#### Non-optimality of the Greedy Algorithm for Subspace Orderings in the Method of Alternating Projections

Non-optimality of the Greedy Algorithm for Subspace Orderings in the Method of Alternating Pro jections
O. Darwin
A. Jha
S. Roy
D. Seifert
R. Steele
L. Stigant
The method of alternating projections involves projecting an element of a Hilbert space cyclically onto a collection of closed subspaces. It is known that the resulting sequence always converges in norm and that one can obtain estimates for the rate of convergence in terms of quantities describing the geometric relationship between the subspaces in question, namely their pairwise Friedrichs numbers. We consider the question of how best to order a given collection of subspaces so as to obtain the best estimate on the rate of convergence. We prove, by relating the ordering problem to a variant of the famous Travelling Salesman Problem, that correctness of a natural form of the Greedy Algorithm would imply that P = NP, before presenting a simple example which shows that, contrary to a claim made in the influential paper (Kayalar and Weinert in Math Control Signals Syst 1(1):43-59, 1988), the result of the Greedy Algorithm is not in general optimal. We go on to establish sharp estimates on the degree to which the result of the Greedy Algorithm can differ from the optimal result. Underlying all of these results is a construction which shows that for any matrix whose entries satisfy certain natural assumptions it is possible to construct a Hilbert space and a collection of closed subspaces such that the pairwise Friedrichs numbers between the subspaces are given precisely by the entries of that matrix. Mathematics Subject Classification. 47J25, 65F10 (68Q25).
Method of alternating projections; orderings; subspaces; rate of convergence; travelling salesman problem; complexity
1. Introduction
Let X be a real or complex Hilbert space, N ≥ 2 an integer, and suppose
that M1, . . . , MN are closed subspaces of X. Furthermore let Pk denote the
orthogonal projection onto Mk, 1 ≤ k ≤ N , and let PM denote the orthogonal
projection onto the intersection M = M1 ∩ . . . ∩ MN . If we let T = PN · · · P1
then it follows from a classical theorem due to Halperin [8] that
T nx − PM x
→ 0,
n → ∞,
(1.1)
for all x ∈ X. It follows easily that, for any x ∈ X, the sequence in X
obtained by starting at X and then projecting cyclically onto the N subspaces
M1, . . . , MN must converge to the point PM x, which is the point in M closest
to the starting vector x. This procedure is known as the method of alternating
projections and has many applications, for instance to the iterative solution of
large linear systems but also in the theory of partial differential equations and
in image restoration; see [3] for a survey.
In view of these applications it is important to understand the rate at
which the convergence in (1.1) takes place; see for instance [1, 2, 6, 7] for
indepth investigations. Recall that the Friedrichs number c(L1, L2) between the
two subspaces L1, L2 of X is defined as
c(L1, L2) = sup |(x1, x2)| : xk ∈ Lk ∩ L⊥ and xk ≤ 1 for k = 1, 2 ,
where L = L1 ∩ L2. The Friedrichs number lies in the interval [0, 1] and may
be thought of as the cosine of the ‘angle’ between the subspaces L1 and L2.
It is shown in [9, Theorem 2] that for N = 2 in the method of alternating
projections we have
When N ≥ 3 no sharp upper bound of this form is known, but it is shown in
[5, Corollary 2.10] that
provided the subspaces are pairwise quasi-disjoint in the sense that Mk ∩
M ∩ M ⊥ = {0} for 1 ≤ k, ≤ N with k = . Moreover, the assumption
on the subspaces cannot be omitted. The same bound was obtained earlier
in [9] in the special case where the subspaces M1 ∩ M ⊥, . . . , MN ∩ M ⊥ are
independent, which is to say that if vectors xk ∈ Mk ∩ M ⊥, 1 ≤ k ≤ N , satisfy
x1 + · · · + xN = 0 then x1 = · · · = xN = 0.
Examples in [5, Section 3] show both that the bound in (1.4) fails to be
sharp in some special cases, thus disproving a conjecture made in [9], and more
generally that it is not possible for N ≥ 3 to obtain a sharp upper bound for
T n − PM , n ≥ 1, which depends only on the pairwise Friedrichs numbers
between the subspaces M1, . . . , MN . Nevertheless, the estimate in (1.3) recovers
the sharp bound in (1.2) when N = 2 and holds with equality in a number of
other cases, for instance if all of the spaces M1, . . . , MN are one-dimensional.
We also see from (1.3) that if the Friedrichs number between a pair of
consecutive subspaces is zero then we have convergence in the method of alternating
projections after at most two steps. Since our interest here is primarily in the
asymptotic rate of convergence as n → ∞, there is no significant loss of
generality in assuming that c(Mk, M ) > 0 for 1 ≤ k, ≤ N with k = . In this
case (1.3) may be recast as
where C = c(M1, MN )−1 and r = kN=1 c(Mk, Mk+1), indices henceforth being
considered modulo N . Since the asymptotic rate of convergence is determined
by the value of r ∈ (0, 1], it is natural to seek the reordering of the subspaces
M1, . . . , MN which leads to the smallest possible value of r. More formally,
given N ≥ 2 we let SN denote the symmetric group on N letters and for each
σ ∈ SN we let rσ = kN=1 c(Mσ(k), Mσ(k+1)), so that for the reordered product
Tσ = Pσ(N) · · · Pσ(
1
) we obtain
T n
σ − PM
≤ Cσrσn,
n ≥ 1,
where Cσ = c(Mσ(
1
), Mσ(N))−1. The objective therefore is to find a
permutation σ ∈ SN such that rσ = r∗, where r∗ = min{rσ : σ ∈ SN }, and to find
such a permutation a version of the following ‘greedy’ algorithm was proposed
in [9, Section 9].
Greedy Algorithm: Given N ≥ 2 independent closed subspaces
M1, . . . , MN of a Hilbert space X whose mutual Friedrichs
numbers are known we obtain permutations σk ∈ SN , 1 ≤ k ≤ N ,
as follows. Let σk(
1
) = k and for j = 2, . . . , N consider as
possible values for σk(j) any previously unused index which minimises
c(Mσk(j−1), M ). If at any stage there is more than one choice of
such an index then proceed by considering all possible choices of
this index and take σk to be that permutation which among those
leading to the least value of rσk comes first in the lexicographical
ordering. Return the permutation σG = σ where ∈ {1, . . . , N } is
the smallest index such that rσ = min{rσk : 1 ≤ k ≤ N }.
If we let rG = rσG , N ≥ 2, then the Greedy Algorithm is correct if and
only if rG = r∗ for all constellations of subspaces. By definition of r∗ it is clear
that r∗ ≤ rG, N ≥ 2. In Sect. 3 we show that if the Greedy Algorithm were
correct then it would follow that P = NP. We then exhibit a simple example
with N = 4 in which r∗ < rG. Both results are obtained as a consequence of
a construction, presented in Sect. 2, which shows that any suitable collection
of numbers in [0, 1] arises as the set of pairwise Friedrichs numbers between
subspaces of some Hilbert space. This result is of independent interest and in
particular implies that the problem of finding an optimal ordering is at least
as hard as solving a multiplicative form of the Travelling Salesman Problem
(TSP). In Sect. 4 we give sharp estimates for the maximal discrepancies
between r∗ and rG. In particular, we show that generically rG < r∗1/2, and that
the estimate is optimal in the sense that for every ε ∈ (
0, 1
) there exists some
N ≥ 2 and a suitable collection of N subspaces of some Hilbert space such
that rG > (1 − ε)r∗1/2. The last step once again requires the construction from
Sect. 2.
2. Friedrichs Matrices
Given N ≥ 2 closed subspaces M1, . . . , MN of a Hilbert space, we may consider
the N ×N -matrix (c(Mk, M ))1≤k, ≤N whose entries are the pairwise Friedrichs
numbers between the various subspaces. We call the matrix arising in this
way the Friedrichs matrix corresponding to the collection of subspaces. It is
clear that any Friedrichs matrix must be symmetric, have zeros along its main
diagonal and elsewhere must have entries lying in the interval [0, 1]. Is every
square matrix which has these three properties a Friedrichs matrix for some
collection of closed subspaces? The following result answers this question in the
affirmative. Here and in what follows we use the same notation as in Sect. 1.
Theorem 2.1. Let F ∈ {R, C} and N ≥ 2, and suppose that C is an N × N
matrix which is symmetric, has zeros along its main diagonal and elsewhere has
entries lying in the interval [0, 1]. Then there exists a Hilbert space X over the
field F and closed subspaces M1, . . . , MN of X such that C is the corresponding
Friedrichs matrix. Furthermore, the subspaces can be constructed in such a way
that Mk ∩ M = {0} for 1 ≤ k, ≤ N with k = and, if N ≥ 3, PkP Pm = 0
for 1 ≤ k, , m ≤ N mutually distinct.
Proof. Let C = (ck, ) and suppose first that 0 ≤ ck, < 1 for 1 ≤ k, ≤ N . Let
{ek, : 1 ≤ k, ≤ N, k = } be an orthonormal basis for the space X = FN(N−1)
endowed with the Euclidean norm, and set
xk, =
ek, ,
c ,ke ,k + (1 − c2,k)1/2ek, , 1 ≤
1 ≤ k < ≤ N,
< k ≤ N.
For 1 ≤ k ≤ N let Bk = {xk, : 1 ≤ ≤ N, = k}, noting that these sets are
orthonormal, and consider the closed subspaces of X given by Mk = span Bk.
By our assumption that the entries of C be strictly smaller than 1 we see that
Mk ∩ M = {0} for 1 ≤ k, ≤ N with k = , and in particular M = {0}.
Furthermore, for 1 ≤ k, k , , ≤ N with k = and k = we have
⎧ 1, k = k ,
⎪
(xk, , xk ) = ⎨ ck, , k = ,
= ,
= k,
⎪⎩ 0,
otherwise,
from which it follows that c(Mk, M ) = ck, for 1 ≤ k, ≤ N with k =
if N ≥ 3, that PkP Pm = 0 for 1 ≤ k, , m ≤ N mutually distinct.
and,
Now consider the general case where 0 ≤ ck, ≤ 1 for 1 ≤ k,
consider the matrix B = (bk, ) with entries
≤ N and
bk, =
ck, , ck, < 1,
0,
ck, = 1,
for 1 ≤ k, ≤ N . By the first part we may find closed subspaces L1, . . . , LN
of FN(N−1) whose Friedrichs matrix is B. Let X = FN(N−1) ⊕ Y , where Y =
1≤ <m≤N 2, and endow X with its natural Hilbert space norm. Moreover,
let U, V be two closed subspaces of 2 such that U + V is not closed. For
1 ≤ k, , m ≤ N with < m define the subspaces Yk ,m of 2 by
⎧ U,
Yk ,m = ⎪⎨ V,
c ,m = 1 and k = ,
c ,m = 1 and k = m,
⎪⎩ {0}, otherwise,
and for 1 ≤ k ≤ N define the closed subspace Mk of X by Mk = Lk ⊕ Yk,
where Yk = 1≤ <m≤N Yk ,m. If 1 ≤ k < ≤ N are such that ck, < 1, then
for 1 ≤ m < n ≤ N we have either Ykm,n = {0} or Y m,n = {0} and therefore
cc(kM,k=, M1. T)h=enc(fLork,1L≤) m= <bk,n =≤ Nck, w. eSsuepeptohsaettYhamt,n1 +≤Ykm<,n = U + V if and
≤ N and that
k
only if k = m and = n, and that otherwise Ykm,n + Y m,n equals either U, V
or {0}. It follows that Yk + Y is not closed, and hence Mk + M is not closed.
By [4, Theorem 9.35] this implies that c(Mk, M ) = 1 = ck, , and hence we
have the required subspaces. Moreover, it is clear from the construction that
Mk ∩ M = {0} for 1 ≤ k, ≤ N with k = and, if N ≥ 3, that PkP Pm = 0
for 1 ≤ k, , m ≤ N mutually distinct.
Remark 2.2. Note that the result in particular provides a new proof of the
fact that in general the optimal value of r in (1.4) cannot be expressed as a
function of pairwise Friedrichs numbers between the subspaces M1, . . . , MN
when N ≥ 3, as was first observed in a particular case in [5, Example 3.7].
Indeed, for any collection of closed subspaces M1, . . . , MN , N ≥ 3, of some
Hilbert space such that in the method of alternating projections we do not have
convergence in one step, by Theorem 2.1 we may find an alternative collection
of closed subspaces M1, . . . , MN of some Hilbert space with the same pairwise
Friedrichs numbers but for which T = PM = 0.
3. Incorrectness of the Greedy Algorithm
In this section we turn to the Greedy Algorithm presented in Sect. 1, and
in particular we ask whether the algorithm is correct in the sense that the
ordering it produces leads to the optimal value of r ∈ [0, 1] in (1.4). We first
consider the connection between our problem of finding an optimal ordering
and the classical TSP, and we show in Corollary 3.3 below that correctness of
the Greedy Algorithm for a sufficiently large class of cases would imply that
P = NP. We then exhibit a simple example in which the Greedy Algorithm
gives a suboptimal ordering.
Recall that in the graph-theoretical formulation of the TSP we are given,
for some N ≥ 2, a complete graph KN with vertices VN = {1, 2, . . . , N } and a
weight function
w : (k, ) ∈ VN2 : k = → R
such that w(k, ) = w( , k) for 1 ≤ k, ≤ N with k = , and the objective is
to find a permutation σ∗ ∈ SN such that Σσ∗ = min{Σσ : σ ∈ SN }, where for
a permutation σ ∈ SN we let
Σσ =
N
k=1
w σ(k), σ(k + 1)
with indices, as usual, considered modulo N . We will be interested primarily in
the multiplicative form of the TSP, denoted by MTSP, in which the objective
is to minimise not the additive cost but instead to find σ∗ ∈ SN such that
Πσ∗ = min{Πσ : σ ∈ SN }, where for a permutation σ ∈ SN we let
N
k=1
Πσ =
w σ(k), σ(k + 1) .
It is clear that TSP and MTSP have the same solution, and indeed one may
pass from one form of the problem to the other simply by replacing the weight
function by its logarithm or its exponential, as appropriate. Furthermore, the
solution of TSP is unaffected by shifting the values of the weight function by a
constant amount, which implies in particular that there is no loss of generality
in considering the MTSP only for weight functions taking values in the range
[0, 1].
It is well known that the TSP, and hence also MTSP, is NP-complete. This
means that it lies in the complexity class NP and is NP-hard, which is to say
that any other problem in NP can be transformed into an instance of the TSP
in polynomial time. Furthermore, by considering the corresponding decision
problems it can be seen that TSP and hence MTSP remain NP-complete if
the weight function is assumed to take distinct values on distinct pairs. Our
first result is an application of Theorem 2.1 showing that the subspace ordering
problem is NP-hard.
Proposition 3.1. The problem of finding an optimal ordering for collections
of independent closed subspaces with pairwise distinct Friedrichs numbers is
NP-hard.
Proof. It suffices to show that every instance of TSP with distinct costs can
be transformed in polynomial time into a subspace ordering problem with
pairwise distinct Friedrichs numbers. However, this follows straightforwardly
from Theorem 2.1. Indeed, given a TSP problem on N ≥ 2 vertices we may
transform it to an instance of MTSP with weight function taking values in the
range [0, 1] in O(N 2) steps. Let C = (ck, )1≤k, ≤N be the symmetric matrix
with zeros along its main diagonal and entries ck, = w(k, ) for 1 ≤ k, ≤ N
with k = . By Theorem 2.1 there exists a Hilbert space X and independent
closed subspaces M1, . . . , MN of X such that C is the associated Friedrichs
matrix. Moreover, it is clear from the proof of Theorem 2.1 that it is possible
to obtain these subspaces in polynomial time. If we find a permutation σ∗ ∈ SN
such that rσ∗ = r∗, then since rσ = Πσ for all σ ∈ SN the permutation σ∗ also
solves our instance of MTSP, and hence the original TSP problem. Since TSP
is known to be NP-hard, our problem is too.
Remark 3.2. Note that the subspaces M1, . . . , MN are not merely independent
but satisfy the much stronger conditions described in Theorem 2.1. In
particular, the result remains true if the subspaces which we are trying to order are
merely pairwise quasi-disjoint in the sense of Sect. 1.
The result shows that the existence of any polynomial-time algorithm
which solves the subspace ordering problem in a sufficiently large number of
cases implies that P = NP. In particular, we obtain the following consequence
for the Greedy Algorithm.
Corollary 3.3. Correctness of the Greedy Algorithm for independent subspaces
with pairwise distinct Friedrichs numbers implies that P = NP.
Proof. It is straightforward to see that if all the pairwise Friedrichs numbers
are distinct then the Greedy Algorithm terminates after O(N 3) steps, where
N ≥ 2 is the number of subspaces we a required to order optimally.
Remark 3.4. The version of the Greedy Algorithm formulated in [9, Section 9]
differs from ours in that it does not consider all possible greedy paths and
hence runs in polynomial time even if the pairwise Friedrichs numbers are
not assumed to be distinct. Note also that, as in the case of Proposition 3.1,
the assumption of independence on the subspaces can be relaxed to pairwise
quasi-disjointness.
Given that the question whether P = NP is a long-standing open
problem, one may view Proposition 3.1 as evidence suggesting that the Greedy
Algorithm does not in general lead to an optimal ordering of the subspaces in
question. This is indeed the case, as the following example illustrates.
Example 3.5. Let F ∈ {R, C} and let X = F4 with the Euclidean norm.
Consider the one-dimensional subspaces Mk = span{xk}, 1 ≤ k ≤ 4, where
x1, . . . , x4 ∈ X are the unit vectors
x1 = (
1, 0, 0, 0
),
1 √3
x2 = ,
2 2
, 0, 0 ,
,
The Friedrichs numbers satisfy c(Mk, M ) = |(xk, x )| for 1 ≤ k, ≤ 4 with
k = , so the associated Friedrichs matrix is given (approximately) by
σG(k) 4k=1 = (
1, 4, 3, 2
),
σ(k) 4k=1 = (
1, 4, 2, 3
)
The permutation σG ∈ S4 produced by the Greedy Algorithm is
which leads to rG ≈ 7.5772 × 10−4. The permutation σ ∈ S4 given by
leads to the optimal value r∗ = rσ ≈ 5.1033 × 10−4, and in particular rG > r∗.
It follows that the Greedy Algorithm is not correct.
Remark 3.6. Example 3.5 disproves a claim made in [9, Section 9], namely
that the Greedy Algorithm always leads to an optimal ordering in the case of
independent subspaces. The examples considered in [9, Section 9] involve only
N = 3 subspaces, a special case in which the Greedy Algorithm performs an
exhaustive search of all possible orderings (up to the direction in which they
are traversed) and in particular is correct. Thus Example 3.5 is minimal in
terms of the number of subspaces involved.
4. Sharp Estimates for the Degree of Suboptimality
Having shown in Sect. 3 that the Greedy Algorithm does not in general lead to
an optimal ordering of the subspaces in the method of alternating projections,
we seek now to quantify how much the result reached by the Greedy Algorithm
can disagree with the optimal result. Given a collection of closed subspaces of a
Hilbert space such that at least one of the pairwise Friedrichs numbers is zero,
we see that for suitable orderings of the subspaces we obtain convergence after
at most two steps in the method of alternating projections. Another essentially
uninteresting case for asymptotic analysis is when all of the pairwise Friedrichs
numbers equal 1, so that no ordering leads to a useful estimate in (1.3). If either
of these two cases holds we shall say that the collection of subspaces involved
is non-generic, and otherwise we call it generic.
Theorem 4.1. Let N ≥ 2 and suppose that M1, . . . , MN are closed subspaces
of a Hilbert space X. Then
r∗ ≤ rG ≤ r∗1/2.
(4.1)
Moreover, the second inequality is strict unless the collection M1, . . . , MN of
subspaces is non-generic
Proof. For 1 ≤ k ≤ N let σk ∈ SN be the permutation produced by running
the Greedy Algorithm with the starting vertex σk(
1
) = k and let rk = rσk .
Then certainly r∗ ≤ rk for 1 ≤ k ≤ N , and hence also r∗ ≤ rG. For 1 ≤ k, ≤
N let
sk( ) = σk σk−1( ) + 1
denote the index of the successor to M in the ordering of the subspaces
determined by σk, noting that sk( ) = 1 if σk( ) = N . Let σ ∈ SN and for
1 ≤ k, ≤ N with k = let w(k, ) = c(Mk, M ). Let 1 ≤ k, ≤ N . If
σ−1(σ( )) < σk−1(σ( + 1)), which is to say that in the ordering determined by
k
σk the subspace Mσ( ) comes before Mσ( +1), then by definition of the Greedy
Algorithm we must have
w σ( ), sk(σ( ) ≤ w σ( ), σ( + 1) ,
while if σk−1(σ( )) > σk−1(σ( + 1)) then
Since w takes values in [0, 1] it follows that
w σ( + 1), sk(σ( + 1) ≤ w σ( ), σ( + 1) .
w σ( ), sk(σ( ) w σ( + 1), sk(σ( + 1) ≤ w σ( ), σ( + 1)
(4.2)
for 1 ≤ k, ≤ N . Thus for 1 ≤ k ≤ N we have
rk2 =
c(Mσk( ), Mσk( +1))2 =
w σk( ), σk( + 1) 2
=
≤
N
=1
N
=1
N
=1
N
=1
N
=1
w σ( ), sk(σ( ) w σ( + 1), sk(σ( + 1)
w σ( ), σ( + 1) =
c(Mσ( ), Mσ( +1)) = rσ.
(4.3)
Since σ ∈ SN was arbitrary we deduce that rk2 ≤ r∗ for 1 ≤ k ≤ N , and in
particular rG2 ≤ r∗, as required.
Now suppose that rG2 = r∗, and let σ∗ ∈ SN be a permutation such that
rσ∗ = r∗. Since rG2 ≤ rk2 ≤ r∗ for 1 ≤ k ≤ N , we see that in fact rk2 = r∗ for
1 ≤ k ≤ N . Now either one of the pairwise Friedrichs numbers is zero or all
of the pairwise Friedrichs numbers are non-zero. In the latter case it is clear
from (4.3) that we must have equality in (4.2) for 1 ≤ k, ≤ N when σ = σ∗.
Taking k = σ∗( ) in (4.2) for 1 ≤ ≤ N , it follows that
w σ∗( ), σ∗( + 1) = min w(σ∗( ), k) : 1 ≤ k ≤ N, k = σ∗( )
for 1 ≤ ≤ N . It follows that σ∗ is itself a permutation considered by the
Greedy Algorithm, and therefore r∗ = rG. Hence r∗2 = r∗, and since r∗ = 0
we have r∗ = 1, which implies that c(Mk, M ) = 1 for 1 ≤ k, ≤ N with
k = . It follows that rG2 < r∗ unless the collection M1, . . . , MN of subspaces
is non-generic.
It remains to be investigated to what extent the second bound in (4.1) is
sharp for generic constellations of subspaces. Our final example shows that it
cannot be improved in the sense that given any ε ∈ (
0, 1
) there exists a generic
constellation of subspaces of some Hilbert space such that
rG > (1 − ε)r∗1/2.
In fact, there exists a constellation of N such subspaces for every even N ≥ 4.
Example 4.2. Given a positive a positive integer n ≥ 2, let N = 2n and
suppose that 0 < δ < c < 1. By Theorem 2.1 there exists a Hilbert space X
and a generic constellation M1, . . . , MN of closed subspaces of X such that for
1 ≤ k, ≤ N with k = we have
⎧ c if k = ± 1 (mod N )
⎪
c(Mk, M ) = ⎨ cδ if k = ± 2 (mod N ) and k is even,
⎪⎩ 1 otherwise.
Let σ0 ∈ SN denote the identity permutation. Then rσ0 = cN . If we think of
the subspaces as the vertices of a complete graph of order N , and we let the
edges have weights given by the pairwise Friedrichs numbers, then rσ ≥ rσ0 for
all permutations σ ∈ SN involving no cδ-edges. Moreover, any cycle σ ∈ SN
which uses at least one of the cδ-edges cannot use more than n − 1 of them,
and must involve at least two 1-edges, so for any such cycle
In particular, if c2 ≤ δn−1 then r∗ = rσ0 . It is easy to that
rσ ≥ cn−1(cδ)n−1 = cN−2δn−1.
rG ≥ c2(cδ)n−1 = cδn−1r∗1/2.
Given ε ∈ (
0, 1
) we deduce that rG > (1 − ε)r∗1/2 provided c, δ ∈ (
0, 1
) are
such that c2 ≤ δn−1 and cδn−1 > 1 − ε. These conditions are satisfied for
instance when (1 − ε)1/3 < c < 1 and δ = c2/(n−1). Furthermore, it is the case
that for any r, ε ∈ (
0, 1
) there exist generic constellations of N subspaces of
a Hilbert space for all sufficiently large even N ≥ 4 with the properties that
rG > (1 − ε)r∗1/2 and r∗ = r.
Acknowledgements
For financial support O.D. thanks Magdalen College, Oxford, A.J. thanks the
Mathematical Institute of the University of Oxford, S.R. and R.S. thank both
St John’s College, Oxford, and the Mathematical Institute, and L.S. thanks
the EPSRC. All authors would further like to express their thanks to Alexis
Chevalier, Stefan Kiefer, Dominik Peters and Zhixuan Wang for useful
discussions.
Open Access. This article is distributed under the terms of the Creative
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source, provide a link to the Creative Commons license, and indicate if changes
were made.
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