Download Steiner Equiangular Tight Frames Redux

Steiner Equiangular Tight Frames Redux
Matthew Fickus, Dustin G. Mixon & Jesse D. Peterson
John Jasper
Department of Mathematics and Statistics
Air Force Institute of Technology
Wright-Patterson Air Force Base, Ohio 45433
Email: [email protected]
Department of Mathematics
University of Missouri
Columbia, MO 65211
Abstract—An equiangular tight frame (ETF) is a set of unit
vectors whose coherence achieves the Welch bound, and so is
as incoherent as possible. ETFs arise in numerous applications,
including compressed sensing. They also seem to be rare: despite
over a decade of active research by the community, only a few
construction methods have been discovered. One known method
constructs ETFs from combinatorial designs known as balanced
incomplete block designs. In this short paper, we provide an
updated, more explicit perspective of that construction, laying
the groundwork for upcoming results about such frames.
Steiner ETFs from combinatorial designs [6], [9]—is less
well-understood. In this paper, we revisit the Steiner ETF
construction. In short, we give a new, explicit, closed-form
expression of the synthesis operator of a Steiner ETF as
the product of several special matrices. We then give a new
proof of the fact that the resulting vectors are tight and
equiangular. This new proof is more formal and symbolic than
that given in [6], and lays the groundwork for some new results
concerning such ETFs that will be presented in [7].
I. I NTRODUCTION
II. BACKGROUND
To see why equality in (1) is achieved if and only if the unit
N
norm vectors {ϕn }N
n=1 are an ETF, note that if {ϕn }n=1 is a
M
unit norm tight frame for F then the tight frame constant α
N
since it satisfies:
is necessarily the redundancy of the frame M
{ϕn }N
n=1
Let M < N be positive integers and let
be a
sequence of unit vectors in FM where the field F is either R
or C. The Welch bound [12] is a lower bound on the coherence
of {ϕn }N
n=1 :
q
N −M
(1)
M (N −1) ≤ max0 |hϕn , ϕn0 i|,
n6=n
and it is well-known that this bound is achieved if and only if
M
{ϕn }N
[10].
n=1 is an equiangular tight frame (ETF) for F
N
M
To be precise, for any vectors {ϕn }n=1 in F , the corresponding synthesis operator is the M × N matrix Φ which
has the vectors {ϕn }N
as its columns, namely the operator
n=1 P
N
Φ : FN → FM , Φy := n=1 y(n)ϕn . Composing Φ with
its N ×M adjoint (conjugate transpose) Φ∗ yields the M ×M
frame operator ΦΦ∗ as well as the N ×N Gram matrix Φ∗ Φ
whose (n, n0 )th entry is (Φ∗ Φ)(n, n0 ) = hϕn , ϕn0 i.
We say {ϕn }N
n=1 is a tight frame if Φ is perfectly conditioned, that is, if there exists α > 0 such that ΦΦ∗ = αI.
This is equivalent to having the rows of Φ be orthogonal and
equal norm. We say {ϕn }N
n=1 is equiangular when each ϕn
is unit norm and the value of |hϕn , ϕn0 i| is constant over all
choices of n 6= n0 , namely when the diagonal entries of Φ∗ Φ
are 1 while the off-diagonal entries have constant modulus.
An ETF is a tight frame whose vectors are equiangular.
Because of their minimal coherence, they are useful in a
number of real-world applications, including waveform design
for wireless communication [10], compressed sensing [4] and
algebraic coding theory [9]. In spite of this fact, only a few
methods for constructing ETFs are known. The most well
known construction—forming harmonic ETFs from difference
sets in finite abelian groups [10], [13], [5]—is elegant and
natural. A more recently discovered construction—forming
c
978-1-4673-7353-1/15/$31.00 2015
IEEE
M α = Tr(αI) = Tr(ΦΦ∗ ) = Tr(Φ∗ Φ) =
N
X
1 = N.
n=1
N
As such, the Frobenius norm of the operator ΦΦ∗ − M
I is
N
one way of quantifying the tightness of {ϕn }n=1 . Moreover,
using similar properties of the trace, we see this nonnegative
quantity can be rewritten as
0 ≤ kΦΦ∗ −
=
=
=
=
=
2
N
M IkFro
N
Tr(ΦΦ∗ − M
I)2
N
Tr(ΦΦ∗ )2 − 2 M
N
Tr(Φ∗ Φ)2 − 2 M
N X
N
X
Tr(ΦΦ∗ ) +
∗
Tr(Φ Φ) +
|hϕn , ϕn0 i|2 −
n=1 n0 =1
N X
N
X
N2
M2
N2
M
Tr(I)
N2
M
N
|hϕn , ϕn0 i|2 − N ( M
− 1).
n=1 n0 =1
n0 6=n
Bounding the above summands by their maximum then gives
the inequality
N
0 ≤ N (N − 1) max0 |hϕn , ϕn0 i|2 − N ( M
− 1).
n6=n
When rearranged, this gives (1). Moreover, in order to have
equality in (1), we necessarily have equality throughout the
N
above analysis, namely that kΦΦ∗ − M
Ik2Fro = 0 (meaning
N
{ϕn }n=1 is a tight frame) and that |hϕn , ϕn0 i|2 is constant
over all n 6= n0 (meaning {ϕn }N
n=1 is equiangular).
To summarize, the columns of an M × N matrix Φ achieve
the Welch bound if and only if Φ has equal-norm orthogonal
rows and unit-norm equiangular columns. ETFs seem to be
very rare. Comparing the number of entries in an ETF’s
synthesis operator Φ against the number of conditions they
must satisfy, it is surprising that several nontrivial infinite
families of ETFs have been discovered. Each of these known
constructions has a remarkable degree of symmetry stemming
from certain special types of combinatorial designs.
Real ETFs in particular are known to be equivalent to a certain type of strongly regular graph. Much of the work in this
area was pioneered by J. J. Seidel and his contemporaries [3];
a good explanation of the equivalence is given in [11]. Here,
the main idea is to construct the Gram matrix Φ∗ Φ of an ETF
rather than construct Φ directly. Indeed, if {ϕn }N
n=1 is a tight
frame then (Φ∗ Φ)2 = Φ∗ (ΦΦ∗ )Φ = αΦ. If we further have
∗
that {ϕn }N
n=1 is an ETF, this means Φ Φ is a scalar multiple
of a orthgononal projection matrix with ones along its diagonal
and off-diagonal entries of constant modulus. Conversely, if G
is any such matrix, taking its spectral decomposition yields a
rank(G) × N matrix Φ such that G = Φ∗ Φ, and this Φ is
necessarily the synthesis operator of an ETF.
From this, we see that finding ETFs is equivalent to finding
N × N self-adjoint matrices G with the properties that (i)
G2 = αG for some α > 0, (ii) G(n, n) = 1 for all n,
and (iii) there exists β ≥ 0 such that |G(n, n0 )| = β for all
n 6= n0 . This, in turn, is equivalent to constructing an N × N
self-adjoint matrix S with zero diagonal and unimodular offdiagonal entries which satisfies
S2 =
α−2
β S
+
α−1
β 2 I.
(2)
This signature matrix S is related to the Gram matrix G
according to G = βS + I.
If we seek real ETFs in particular, even more simplification
is possible. In this case, the off-diagonal entries of S are either
1 or −1. In fact, we can assume without loss of generality the
the off-diagonal entries in the first row and column of S are 1;
this equates to multiplying our frame vectors by unimodular
scalars to ensure they have a positive inner product with the
first frame vector. Removing this first row and column then
yields an (N − 1) × (N − 1) reduced signature matrix R.
Having S satisfy (2) is equivalent to having α−1
β2 = N − 1
α−2
where the all-ones vector 1 satisfies R1 = β 1 and
R2 =
α−2
β R
+ (N − 1)I − J.
Here, J := 11∗ is an all-ones matrix. Continuing, any
symmetric (N − 1) × (N − 1) matrix R with zero diagonal
and off-diagonal entries of ±1 corresponds to the adjacency
matrix A of a graph by letting A = 12 (R − I + J). Under
this identification, our two conditions on R are equivalent to
a single condition on A, namely
A2 = (L −
N
2 )A
+
L
2I
+
L
2J
where L := 12 (N − 2 + α−2
β ). For the adjacency matrix A of
any graph, the quantity A2 (n, n0 ) counts the number of twostep paths from vertex n to n0 . As such, the above relation is
equivalent to having that (i) every vertex has L neighbors (the
graph is regular), (ii) any two neighbors have 3L−N
neighbors
2
in common and (iii) any two nonneighbors have L2 neighbors
in common.
In general, any regular graph which such properties—
neighbors have a constant number of common neighbors and
so do nonneighbors—is called strongly regular. By the above
rationale, the existence of a real M × N ETF is equivalent
to the existence of a strongly regular graph with parameters
N
, L2 ) where L := 12 (N −2+ α−2
srg(N −1, L, 3L−N
2
β ), α = M ,
p
β = (N − M )/M (N − 1). This equivalence allows us to
comb the rich literature of strongly regular graphs, like [1]
and [2] for example, for “new” constructions of ETFs. For
example, in [6], a method was given for constructing the
synthesis operators of real and complex ETFs; it turns out
that essentially the same idea was used to construct strongly
regular graphs many years earlier in [8]. In the next section,
we detail this construction method.
III. S TEINER ETF S
Given positive integers K < V , a corresponding (V, K, 1)balanced incomplete block design (BIBD)—also known as
a (2, K, V )-Steiner system—is a collection of K-element
subsets (blocks) of {1, . . . , V } with the property that every
point is contained in the same number of blocks and that every
pair of distinct points is contained in exactly one block. There
are several known infinite families of BIBDs, the most natural
being finite affine and projective geometries: each block is a
discrete line, where every line contains the same number of
points, every point is contained in the same number of lines,
and any pair of distinct points determines a unique line. We
denote the total number of blocks by B, and denote the number
of blocks that contain a given point by R.
The easiest way to verify that a given set of blocks is a
BIBD is to form the incidence matrix of the design: writing
the indicator function of each block as the row of a B × V
matrix E, we need that (i) each row of E sums to K, (ii)
each column of E sums to R, and (iii) the dot product of any
two distinct columns of E is 1. For example, the set of all
2-element subsets of {1, 2, 3, 4} has


+ +

+ +


+
+ 

,
E=
(3)
+
 +

+
+
+ +
and corresponds to a (4, 2, 1)-BIBD with B = 6 and R = 3;
here, for the sake of readability, we denote matrix entries of
1, −1 and 0 by +, − and an empty space, respectively.
When discussing (V, K, 1)-BIBDs, the reason we do not
specify B and R is that they are uniquely determined by V
and K. Indeed, since the entries of each of the B rows of E
sum to K while the entries of each of its V columns sum to R
we see that BK = V R. Moreover, note that the entries of B
which lie to the right of a “1” in the first column of E sum to
R(K − 1) (since there are R such rows, each summing to K)
as well as to V − 1, since any two distinct columns overlap
in exactly one row. Since BK = V R and R(K − 1) = V − 1
V (V −1)
V −1
we see that R = K−1
and B = K(K−1)
.
In [6], the B × V (R + 1) synthesis operator of an ETF is
constructed from a tensor-like product of a BIBD’s incidence
matrix E with the synthesis operator F of a unimodular regular
simplex in FR . Such an F is obtained by removing a row from
an (R + 1) × (R + 1) (possibly complex) Hadamard matrix,
such as a DFT. For example, the incidence matrix (3) has
R = 3, and we can form a R×(R+1) = 3×4 regular simplex
(tetrahedron) by removing the top row from the canonical 4×4
Hadamard matrix:


+ − + −
F = + + − −.
(4)
+ −− +
To form a Steiner ETF, each of the V columns of E is
expanded into a B × (R + 1) matrix: each nonzero entry in a
given column is replaced by a distinct row of F, while each
zero entry is replaced by a row of zeros. The resulting matrix
is then scaled by a factor of √1R so as to make its columns unit
norm. For the E and F given in (3) and (4) this construction
yields the following 6 × 16 matrix Φ:


+− + − + −+ −

+ − + − + − + −



1 
+
+
−
−
++ −−

 . (5)
√ 

+
+
−
−
+
+
−
−
3

+ − − +
+ − − +
+ −−+ + −−+
Here, the vertical lines delineate the block submatrices of
Φ that arise from each of the four columns of E. One can
easily see that this particular Φ is the synthesis operator of
a 16-element ETF for R6 . Indeed, since the rows of F are
orthogonal, the rows of Φ are as well. Also, the rows of Φ
are equal norm since the nonzero part of any such row consists
of exactly K = 2 of the equal-norm rows of F. Together, these
facts imply that Φ is tight. Moreover, the columns of Φ have
unit norm since
exactly R = 3 nonzero values of
√ each contains
√
magnitude 1/ R = 1/ 3. To see that Φ is equiangular, note
that the inner product of any two columns of Φ in the same
B × (R + 1) = 6 × 4 block submatrix is always −1/R since it
arises from the (−1)-valued inner product of the corresponding
two columns of F. Meanwhile, the inner product of any two
columns from distinct block submatrices is ±1/R since the
supports of the corresponding columns of E overlap in exactly
one row and the entries of F are unimodular.
From this example, it is straightforward to see that this
argument works in general. That is, combining any (V, K, 1)BIBD matrix E with any R × (R + 1) unimodular regular
simplex F in this manner yields an ETF of V (R + 1) vectors
for FB . This result is formally proven in [6], but the proof
mostly uses words rather than symbols, making it difficult
to generalize and expand upon. The same presentation issue
arises in [8], in which essentially the same method is used to
construct certain strongly regular graphs. In the next section,
we introduce a new concept—the permutation matrix of a
BIBD—that leads to an explicit formula for Φ and a new
proof of this same result. Though this proof will be less
intuitive and more complicated than that given in [6], it will
also be more formal and much more symbolic, laying the
necessary theoretical groundwork for upcoming results about
these frames [7].
IV. S TEINER ETF S R EDUX
In this section, we give a more explicit construction of the
Steiner ETFs discussed in the previous section. As before, let
E be the B × V incidence matrix of a (V, K, 1)-BIBD. We
construct a BK × V R matrix V by replacing each of the
entries of E with a K × R block.
Specifically, for any b = 1, . . . , B and v = 1, . . . , V , if
E(b, v) = 0 we let Vb,v be the K × R block of zeros. If,
on the other hand, E(b, v) = 1, we know that the bth block
of the design contains the vth point. For example, for the
(4, 2, 1)-BIBD whose incidence matrix E given in (3), the
fact that E(5, 4) = 1 implies that the fifth block of the design
contains the fourth point. Now recall that there are K points
in this particular block. Let k(b, v) ∈ {1, . . . , K} denote the
order of the vth point as an element of the bth block, that is,
indicate which of the K points of this block it is. For example,
since E(5, 4) = 1 in (3), and since the fourth point (column)
is only the second point in block five (i.e., this is only the
second 1 in row five) we have k(5, 4) = 2. Similarly, we let
r(b, v) ∈ {1, . . . , R} denote the order of the bth block as a
set that contains the vth point. Continuing our example, since
E(5, 4) = 1 and since block five is the third block to contain
the fourth point (i.e., this is the third one in column four) we
have r(5, 4) = 3. For any (b, v) for which E(b, v) = 1, we
let Vb,v be a K × R matrix whose entries are 0 except for
a single 1 at (k(b, v), r(b, v)). For example, for the incidence
matrix E is given in (3) we have the 2 × 3 block
V5,4 =
.
+
Assembling these B × V blocks of size K × R yields a
BK × V R matrix V; this matrix is square since BK = V R.
For example, (3) begets the 12 × 12 matrix


+


+




+




+



 +




+
.

V=

+



+ 




+



+




+
+
This matrix is unitary, being a permutation matrix. Indeed,
there is a single 1 in each row: writing a given row index as
(b − 1)K + k for some b = 1, . . . , B and k = 1, . . . , K, it
arises from the kth point in the bth block of the design. We
dub V the permutation matrix of the BIBD.
To form a Steiner ETF from such a permutation matrix,
we multiply it on the left and right by special block-diagonal
matrices. In particular, on the left we multiply by a B × BK
block-diagonal matrix U which consists of B × B blocks of
size 1 × K where each diagonal block is a 1 × K row of ones.
To be precise, letting 0, 1 ∈ FK denote column vectors of
zeros and ones, respectively, we let
∗
1 , b = b0 ,
Ub,b0 :=
0∗ , b 6= b0 .
For example, when B = 6 and K = 2 as in (3), U is the
6 × 12 matrix


++


++




++
.

U=

++




++
++
This is a special example of what is known as a decimation operator in the signal processing community: by replacing each
block of K neighboring values with their sum, it essentially
downsamples a low-passed version of a given signal.
Meanwhile, we multiply V on the right by a V R×V (R+1)
block diagonal matrix W which consists of V × V blocks of
size R × (R + 1) where each diagonal block is a unimodular
regular simplex F. Specifically, let
F, v = v 0 ,
Wv,v0 :=
0, v 6= v 0 ,
R+1
where F is the synthesis operator of vectors {fs }s=1
in FR
with the properties that |fs (r)| = 1 for all r and s, and where
|hfr , fr0 i| = 1 for all r 6= r0 . For example, when V = 4 and
R = 3 as in (3) and F is chosen as in (4), we have


+ − +−
+ + − −



+ − − +





+ − +−




++ −−




+ −−+

.
W=

+ − + −




++ −−




+ −− +



+ − + −



+ + − −
+ − − +
this holds in general, note that since U and W are block
diagonal, the (b, v)th block of Φ is the 1 × (R + 1) matrix
Φb,v =
B X
V
X
√1
R
Ub,b0 Vb0 ,v0 Wv0 ,v
b0 =1 v 0 =1
=
=
√1 Ub,b Vb,v Wv,v
R
√1 1∗ Vb,v F.
R
(6)
Here, 1∗ Vb,v is the 1 × R matrix obtained by summing the
rows of Vb,v . This row vector is zero if the vth point of the
design does not lie in the bth block. On the other hand, if this
point is in this block, the definition of Vb,v implies 1∗ Vb,v =
δ ∗r(b,v) where δ r is the rth standard basis (column) vector in
FR , implying Φb,v is a scaled version of the r(b, v)th row of
F; this is precisely the Steiner ETF construction of [6]. For
the remainder of this paper, we use (6) and the properties of
the BIBD’s permutation matrix V to reprove that Φ is indeed
the synthesis operator of an ETF.
We first verify that Φ is tight, namely that ΦΦ∗ = αI for
some α > 0. This immediately follows from the fact that it is
a product of the tight matrices U, V and W. Alternatively,
we can prove this from basic principles: using (6) and the fact
that V is unitary, the (b, b0 )th block of ΦΦ∗ is the scalar
(ΦΦ∗ )b,b0 =
=
=
=
=
V
X
Φb,v (Φb0 ,v )∗
v=1
V
X
1
1∗ Vb,v FF∗ Vb∗0 ,v 1
R
v=1
X
V
∗
R+1 ∗
Vb,v Vb0 ,v 1
R 1
v=1
∗
R+1 ∗
0
R 1 (VV )b,b 1
∗
1 I1, b = b0
R+1
∗
0
R
1 01, b 6= b
=
(R+1)K
,
R
0,
b = b0 ,
b 6= b0 .
Thus, ΦΦ∗ = (R+1)K
I, as desired. As an aside, note this
R
constant is the redundancy of the frame: Φ is B × V (R + 1)
N
and since BK = V R we have (R+1)K
= V (R+1)
= M
; this
R
B
is a well-known property of all unit norm tight frames.
Next, we verify that the diagonal entries of Φ∗ Φ are 1 while
the off-diagonal entries have constant modulus. From (6), the
(v, v 0 )th block of Φ∗ Φ is the (R + 1) × (R + 1) matrix
(Φ∗ Φ)v,v0 =
B
X
Φ∗b,v Φb,v0
b=1
More generally, one can choose a different unimodular simplex
in each diagonal block.
Now let Φ = √1R UVW, which is a B × V (R + 1) block
matrix consisting of B × V blocks of size 1 × (R + 1). We
claim Φ is a Steiner ETF. For instance, multiplying the above
examples together yields the 6 × 16 ETF given in (5). To see
=
1
R
B
X
∗
F∗ Vb,v
11∗ Vb,v0 F
b=1
=
1 ∗
RF
X
B
b=1
∗
Vb,v
JVb,v0 F.
(7)
To simply this further, note the (r, r0 )th entry in the middle
term of (7) is
X
B
∗
Vb,v
JVb,v0 (r, r0 )
(Φ∗ Φ)v,v0 (s, s0 )
b=1
=
=
B X
K X
K
X
b=1 k=1 k0 =1
B X
K X
K
X
∗
Vb,v
(r, k)J(k, k 0 )Vb,v0 (k 0 , r0 )
Vb,v (k, r)Vb,v0 (k 0 , r0 )
b=1 k=1 k0 =1
=
B X
K
X
b=1
k=1
X
K
0 0
Vb,v (k, r)
Vb,v0 (k , r ) .
(8)
k0 =1
At this point, recall that V is a permutation matrix and that
Vb,v (k, r) = 1 if and only if the vth point of the design is
contained in the bth block and moreover that this is the kth
point in that blockP
and the rth block that contains that point.
K
As such, the sum k=1 Vb,v (k, r) of these values over all k
is either 0 or 1, and equals 1 if and only if the vth point of
the design is contained in the bth block and moreover that this
is the rth block that contains that point. This, in turn, implies
that the product
X
X
K
K
0 0
Vb,v0 (k , r )
(9)
Vb,v (k, r)
k=1
At this point, recall that F is the synthesis operator of a
R
unimodular regular simplex {fs }R+1
s=1 for F . As such, for any
0
0
s, s = 1, . . . , R + 1, the (s, s )th entry of the (v, v 0 )th block
of the Gram matrix of Φ is
k0 =1
is either 0 or 1, the latter occurring if and only if both the vth
and v 0 th point of the design lie in the bth block, and that this
block is the rth block to contain point v as well as the r0 th
block to contain point v 0 .
At this point, we consider cases. When v = v 0 , (9) is 1
precisely when we simultaneously have that (i) the vth point
lies in the bth block, (ii) this block is the rth block to contain
v, and (iii) r = r0 . Thus, in this case,
these values (9)
PBsumming
∗
over all b as in (8), we see that ( b=1 Vb,v
JVb,v )(r, r0 ) = 1
precisely when r = r0 : when r = r0 , the rth block
to contain v corresponds to exactly one value of b; when
rP6= r0 , all the summands are zero. Altogether, this implies
B
∗
b=1 Vb,v JVb,v = I.
In the remaining case where v 6= v 0 , recall that by the
definition of (V, K, 1)-BIBD, there is exactly one block that
contains both v and v 0 . As such, in this case (9) can be only 1
for one particular choice of b. Moreover, even for this b, (9)
is only 1 for one choice of r and r0 , denoted r1 (v, v 0 ) and
r2 (v, v 0 ), respectively. In P
particular, if v 6= v 0 , summing (9)
B
∗
JVb,v0 )(r, r0 ) = 1 when
over all b as in (8) gives ( b=1 Vb,v
0
0
0
r = r1 (v, v ) and r = r2 (v, v ); otherwise, this sum is 0.
In summary, the middle term of (7) is:
B
X
I,
v = v0 ,
∗
Vb,v JVb,v0 =
(10)
∗
δ r1 (v,v0 ) δ r2 (v,v0 ) , v 6= v 0 .
b=1
Returning to (7), we thus have
F∗ F,
v = v0 ,
∗
1
(Φ Φ)v,v0 = R
∗
∗
F δ r1 (v,v0 ) δ r2 (v,v0 ) F, v =
6 v0 .
= hδ s , (Φ∗ Φ)v,v0 δ s0 i
(
hδ s , F∗ Fδ s0 i,
v = v0
= R1
∗
∗
hδ s , F δ r1 (v,v0 ) δ r2 (v,v0 ) Fδ s0 i, v 6= v 0
(
hfs , fs0 i,
v = v0
1
=R
∗
hfs , δ r1 (v,v0 ) δ r2 (v,v0 ) fs0 i, v 6= v 0

1,
v = v 0 , s = s0 ,


1
0
v = v 0 , s 6= s0 ,
=
R hfs , fs i,

 1
0
∗
0
0
R [fs (r1 (v, v ))] fs (r2 (v, v )), v 6= v .
Thus, the diagonal entries of Φ∗ Φ are indeed one. Moreover,
the off-diagonal entries indeed have modulus R1 since {fs }R+1
s=1
is a sequence of vectors with unimodular entries with the
property that |hfs , fs0 i| = 1 for all s 6= s0 . Altogether, we
see that Φ is indeed the synthesis operator of an ETF where
the Welch bound is R1 .
ACKNOWLEDGMENTS
This work was partially supported by NSF DMS 1321779.
The views expressed in this article are those of the authors and
do not reflect the official policy or position of the United States
Air Force, Department of Defense, or the U.S. Government.
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