = |Aw2 | .
(3.12)
We set the parameter γ according to the following proposition which shows a lower bound on |E 0 | using
the smoothness property. Recall that t is a function of ζ only.
Proposition 3.7. There exists an absolute constant c0 > 0 such that for any γ ≤ c0 ε 8 β 4 /t 4 , the set E 0 has
cardinality |E 0 | ≥ (ε 4 /4)|E| .
Proof. Consider any vertex v ∈ V0 . By the smoothness property of Theorem 2.1 and a union bound over
all distinct pairs i, j ∈ A2v , the fraction of edges e ∈ E incident on v that do not satisfy
|πev (Av2 )| = |Av2 | (3.13)
is at most
γ |Av2 |2 1 ε 8β 4 162 · t 4 ε4
≤ = ,
2 2 210 · t 4 ε 4β 4 8
via an appropriate setting of c0 . Therefore, the number of edges in E that are incident on some v ∈ V0 and
do not satisfy (3.13) is at most
ε4 ε4 ε4
∑ 8 deg(v) ≤ 8 ∑ deg(v) ≤ 4 |E| .
v∈V0 v∈V
Thus,
|E 0 | ≥ |E(V0 )| − (ε 4 /4)|E| ≥ (ε 4 /4)|E| ,
by Equation (3.9).
The following proposition shows that for an edge e = (v, w) ∈ E 0 , the label sets Av1 and Aw1 intersect
under projections given by e.
Proposition 3.8. For every edge e = (v, w) ∈ E 0 , we have πev (Av1 ) ∩ πew (Aw1 ) 6= 0.
/
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Proof. From Proposition 3.5, let i∗ ∈ [n] be such that |bv (i∗ )| ≥ β . Note that i∗ ∈ Av1 . Let j∗ = πev (i∗ ).
Clearly it suffices to show that there exists an i0 ∈ Aw1 such that πew (i0 ) = j∗ , as this implies that j∗ ∈
πev (Av1 ) ∩ πew (Aw1 ).
Recall that since b ∈ H, the vector b satisfies the constraint (3.1), in particular,
∑ bv (i) = ∑ bw (i) . (3.14)
−1 ∗ −1 ∗
i∈πev (j ) i∈πew (j )
We show that because i∗ ∈ Av1 , the left-hand side must be large. Therefore the right hand side is also large,
from which we conclude that there must exist a coordinate i0 ∈ πew −1 ( j ∗ ) such that |b (i0 )| is large, and so
w
0 w
i ∈ A1 .
Recall from the second structural property in Theorem 2.1 that |πev −1 ( j ∗ )| ≤ t. Moreover, since π
ev
acts injectively on the set A2 and since i ∈ A2 , no index i 6= i such that πev (i) = πev (i∗ ) can belong to Av2 .
v ∗ v ∗
Hence, by the triangle inequality, the left-hand side of (3.14) is at least
∗ β 3β
|bv (i )| − ∑ |bv (i)| ≥ β − t · = . (3.15)
i∈π −1 ( j∗ )
4t 4
ev
i6=i∗
Combining (3.14), (3.15), and the triangle inequality lets us bound the right-hand side of (3.14) by
3β
≤ ∑ bw (i)
4 −1 ∗
i∈πew (j )
≤ ∑ |bw (i)| + ∑ |bw (i)|
−1 ∗ −1 ∗
i∈πew ( j )∩Aw
2 i∈πew ( j )rAw
2
β
≤ ∑ |bw (i)| + t , (3.16)
−1 ∗ 4t
i∈πew ( j )∩Aw
2
where the last inequality uses the same facts as above. Since πew acts injectively on Aw2 , there is at most
−1 ( j ∗ ) that also belongs to Aw , meaning that the sum in (3.16) consists of at most one term.
one index i ∈ πew 2
We see that sum must is at least β /2 and in particular, there is an i0 ∈ πew −1 ( j ∗ ) such that |b (i0 )| ≥ β /2.
w
We conclude that i0 ∈ Aw1 and πew (i0 ) = j∗ = πev (i∗ ), proving the claim.
Proof of Lemma 3.6. By Proposition 3.8 and Equation (3.11) any edge e = (v, w) ∈ E 0 is satisfied by the
assignment A with probability at least 1/(|Av1 ||Aw1 |) ≥ (ε 4 β 4 )/256. Since by Proposition 3.7, we have
|E 0 | ≥ (ε 4 /4)|E|, the expected fraction of satisfied edges is at least ε 8 β 4 /1024.
4 The commutative case
Recall that the commutative Little Grothendieck problem asks for the norm of a linear operator F : L2 → L1 .
In this section we use Theorem 3.1 to prove Theorem 1.3, the tight hardness result for this problem. We
first consider the real case of Theorem 1.3, and then the complex case in Section 4.2.
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4.1 The real case
The real case of Theorem 1.3 follows easily by combining Theorem 3.1 with the following simple lemma.
Lemma 4.1. For every positive integer n there exists a map f : Rn → L1 with the following properties:
• For any vector a ∈ Rn , we have k f (a)kL1 ≤ kak`2 .
p
• For each standard basis vector ei , we have k f (ei )kL1 = 1. If k f (a)kL1 > ( 2/π + ε)kak`2
then kak`4 > (ε/K)kak`2 , where K < ∞ is a universal constant.
This shows that there
p is an L1 -valued function f that satisfies the conditions of the real variant of
Theorem 3.1 for τ = 2/π, η = 1 and δ (ε) = (ε/K). p Hence, it is NP-hard to approximate the norm of a
linear operator F : L2 → L1 (L1 ) over R to a factor 2/π + ε for any ε > 0. The real case of Theorem 1.3
then follows from the fact that L1 (L1 ) is isometrically isomorphic to L1 (i. e., there is a bijective isometry
between the two).
The proof of Lemma 4.1 uses the following version of the Berry–Esséen Theorem (see for example [30,
Chapter 5.2, Theorem 5.16]).
Theorem 4.2 (Berry–Esséen Theorem). There exists a universal constant K < ∞ such that the following
holds. Let n be a positive integer and let Z1 , . . . , Zn be independent centered {−1, 1}-valued random
variables. Then, for any ε > 0 and for any vector a ∈ Rn such that kak`∞ ≤ εkak`2 , we have
r
n
h i 2
E ∑ Zi ai − π
kak`2 ≤ Kεkak`2 .
i=1
Proof of Lemma 4.1. Endow {−1, 1}n with the uniform probability measure and define the function
f : Rn → L1 ({−1, 1}n ) by
n
f (a) (Z1 , . . . , Zn ) = ∑ ai Zi .
i=1
The first property follows since
!1/2
h n 2i
k f (a)kL1 ≤ k f (a)kL2 = E ∑ ai Zi = kak`2 .
i=1
The second property is trivial. The third property follows from Theorem 4.2. Indeed, the theorem implies
that if for some ε > 0, we have
r !
h n i 2
k f (a)kL1 = E ∑ ai Zi > + ε kak`2 ,
i=1 π
then kak`∞ > (ε/K)kak`2 . Since kak`4 ≥ kak`∞ the last property follows.
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4.2 The complex case
A similar argument to the one above shows the complex case of Theorem 1.3. This follows from the
following complex analogue of Lemma 4.1.
Lemma 4.3. For every positive integer n there exists a map f : Cn → L1 with the following properties:
• For any vector a ∈ Cn , we have k f (a)kL1 ≤ kak`2 .
p
• For each standard basis vector ei , we have k f (ei )kL1 = 1. If k f (a)kL1 > ( π/4 + ε)kak`2
then kak`4 > (ε 2 /K)kak`2 , where K < ∞ is a universal constant.
p This shows that there is an L1 -valued function f that satisfies the conditions of Theorem 3.1 for τ =
p Hence, it is NP-hard to approximate the norm of a linear operator F :
2
π/4, η = 1 and δ (ε) = (ε /K).
L2 → L1 (L1 ) over C to a factor π/4 + ε for any ε > 0. The complex case of Theorem 1.3 then follows
as before from the fact that L1 (L1 ) is isometrically isomorphic to L1 .
The proof of Lemma 4.3 is based on the following complex analogue the Berry–Esséen Theo-
rem. Since we could not find this precise formulation in the literature we include a proof below for
completeness.
Lemma 4.4 (Complex Berry–Esséen Theorem). There exists a universal constant K < ∞ such that the
following holds. Let Z1 , . . . , Zn be independent uniformly distributed random variables over {1, i, −1, −i}.
Then, for any vector a ∈ Cn such that kak`∞ ≤ εkak`2 , we have
h n i rπ √
E ∑ Z ja j − kak`2 ≤ K εkak`2 .
j=1 4
The proof is based on the following multi-dimensional version of the Berry–Esséen theorem due to
Bentkus [4, Theorem 1.1].
Theorem 4.5 (Bentkus). Let X1 , . . . , Xn be independent Rd -valued random variables such that E[X j ] = 0
for each j ∈ [n]. Let S = X1 + · · · + Xn and assume that the covariance matrix of S equals 1d . Let g ∼
N(0, 1d ) be a standard Gaussian vector in Rd with the same covariance matrix as S. Then, for any
measurable convex set A ⊆ Rd , we have
n
Pr[S ∈ A] − Pr[g ∈ A] ≤ c(d) ∑ E kX j k3`2 ,
j=1
where c(d) = O(d 1/4 ).
We also use the following standard tail bounds.
Lemma 4.6 (Gaussian tail bound [5]). Let g ∼ N(0, 1d ) be a standard Gaussian vector in Rd . Then, for
any t > 0, we have √ 2
Pr kgk`2 − d > t ≤ 2e−t /2 .
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Lemma 4.7 (Hoeffding’s inequality [17]). Let X1 , . . . , Xn be independent real-valued random variables
such that for each i ∈ [n], Xi ∈ [ai , bi ] for some ai < bi . Let S = X1 + · · · + Xn . Then, for any t > 0,
2 n 2
Pr |S − E[S]| > t ≤ 2e−2t / ∑i=1 (bi −ai ) .
Proof of Lemma 4.4. Let a ∈ Cn be some vector. By homogeneity we may assume that kak`2 = 1.
√
Set ε = kak`∞ . For each j ∈ [n] define the random vector X j ∈ R2 by X j = 2[ℜ(Z j a j ), ℑ(Z j a j )]T and
√ √ √
note that kX j k`2 = 2|a j | ≤ 2ε. Let S = X1 + · · · + Xn , and let T ≥ 8 be some number to be set later.
We have Z ∞ Z T Z ∞
E[kSk`2 ] = Pr[kSk`2 > t] dt = Pr[kSk`2 > t] dt + Pr[kSk`2 > t] dt . (4.1)
0 0 T
We now analyze each integral separately. Notice that E[X j ] = 0, and E[X j X jT ] = |a j |2 12 . It follows
that the covariance matrix of S equals 12 . If we let g ∼ N(0, 12 ) be a standard Gaussian vector in R2 ,
then it follows from Theorem 4.5 (for d = 2) that for any t > 0, we have
n
Pr kSk`2 > t − Pr kgk`2 > t ≤ c ∑ E kX j k3`2
j=1
√ n
≤ 2cε ∑ E kX j k2`2
j=1
√
≤ 2 2cε . (4.2)
Therefore, the first integral in (4.1) satisfies
Z T Z T
Pr kSk`2 > t dt − Pr kgk`2 > t dt
0 0
Z T
≤ Pr kSk`2 > t − Pr kgk`2 > t dt
0
√
≤ 2 2cεT . (4.3)
Since kgk`2 is distributed according to a χ2 distribution, we have
Z ∞ p
Pr kgk`2 > t dt = E[kgk`2 ] = π/2 . (4.4)
0
Moreover, it follows from Lemma 4.6 (for d = 2) and our assumption on T that
Z ∞ √ √ Z ∞ √ 2
2e−(t− 2) /2 dt
Pr kgk`2 − 2 > t − 2 dt ≤
T
ZT∞
2
≤ 2te−t /8 dt
T
2
= 8e−T /8 , (4.5)
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√ √
where we used that (t − 2)2 ≥ t 2 /4 for t ≥ 8. Combining (4.3), (4.4), and (4.5), we obtain that the
first integral in (4.1) satisfies
Z T √ 2
Pr kSk`2 > t dt − π/2 ≤ 2 2cεT + 8e−T /8 .
p
(4.6)
0
We now bound the second integral in (4.1), √ which is clearly nonnegative. The first coordinate S1
is a sum
√ of independent random variables, 2ℜ(Z j a j ), which are centered and have magnitude at
most 2|a j |. Similarly, the same holds for S2 . Lemma 4.7 therefore gives,
Z ∞ Z ∞ √ Z ∞ √
Pr[kSk`2 > t] dt ≤ Pr[|S1 | > t/ 2] dt + Pr[|S2 | > t/ 2] dt
T T T
Z ∞
−t 2 /8
≤4 e dt
ZT∞
2
≤4 te−t /8 dt
T
−T 2 /8
= 16e , (4.7)
where in the last inequality
p we used the assumption T ≥ 1.
Now set T = 8/ε. Combining (4.1), (4.6), and (4.7), we get
h n i rπ 1 √ √
r
1 π 2
≤ √ 2 2cεT + 24e−T /8 ≤ K ε .
E ∑ Z ja j − = √ E kSk`2 −
j=1 4 2 4 2
The proof of Lemma 4.3 is nearly identical to that of Lemma 4.1, now based on Lemma 4.4
n n
and the function f : C → L1 ({1, i, −1, −i} ) given by f (a) (Z1 , . . . , Zn ) = a1 Z1 + · · · + an Zn , where
{1, i, −1, −i}n is endowed with the uniform probability measure.
5 The non-commutative case
In this section we complete the proof of our main theorem (Theorem 1.2). The following lemma gives the
linear matrix-valued map f mentioned in the introduction.
Lemma 5.1. Let n be a positive integer and let d = 22n+dn/2e . Then, there exists a linear operator
f : Cn → Cd×d such that for any vector a ∈ Cn , we have
s
kak2`2 + kak2`4
k f (a)kS1 ≤ .
2
√
In particular, k f (a)kS1 ≤ (kak`2 + kak`4 )/ 2. Moreover, for each basis vector ei we have k f (ei )kS1 = 1.
Theorem 1.2 now follows easily by combining the above lemma with Theorem 3.1.√Indeed, Lemma 5.1
shows that the conditions of Theorem 3.1 hold for τ = 2−1/2 , η = 1 and δ (ε) = 2ε. It is√therefore
NP-hard to approximate the norm of a linear operator F : L2 → L1 (S1 ) to within a factor 1/ 2 + ε for
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any ε > 0. This implies the theorem because L1 (S1 ) embeds isometrically into S1 . To see the last fact, we
use the map that takes a matrix-valued function g on a finite measure space U to a block diagonal matrix
with blocks proportional to g(u) for u ∈ U and use the fact that the trace norm of a block diagonal matrix
is the average trace norm of the blocks.
The rest of this section is devoted to the proof of Lemma 5.1. For a complex vector a ∈ Cn define
q
2
Λ(a) = kℜ(a)k2`2 kℑ(a)k2`2 − ℜ(a), ℑ(a) . (5.1)
Note that this value is the area of the parallelogram in Rn generated by the vectors ℜ(a) and ℑ(a).
0 0
Lemma 5.2. Let n be a positive integer and let d 0 = 2dn/2e . Then, there exists a operator C : Cn → Cd ×d
such that for any vector a ∈ Cn , we have
1 1
q q
2
kC(a)kS1 = kak`2 + 2Λ(a) + kak2`2 − 2Λ(a) . (5.2)
2 2
Though we will not use it here, let us point out that the map C becomes an isometric embedding if we
restrict it to Rn , since Λ(a) = 0 for real vectors.
Proof. We begin by defining a set of pairwise anti-commuting matrices as follows. The Pauli matrices
are the four Hermitian matrices
1 0 0 1 0 −i 1 0
I= , X= , Y= , Z= .
0 1 1 0 i 0 0 −1
0 0
Using these we define 2dn/2e matrices in Cd ×d by
C2 j−1 = Z ⊗ · · · ⊗ Z ⊗ X ⊗ I ⊗ · · · ⊗ I ,
| {z } | {z }
j − 1 times dn/2e − j times
C2 j = Z ⊗ · · · ⊗ Z ⊗Y ⊗ I ⊗ · · · ⊗ I ,
| {z } | {z }
j − 1 times dn/2e − j times
for each j ∈ [dn/2e]. It is easy to verify that these matrices have trace zero, that they are Hermitian, unitary,
and that they pairwise anti-commute. In particular, they satisfy C2j = I. For a vector a ∈ Cn we define the
map C by C(a) = a1C1 + · · · + anCn . Note that for a real vector x ∈ Rn , the matrix C(x) is Hermitian and
that it satisfies C(x)2 = kxk2`2 I. If a real vector z ∈ Rn is orthogonal to x then by expanding the definitions
of the matrices C(x) and C(z) and using the above properties we find that they anti-commute:
C(x)C(z) = hx, ziI + ∑ x j zkC jCk
j6=k
= 0 − ∑ x j zkCkC j
j6=k
= −C(z)C(x) .
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This shows that the matrix C(x)C(z) is skew-Hermitian, which implies that it has purely imaginary
∗
eigenvalues. Since this matrix has trace zero and satisfies C(x)C(z) C(x)C(z) = kxk2`2 kzk2`2 I, half the
eigenvalues equal ikxk`2 kzk`2 and the other half equal −ikxk`2 kzk`2 .
We show that C satisfies (5.2). Let x = ℜ(a) and y = ℑ(a) so that C(a) = C(x) + iC(y). Write y =
yk + y⊥ where yk is parallel to x and y⊥ is orthogonal to x. Then,
C(a)C(a)∗ = C(x) + iC(y) C(x) − iC(y)
= kak2`2 I − i C(x)C(y) −C(y)C(x)
= kak2`2 I − 2iC(x)C(y⊥ ) ,
where in the last line we used the fact that C(yk ) commutes with C(x) while C(y⊥ ) anti-commutes
with C(x). Using what we deduced above for the matrix C(x)C(y⊥ ) we see that half of the eigenvalues
of C(a)C(a)∗ equal kak2`2 + 2kxk`2 ky⊥ k`2 and the other half equal kak2`2 − 2kxk`2 ky⊥ k`2 . Hence,
1 1
q q
kC(a)kS1 = kak2`2 + 2kxk`2 ky⊥ k`2 + kak2`2 − 2kxk`2 ky⊥ k`2 .
2 2
The claim now follows because kxk`2 ky⊥ k`2 is precisely the area of the parallelogram generated by the
vectors x and y.
We denote the entry-wise product of two vectors a, b ∈ Cn by a ◦ b = (a1 b1 , . . . , an bn ).
Proposition 5.3. Let ω be a vector chosen uniformly from {1, i, −1, −i}n . Then, for any a ∈ Cn , we have
4Eω Λ(a ◦ ω)2 = kak4`2 − kak4`4 .
Proof. Fix a vector a ∈ Cn . Define the random vectors xω = ℜ(a ◦ ω) and yω = ℑ(a ◦ ω). Then
Λ(a ◦ ω)2 = kxω k2`2 kyω k2`2 − hxω , yω i2 . For each j ∈ [n] we factor a j = α j eiφ j and ω j = eiψ j , where
α j ∈ R+ and φ j , ψ j ∈ [0, 2π]. Note that ψ1 , . . . , ψn are independent uniformly distributed random phases
in {0, π/2, π, 3π/2}. Then,
n
kxω k2`2 = ∑ α 2j cos2 (φ j + ψ j )
j=1
n
kyω k2`2 = ∑ α 2j sin2 (φ j + ψ j )
j=1
n
hxω , yω i = ∑ α 2j cos(φ j + ψ j ) sin(φ j + ψ j ) .
j=1
With this it is easy to verify that
kxω k2`2 kyω k2`2 − hxω , yω i2 = ∑ α 2j αk2 cos2 (φ j + ψ j ) sin2 (φk + ψk )−
j6=k
∑ α 2j αk2 cos(φ j + ψ j ) sin(φ j + ψ j ) cos(φk + ψk ) sin(φk + ψk ) . (5.3)
j6=k
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By independence of ψ j and ψk when j 6= k and the elementary identities E[cos2 (φ j + ψk )] = 1/2,
E[sin2 (φ j + ψk )] = 1/2 and E[cos(φ j + ψ j ) sin(φ j + ψ j )] = 0, the expectation of (5.3) equals
1 1
Eω Λ(a ◦ ω)2 = ∑ a2j a2k = kak4`2 − kak4`4 .
4 j6=k 4
We remark that in the above proof, it suffices if ω ∈ {1, i, −1, −i}n is chosen from a pairwise
independent family. Using this in the proof below, allows one to prove Lemma 5.1 with a smaller
parameter d.
Proof of Lemma 5.1. Let C be the map given by Lemma 5.2. Define the map
M
f (a) = C(a ◦ w)
ω
where ω ranges over over {1, i, −1, −i}n . By convexity of the square function, Jensen’s inequality, and
the fact that ka ◦ ωk`2 = kak`2 , we have
2
k f (a)k2S1 = Eω kC(a ◦ ω)kS1
q 2
1 1
q
Lemma 5.2 2 2
= Eω ka ◦ ωk`2 + 2Λ(a ◦ ω) + ka ◦ ωk`2 − 2Λ(a ◦ ω)
2 2
" q 2 #
1 1
q
2 2
≤ Eω kak`2 + 2Λ(a ◦ ω) + kak`2 − 2Λ(a ◦ ω)
2 2
kak2`2 1
q
= + Eω kak4`2 − 4Λ(a ◦ ω)2 . (5.4)
2 2
Concavity of the square-root function, Jensen’s inequality and Proposition 5.3 gives that the expectation
in (5.4) is at most
1/2 4 1/2
kak4`2 − 4E Λ(a ◦ ω)2 = kak`2 − kak4`2 + kak4`4 = kak2`4 .
Hence,
s
kak2`2 + kak2`4
k f (a)kS1 ≤ .
2
For the second claim observe that for any standard basis vector e j and ω ∈ {1, i, −1, −i}n , the vector e j ◦ω
is either purely real or purely imaginary. This implies Λ(e j ◦ ω) = 0. Hence, by Lemma 5.2,
q
1
q
k f (ei )kS1 = Eω ke j ◦ ωk2`2 + 2Λ(e j ◦ ω) + ke j ◦ ωk2`2 − 2Λ(e j ◦ ω) = 1 .
2
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J OP B RI ËT, O DED R EGEV, AND R ISHI S AKET
5.1 The real and Hermitian variants
We end this section by showing that our hardness result of Theorem 1.2 also holds for two variants of
the Little NCG, the real variant and the Hermitian variant. Both variants were introduced (in the context
of the “big” NCG) in [28], partly for the purpose of using them in applications. The real variant asks
for the operator norm of a linear map F from Rn to a space Rd×d endowed with the Schatten-1 norm; in
the Hermitian variant, the linear map is from Rn to the space Hd ⊆ Cd×d of Hermitian matrices, again
endowed with the Schatten-1 norm. In both cases the operator norm is given by kFk = supa kF(a)kS1
with the supremum over real unit vectors a. Both the real and Hermitian variants follow directly by
combining the lemma shown below and the real version of Theorem 3.1. Let us denote by Sd×d ⊆ Rd×d
the space of real symmetric matrices.
Lemma 5.4. Let n be a positive integer and let d be as in Lemma 5.1. Then, there exists a linear operator
f : Rn → S4d×4d satisfying the conditions stated in Lemma 5.1 (with a ∈ Rn ).
The lemma follows by applying the map ρ of the elementary claim below to the restriction of the
operator f of Lemma 5.1 to Rn .
Claim 5.5. For every positive integer d there exists a map ρ : Cd×d → S4d×4d such that for any matrix
A ∈ Cd×d , we have kρ(A)kS1 = kAkS1 . Moreover, ρ is linear over the real numbers, that is, for any α ∈ R
and A, B ∈ Cd×d , we have ρ(αA) = αρ(A) and ρ(A + B) = ρ(A) + ρ(B).
Proof. The proof follows by combining two standard transformations taking complex matrices to Hermi-
d×d be a matrix with singular values σ ≥ · · · ≥ σ .
tian matrices and real matrices, respectively.
0Let
A
A ∈ C 1 d
The first transformation is given by A 7→ A∗ 0 . By [18, Theorem 7.3.3], the last matrix has eigenval-
ues σ1 ≥ · · · ≥ σd ≥ −σd ≥ · · · ≥ −σ1 . Notice that this transformation is linear over the reals since
the adjoint is such. Let B ∈ Cd×d be a Hermitian matrix with eigenvalues λ1 ≥ · · · ≥ λd . The second
transformation is given by
ℜ(B) ℑ(B)
B 7→ .
−ℑ(B) ℜ(B)
Then the last matrix is symmetric and by [18, 1.30.P20 (g), p. 71], that matrix has the same eigenvalues
as B but with doubled multiplicities, that is, the matrix has eigenvalues λ1 ≥ λ1 ≥ · · · ≥ λd ≥ λd . Notice
that this transformation is also linear over the reals. Let ρ be the composition of these maps. Then
the matrix ρ(A) has the same singular values as A but with quadrupled multiplicities, which implies
that kρ(A)kS1 = kAkS1 , and ρ is linear over the reals.
6 Little versus big Grothendieck theorem
For completeness, we include here the well-known relation between the little and big Grothendieck
problems. We focus on the non-commutative case; the commutative case is similar and can be found in,
e. g., [32, Section 5]. This discussion clarifies how to derive Theorem 1.1 from Theorem 1.2.
Consider a linear map F : Cn → S1d . A standard and easy-to-prove fact is that for two finite-dimensional
Banach spaces X,Y , the operator norm of a linear map G : X → Y equals the norm of its adjoint G∗ :
Y ∗ → X ∗ . As a result, kFk = kF∗ k. Notice that since Hilbert space is self-dual and the dual of S1 is the
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T IGHT H ARDNESS OF THE N ON -C OMMUTATIVE G ROTHENDIECK P ROBLEM
space S∞ of matrices endowed with the Schatten-∞ norm (i. e., the maximum singular value), we have
that F∗ : S∞
d → Cn . In particular,
kF∗ k = supkF∗ (A)k2 ,
where the supremum is taken over all A of Schatten-∞ norm at most 1. Equivalently, since any matrix
with Schatten-∞ norm at most 1 lies in the convex hull of the set of unitary matrices, we could take the
supremum over all unitary matrices A. Next, recall that in the NCG problem we are given a bilinear
form T : Cd×d × Cd×d → C, and asked to compute OPT(T ) = supA,B T (A, B) , where the supremum
ranges over unitary matrices. Define the bilinear form T (A, B) = hF∗ (A), F∗ (B)i. By Cauchy-Schwarz,
OPT(T ) = supkF∗ (A)k22 = kF∗ k2 = kFk2 ,
A
where the supremum is over all unitary A, showing that the Little NCG is a special case of the “big” NCG.
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J OP B RI ËT, O DED R EGEV, AND R ISHI S AKET
AUTHORS
Jop Briët
Assistant professor
CWI, Amsterdam
The Netherlands
j.briet cwi nl
http://homepages.cwi.nl/~jop/
Oded Regev
Professor
Courant Institute of Mathematical Sciences
New York University
New York, N.Y.
regev cims nyu edu
http://www.cims.nyu.edu/~regev/
Rishi Saket
Researcher
IBM Research
Bangalore, India
rissaket in ibm com
http://researcher.ibm.com/researcher/view.php?person=in-rissaket
ABOUT THE AUTHORS
J OP B RIËT graduated from CWI in 2011; his advisor was Harry Buhrman. He enjoys
problems at the intersection of theoretical computer science and pure mathematics. After
a stint as a yogi during a postdoc at the Courant Institute in New York City, he returned
to rock climbing in the flattest country in the world.
O DED R EGEV graduated from Tel Aviv University in 2001 under the supervision of Yossi
Azar. He spent two years as a postdoc at the Institute for Advanced Study, Princeton,
and one year at the University of California, Berkeley. He is currently with the Courant
Institute of Mathematical Sciences, and enjoys life in NYC. His research interests include
computational and mathematical aspects of lattices, quantum computation, and other
topics in theoretical computer science.
R ISHI S AKET completed his Ph. D. from Georgia Tech in 2009 under the supervision of
Subhash Khot. After post-doctoral stints at CMU, Princeton University, and IBM T. J.
Watson, he joined IBM Research, Bangalore, India in 2013 where he is currently a
researcher. His interests are in hardness of approximation, approximation algorithms,
optimization, learning theory, and related areas of theoretical and applied computer
sciences.
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