A set of vectors is a frame for if there exist such that
where denotes the Euclidean norm. If, in addition, each is unit-norm, we say that is a unit-norm frame. is called tight if . A tight unit-norm frame is called a finite unit-norm tight frame (FUNTF). One attractive feature of FUNTFs is the fact that they can be used to decompose and reconstruct any vector via the following formula:
Frames in general, and FUNTFs in particular, are routinely used in many applications, especially in signal processing. For more on the theory and the applications of frames we refer to [9, 16, 17, 18].
A frame is said to be equiangular if there exists such that
If in addition is tight, then is called an equiangular tight frame (ETF). It follows from [7, Proposition 1.2] that the vectors of an ETF have necessarily equal norm. Consequently, and without loss of generality, all ETFs considered in the sequel will be unit-norm frames, i.e., FUNTFs.
Let be the collection of all sets of unit-norm vectors. For any , the -frame potential of is defined as
The definition of the -frame potential above differs from the one given in  as (3) excludes self inner products. As will be seen in Section 2, the present definition will allow us to state our results in a more concise manner. The subscripts are a little redundant since they are suggested by the input , but they will come handy when we want to emphasize the dimension or the number of points. We are interested in finding the infimum of the -frame potential among all -point configurations in . It is a standard argument to show that this infimum can be achieved due to the compactness of the sphere and the continuity of the function, so we can replace infimum by minimum and define
In situations when are both fixed, we will simply use for , and for . Similarly we use the notations if and are fixed. Any minimizer of (4) will be called an optimal configuration of the -frame potential. We observe that if
is optimal, then with any orthogonal matrix, any permutation , and any ,
is optimal too. In other words, the optimal configuration is an equivalence class with respect to orthogonal transformations, permutations or sign switches. So when we say an optimal configuration is unique, we mean that it is unique up to this equivalence relation.
Note that in the definition of the frame potential, does not necessarily need to be a frame of , but we will show in Proposition 2.1 that the minimizers of the -frame potential must be a frame, as expected. Therefore problem (4) remains the same if we had restricted to be a unit-norm frame with frame vectors.
The name “frame potential” originates from the special case ,
which was studied by Benedetto and Fickus . They proved that is an optimal configuration of if and only if is a FUNTF.
Another important special case is . In this case, the quantity
and the equality in (7) holds if and only if is an ETF, which is only possible if . The coherence minimization problem corresponds to because it appears to be the limiting case when grows to infinity; see Proposition 2.2.
When is an even integer, the minimizers of have long been investigated in the setting of spherical designs, see [13, 24]. A set of points (the unit sphere in ) is called a spherical design if for every homogeneous polynomial of degree or less,
where is the normalized surface measure on . For example, a spherical -design is a set of points whose center of mass is at the origin. More generally, as shown in  or [24, Theorem 8.1], if is an even integer and is symmetric, that is , then
and equality holds if and only if is a spherical -design.
Optimal configurations of (4) are often not symmetric since and are considered the same points as far as frame potential is concerned. However, we can still use (8) by symmetrizing a frame. Given such that its coherence (i.e. no repeated vectors or opposite vectors), we let
Some straightforward computations result in
which combined with (8), can be used to prove
Let be an even integer, then
and equality holds if and only if is a spherical -design.
Not only is Proposition 1.1 limited to even ’s, but it is also not trivial to find spherical -designs for large . More generally, and to the best of our knowledge, little is known about the complete solutions to (4) even in the simplest case . When , a solution is given in  for all positive . See also [6, 19] for related results. For any and , it is shown in  that the Grassmannian frame is
which can be viewed as equally spaced points on the half circle. The main result of this paper establishes that the unique optimal configuration when , , and is , where is the largest integer that does not exceed . Moreover for , our result is sharper as we prove this is the case for . Such a result is expected since optimal configurations for large are approaching the Grassmannian frame. Moreover, we are able to show that is the optimal configuration for a big class of kernel functions. See Theorem 3.5. The phenomenon that a given configuration is the optimal configuration for a large range of functions is what we call universal. Such a name stems from the work . In addition to these results, we present numerical results for all other values of and when . Finally, we also consider the special case of and and state a conjecture regarding the function for . Based on the results of the present paper, Table 1 gives the state of affairs concerning the solutions of (4) and is an invitation to initiate a broader discussion on the problem.
The rest of the paper is organized as follows. Section 2 states some basic results of the -frame potential including some asymptotic results as . Section 3 presents the results for . Section 4 presents conjectures and numerical results for the case . Throughout the paper, we will use for the index set .
|: ONB+ ||: ONB+ |
|: ETF ||: see Conjecture 4.5|
|: copies of ONB ||: copies of ONB |
|(Theorem 3.7)||ETF if exists [13, 20]|
|Any : |
|: (Theorem 3.7)|
ONB+ refers to an orthonormal basis with a repeated vector. See Definition 4.1(a).
2. Some basic results
Intuitively, minimizing the frame potential amounts to promoting big angles among vectors. Consequently, it is expected that the optimal configurations will be at least a frame whose vectors are reasonably spread out in the sphere. If is not a frame, then one can always find a vector that is orthogonal to , and replacing any vector in by won’t increase the frame potential. In other words, it is trivial to show that problem (4) might as well be restricted to frames. The following result shows something stronger, that is, it excludes the possibility that a minimizer doesn’t span .
For , any optimal configuration of (4) is a frame of .
We first consider the case . Suppose not, and say is a minimizer so that is a strict subset of . Because there are vectors, it is possible to select two indices and such that . Finally, select any unit-norm vector and replace with ; i.e., define . A direct computation shows that .
Now consider the case and let be a minimizer of . Suppose that the dimension of . Choose a unit vector . There could be multiple pairs of vectors that achieve the maximal inner product . Without loss of generality, we assume these vectors are among the first vectors, that is,
We will construct that will have smaller coherence.
For , let , where . Define
If we choose such that
We will pick iteratively to satisfy (12):
Step 1: pick arbitrarily.
Step : given , pick such that . This is possible because for all .
This is a contradiction, so the optimal configuration must be a frame. ∎
Now we establish the relationship between large and .
. Moreover, if is an optimal configuration for (4) when and is a cluster point of the set , then optimizes the coherence as .
On one hand, we have
On the other hand,
Taking the limit of both inequalities gives us the desired limit.
Next, we establish a continuity result of .
The minimal frame potential is a continuous and non-increasing function of .
We first prove that the function is non-increasing. Letting , for any ,
For continuity, we have
which comes from applying the inequality for to every nonzero term in the frame potential.
Therefore , which implies the continuity of . ∎
Next, for fixed , we consider the asymptotics of as the number of points grows. In particular, we show that , see Proposition 2.6. We note that this behavior was numerically observed in . We begin by establishing some preliminary results.
Given , and , the sequence is a non-decreasing sequence.
Let be an optimal configuration for . For each ,
Summing (16) over , we obtain
It follows that exists. In fact, in the minimal energy literature, is called the transfinite diameter due to Fekete. Furthermore, is related to the continuous version of the frame potential, which is introduced in . More specifically, given a probabilistic measure on the sphere, the probabilistic frame potential is defined as
Let be the collection of all probabilistic measures on the sphere. Simple compactness and continuity arguments show that
Given any point configuration , its normalized counting measure is defined as
Consequently, if is an optimal configuration, i.e., , then by (19), it is plausible that . This is indeed the case, and it was proved in a more general setting by Farkas and Nagy . For the sake of completeness, we reproduce their proof below.
Given and , .
Let be the optimal probabilistic measure, that is,
The result follows by dividing on both sides and taking the limit. ∎
We can now state and prove that as .
Given and , we have . Moreover, if is a sequence of -point configurations such that , then every weak star cluster point of the normalized counting measure solves (18), that is . In particular, this holds for any sequence of the optimal configurations of .
The exact value of can be found in many cases. We list two examples in the following corollary.
(a) When and , we have .
(b) When and is an even integer, we have .
(b) In dimension , it is known that equally spaced points on the unit circle are spherical -design ([24, Section 4]), so Proposition 1.1 implies that is an optimal configuration if is an even integer. In other words, with fixed even integer , when is large enough, is going to be a -design (hence -design), so the equality in Proposition 1.1 holds and we get the desired result. ∎
3. Optimal configurations in dimension
This section focuses on the case , when the points are on the unit circle .
3.1. A class of minimal energy problems
We recall that when is even and , the solution to (4) was given in [13, Theorem 3.5], where it was established that the minimizers are copies of any orthonormal basis of . The case was settled by Benedetto and Fickus . In order to address other values of , we will consider a more general problem
where is a nonnegative and decreasing function, and is a dimensional circle with radius . This circle does not need to be centered at 0 and could be in any dimension. It will become clear later why we require points on a general circle instead of the usual .
The first result only requires to be convex, but it only works for up to 4 points.
Given , let be a decreasing convex function. Any configuration of equally spaced points on is an optimal configuration of (20). If in addition, is strictly convex, then no other -point configuration is optimal.
Let be an arbitrary configuration with ordered counter clockwise. Let be the angle between and for any . The index of the vectors is cyclic as . Then It is evident that . Using the convexity of ,
Next, let . In order to minimize the right hand side of (21), we solve
When , we let for short. Using Lagrange multipliers, we have , which implies that
If we are in the case that (or any pair with ), then . If we are in the other case that , then So for ,
and the equality holds when for some .
When , it is obvious that
with equality at , for all . This implies that for some .
When , which reduces to the case.
In summary, for any ,
and the equality holds simultaneously when , or equivalently .
Following (21), we have
It is easy to check that four equally spaced points on achieve this minimum.
If is strictly convex, then the inequality of (21) becomes equality if for every , which only holds for equally spaced points. ∎
The proof of Theorem 3.1 breaks down for because is not maximized at equally spaced points.
Our second result regarding (20) is a variation of the main result of the work by Cohn and Kumar [11, Theorem 1.2]. Let be a positive integer. An -sharp configuration is a spherical -design with inner products between its distinct points. It was proven in  that sharp configurations are the unique universal optimal configurations of the problem
for completely monotonic functions . A function is called -completely monotonic if for all and all , and strictly -completely monotonic if strict inequality always holds in the interior of . The notion -completely monotonic is simply called completely monotonic as traditionally defined, which means for all and all . A list of known sharp configurations was given in . For example, equally spaced points on is an -sharp configuration.
Another notion that we will need is that of absolutely monotonic functions. A function is called -absolutely monotonic if for all and all . Similarly, -absolutely monotonic means the inequality is true for all nonnegative integers , and will be simply referred to as absolutely monotonic. It is straightforward that being completely monotonic is equivalent to being absolutely monotonic.
As remarked by , the complete monotonicity on can be weakened slightly. To ensure a good flow of the paper, the proof of the next result which is a variation of [11, Theorem 1.2] will be given in the appendix.
A direct consequence of Theorem 3.3 for dimension is that equally spaced points are optimal configurations if the energy kernel function