A Delsarte-Style Proof of the Bukh-Cox Bound

02/03/2019 ∙ by Mark Magsino, et al. ∙ 0

The line packing problem is concerned with the optimal packing of points in real or complex projective space so that the minimum distance between points is maximized. Until recently, all bounds on optimal line packings were known to be derivable from Delsarte's linear program. Last year, Bukh and Cox introduced a new bound for the line packing problem using completely different techniques. In this paper, we use ideas from the Bukh--Cox proof to find a new proof of the Welch bound, and then we use ideas from Delsarte's linear program to find a new proof of the Bukh--Cox bound. Hopefully, these unifying principles will lead to further refinements.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

I Introduction

The last decade has seen a surge of progress in the line packing problem, where the objective is to pack points in or so that the minimum distance is maximized. Indeed, while instances of this problem date back to Tammes [1] and Fejes Tóth [2], the substantial progress in recent days has been motivated by emerging applications in compressed sensing [3], digital fingerprinting [4], quantum state tomography [5], and multiple description coding [6]. Most progress in this direction has come from identifying new packings that achieve equality in the so-called Welch bound (see [7] for a survey), but last year, Bukh and Cox [8] discovered a completely different bound, along with a large family of packings that achieve equality in their bound.

Focusing on the complex case, let

be a sequence of unit vectors in

, and define coherence to be

Then the Buhk–Cox bound guarantees

provided . The Bukh–Cox bound is the best known lower bound on coherence in the regime where . While the other lower bounds can be proven using Delsarte’s linear program [9]

, the proof of the Bukh–Cox bound is completely different: it hinges on an upper bound on the first moment of isotropic measures.

In the present paper, we provide an alternate proof of the Bukh–Cox bound. We start by isolating a lemma of Bukh and Cox that identifies a fundamental duality between the coherence of unit vectors in dimensions and the entrywise -norm of the Gram matrix of -tight frames of vectors in dimensions. Next, we illustrate the power of this lemma by using it to find a new proof of the Welch bound. Finally, we combine the lemma with ideas from Delsarte’s linear program to obtain a new proof of the Bukh–Cox bound. This new proof helps to unify the existing theory of line packing, and hopefully, it will spur further improvements (say, by leveraging ideas from semidefinite programming [10]).

Ii The Bukh–Cox Lemma

Let denote any sequence in . By abuse of notation, we identify with the matrix whose th column is . We say is a -tight frame if . Let denote the set of matrices in with unit norm columns, and let denote the set of matrices in corresponding to -tight frames. Define

(Indeed, the maximum exists by a compactness argument.) We say is equiangular if there exists a constant such that for every . With this nomenclature, we are ready to state the following lemma of Bukh and Cox:

Lemma 1.

thm: minmax coherence Let . Then every satisfies

(1)

Furthermore, minimizes over if is equiangular and there exists such that

  1. maximizes over ,

  2. , and

  3. for .

Proof.

Given , select such that . Then,

(2)

Bringing to the left hand side of (2) and taking absolute values, we have that

(3)
(4)

where (3) uses the triangle inequality and (4) is by the definition of coherence. Using (4) and ,

Thus, we conclude that

(5)

This proves the bound. Considering (2), equality occurs in (3), (4) and (5) when is equiangular and (i)–(iii) holds. ∎

Iii The Welch Bound

Theorem 2.

lemma: welch y1 For all we have

(6)

Equality is achieved if and only if is an equiangular tight frame.

Proof.

First we separate the diagonal part of and use the Cauchy–Schwarz inequality,

(7)

Noting that the the sum in (7) is (almost) and once again using the Cauchy–Schwarz inequality,

(8)
(9)

Putting this back into (7) we obtain the inequality

(10)

Equality is achieved in the Cauchy–Scwharz inequality if and only if the vectors are scalar multiples. For the first instance of Cauchy–Schwarz, this occurs if and only if is equiangular. For the second instance of Cauchy–Schwarz, equality is achieved if and only if is constant, that is . Thus, equality is achieved in (6) if and only if is an equiangular tight frame. ∎

Equality in (6) depends on the existence of equiangular tight frames of vectors in . The Gerzon bound says that if forms an equiangular tight frame of vectors in , then  [7]. This gives the bound as a necessary condition for to be an equiangular tight frame. On the other hand, if in Lemma 1 is an equiangular tight frame, then is also an equiangular tight frame since and are Naimark complements [11]. In particular, this gives the upper bound as a necessary condition for to be equiangular, because the Gerzon bound applied to gives the requirement that .

Being within the range is not a sufficient condition. The existence of equiangular tight frames of vectors in dimensions for is an open question. Some equiangular tight frames are known to exist, by construction, for certain values of and . An overview of the known constructions can be found in [7]. On the other hand, there are known values of and for which equiangular tight frames cannot exist. One such example is the case when and [12]. In particular, equality in the Welch bound is achieved for some values of and which satisfy , but not necessarily achieved at all values of and which satisfy that inequality.

By combining Lemma LABEL:thm:_minmax_coherence with Theorem LABEL:lemma:_welch_y1, we obtain

Corollary 3 (Welch).

Let . For all ,

(11)

Iv Bukh–Cox Bound via Linear Programming

We now turn our attention to the range . Since the Gerzon bound prevents from being an equiangular tight frame in this range, equality in (6) cannot be achieved and a sharper bound can be obtained. The Bukh–Cox bound is an improvement in this range, and is sharp if is given by concatenated copies of vectors in which forms an equiangular tight frame. In order to apply Delsarte’s linear programming ideas, we require the following special polynomials [9]:

Theorem 4.

lemma: bukh-cox y1 For all we have

(12)

Equality is achieved when is of the form where is an equiangular tight frame.

Fig. 1: The coherence of best known line packings in for , along with the best known lower bounds. The black dots correspond to packings found in Sloane’s database [13]. The Bukh–Cox bound is displayed in green, the Welch bound in blue, the orthoplex bound [14] in pink, and the Levenshtein bound [15] in red. In this setting, the Bukh–Cox bound is the best known lower bound for .
Proof.

By continuity, we may assume for every without loss of generality. First, we normalize the columns of , , by defining . We obtain the desired bound considering the following linear program, inspired by Delsarte’s LP bound,

minimize
subject to

Suppose we have a feasible . Then,

(13)

We now establish an upper bound for each innermost term for . For , since , we have

(14)

For , we have

(15)
(16)

To bound the first term of (16), we use Cauchy–Schwarz and the fact that is an -tight frame:

(17)

Overall this gives the following bound for the case:

(18)

Last, we need a bound for the case. Let be any orthonormal basis for the (finite) -dimensional vector space spanned by degree- projective harmonic polynomials in variables. Then, by the addition formula, there is a constant , which depends on and , such that

(19)

Multiplying both sides of (19) by then gives

(20)

Finally, returning to (13) we have

(21)

where we have used that . The bound (12) comes from observing that the following choice of is feasible:

This feasible choice comes from forcing , , and , and solving for . Equality is achieved in inequalities (13), (18), (20), (21) when

  1. ,

  2. ,

  3. ,

  4. .

All four are achieved if is multiple copies of an equiangular tight frame of vectors in . ∎

The proof of lemma: bukh-cox y1 actually generates a bound for any feasible in the described linear program. Minimizing gives the best possible bound generated by this method. Computational experiments suggest that this particular feasible gives the minimum . Although equality is achieved when is multiple copies of an equiangular tight frame of vectors in , the existence of such frames is an open question, and is known as Zauner’s conjecture [16].

By combining Lemma LABEL:thm:_minmax_coherence with Theorem LABEL:lemma:_bukh-cox_y1, we obtain

Corollary 5 (Bukh–Cox).

Let . For all ,

Bukh and Cox also provide a new bound for the case of vectors in . For the real case, it suffices to adjust the special polynomials in the proof of lemma: bukh-cox y1 [9]. and remain the same, but is replaced with:

This adjustment changes the feasible region for the linear program and leads to a different optimal , and thus a different bound for the case. Fig. 1 demonstrates the Bukh–Cox bound improvement over the Welch bound for small values of in the case where the vectors are in . (We illustrate the real case since, in this case, packing data is available and provided in [13].)

Acknowledgements

We thank the anonymous reviewers for helpful comments that improved the presentation of our results. MM and DGM were partially supported by AFOSR FA9550-18-1-0107. DGM was also supported by NSF DMS 1829955 and the Simons Institute of the Theory of Computing.

References

  • [1] P. M. L. Tammes, “On the origin of number and arrangement of the places of exit on the surface of pollen-grains,” Recueil des travaux botaniques néerlandais, vol. 27, no. 1, pp. 1–84, 1930.
  • [2] L. Tóth, “Distribution of points in the elliptic plane,” Acta Mathematica Hungarica, vol. 16, no. 3-4, pp. 437–440, 1965.
  • [3] A. S. Bandeira, M. Fickus, D. G. Mixon, and P. Wong, “The road to deterministic matrices with the restricted isometry property,” Journal of Fourier Analysis and Applications, vol. 19, no. 6, pp. 1123–1149, 2013.
  • [4] D. G. Mixon, C. J. Quinn, N. Kiyavash, and M. Fickus, “Fingerprinting with equiangular tight frames,” IEEE Transactions on Information Theory, vol. 59, no. 3, pp. 1855–1865, 2013.
  • [5] J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric informationally complete quantum measurements,” Journal of Mathematical Physics, vol. 45, no. 6, pp. 2171–2180, 2004.
  • [6] T. Strohmer and R. W. Heath Jr, “Grassmannian frames with applications to coding and communication,” Applied and computational harmonic analysis, vol. 14, no. 3, pp. 257–275, 2003.
  • [7] M. Fickus and D. G. Mixon, “Tables of the existence of equiangular tight frames,” arXiv preprint arXiv:1504.00253, 2015.
  • [8] B. Bukh and C. Cox, “Nearly orthogonal vectors and small antipodal spherical codes,” arXiv preprint arXiv:1803.02949, 2018.
  • [9] P. Delsarte, J. Goethals, and J. Seidel, “Bounds for systems of lines, and Jacobi polynomials,” in Geometry and Combinatorics.   Elsevier, 1991, pp. 193–207.
  • [10] D. de Laat and F. Vallentin, “A semidefinite programming hierarchy for packing problems in discrete geometry,” Mathematical Programming, vol. 151, no. 2, pp. 529–553, 2015.
  • [11] P. G. Casazza, M. Fickus, D. G. Mixon, J. Peterson, and I. Smalyanau, “Every Hilbert space frame has a Naimark complement,” Journal of Mathematical Analysis and Applications, vol. 406, no. 1, pp. 111–119, 2013.
  • [12] F. Szöllősi, “All complex equiangular tight frames in dimension 3,” arXiv preprint arXiv:1402.6429, 2014.
  • [13] N. J. A. Sloane, “Packings in Grassmannian spaces,” http://neilsloane.com/grass/.
  • [14] J. H. Conway, R. H. Hardin, and N. J. Sloane, “Packing lines, planes, etc.: Packings in Grassmannian spaces,” Experimental mathematics, vol. 5, no. 2, pp. 139–159, 1996.
  • [15] J. I. Haas, N. Hammen, and D. G. Mixon, “The Levenstein bound for packings in projective spaces,” in Wavelets and Sparsity XVII, vol. 10394.   International Society for Optics and Photonics, 2017, p. 103940V.
  • [16] G. Zauner, “Grundzüge einer nichtkommutativen Designtheorie,” Ph.D. dissertation, PhD thesis, University of Vienna, 1999. Published in English translation  , 2011.