[Paper Review] New Upper Bounds for Equiangular Lines by Pillar Decomposition

Emily J. King, Xiaoxian Tang|arXiv (Cornell University)|Jun 10, 2016

graph theory and CDMA systems3 citations

TL;DR

This paper introduces a novel pillar decomposition method to derive tighter upper bounds on the maximum number of equiangular lines in ℝʳ, improving upon both Gerzon's bound and existing semidefinite programming (SDP) bounds. By analyzing inner products with negative cliques and leveraging combinatorial and linear algebra techniques, the authors establish explicit, non-trivial bounds that are significantly smaller than SDP results for larger dimensions, particularly for angles arccos(1/5) and arccos(1/7).

ABSTRACT

We derive a procedure for computing an upper bound on the number of equiangular lines in various Euclidean vector spaces by generalizing the classical pillar decomposition developed by (Lemmens and Seidel, 1973); namely, we use linear algebra and combinatorial arguments to bound the number of vectors within an equiangular set which have inner products of certain signs with a negative clique. After projection and rescaling, such sets are also certain spherical two-distance sets, and semidefinite programming techniques may be used to bound the size. Applying our method, we prove new relative bounds for the angle arccos(1/5). Experiments show that our relative bounds for all possible angles are considerably less than the known SDP bounds for a range of larger dimension r. Our computational results also show an explicit bound on the size of a set of equiangular lines regardless of angle, which is strictly less than the well-known Gerzon's bound if r+2 is not a square of an odd number.

Motivation & Objective

• To develop a generalized pillar decomposition method to compute tighter upper bounds on the maximum number of equiangular lines in ℝʳ.
• To improve upon the classical Gerzon bound and existing semidefinite programming (SDP) bounds for equiangular line sets.
• To provide explicit, computable upper bounds for sₐ(r) that are strictly less than Gerzon’s bound when r+2 is not the square of an odd integer.
• To demonstrate that the new bounds are consistently tighter than SDP bounds for larger dimensions, particularly for α = 1/5 and α = 1/7.
• To establish a framework that combines linear algebra, combinatorics, and semidefinite programming to bound spherical two-distance sets derived from equiangular lines.

Proposed method

• Generalizes Lemmens and Seidel’s classical pillar decomposition by analyzing vectors with specific inner product signs relative to a negative clique.
• Uses projection and rescaling to relate equiangular line sets to spherical two-distance sets, enabling application of semidefinite programming techniques.
• Applies combinatorial arguments to bound the size of equiangular sets based on inner product structure and clique properties.
• Derives explicit upper bounds via algebraic expressions involving m, the largest integer such that (2m+1)² ≤ r+2.
• Employs SDP solvers (e.g., sdpt3 via CVX 3.0) to compute and compare bounds, with results rounded to the nearest integer.
• Validates results through experiments across dimensions 44 ≤ r ≤ 400, comparing the new bounds with standard SDP bounds and Gerzon’s bound.

Experimental results

Research questions

• RQ1 Can pillar decomposition be generalized to yield tighter upper bounds on equiangular lines than existing semidefinite programming methods?
• RQ2 For which dimensions r is the new bound strictly less than Gerzon’s bound, and under what conditions does this occur?
• RQ3 How do the new bounds compare to SDP bounds for α = 1/5 and α = 1/7 across large r (e.g., r ≥ 44)?
• RQ4 Can the method produce non-trivial, explicit upper bounds for sₐ(r) that are computationally feasible and superior to SDP in high dimensions?
• RQ5 What is the performance of the new method when SDP solvers fail (e.g., for r > 250)?

Key findings

• For r = 137, the new bound is 2015, while the SDP bound is 9528, demonstrating a substantial improvement.
• The new bound is strictly less than Gerzon’s bound for all r between 44 and 400 where r+2 is not the square of an odd integer.
• For r = 44 to 46, 76 to 78, 117 to 118, 166, 222, 286, and 358, the bound is given by 4r(m+1)(m+2)/((2m+3)²−r), where m is the largest integer with (2m+1)² ≤ r+2.
• For other r between 44 and 400, the bound is ( (2m+1)²−2 ) ( (2m+1)²−1 ) / 2, again with m defined as above.
• The new bounds are consistently smaller than SDP bounds for r ≥ 94 (for α = 1/5) and r ≥ 235 (for α = 1/7), with the gap widening in higher dimensions.
• For α < 1/7, SDP bounds remain below Gerzon’s bound in the tested range (r ≤ 400), suggesting that the new method’s advantage is most pronounced for larger angles.