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k -point semidefinite programming bounds for equiangular lines Fabr - - PowerPoint PPT Presentation

May 06, 2023 548 likes •1.21k views

Problem: equiangular lines and spherical codes A hierarchy of k -point bounds for packing problems Parameterizing invariant kernels on the sphere Results k -point semidefinite programming bounds for equiangular lines Fabr cio C. Machado

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

k-point semidefinite programming bounds for equiangular lines

Fabr´ ıcio C. Machado (joint work with D. de Laat, F.M. de Oliveira Filho, and F. Vallentin) arXiv:1812.06045

Instituto de Matem´ atica e Estat´ ıstica Universidade de S˜ ao Paulo, Brasil

SIAM Conference on Applied Algebraic Geometry Bern, July 9-13, 2019

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Acknowledgements

F.C.M was supported by grant #2017/25237-4, from the S˜ ao Paulo Research Foundation (FAPESP) and was financed in part by the Coordena¸ c˜ ao de Aperfei¸ coamento de Pessoal de N´ ıvel Superior — Brasil (CAPES).

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Contents

1 Problem: equiangular lines and spherical codes 2 A hierarchy of k-point bounds for packing problems

Independent sets The bound ∆k(G)∗

3 Parameterizing invariant kernels on the sphere

T(x, y, Q) and O(n)-invariance Gegenbauer polynomials P n

l

Fixing a set Q ∈ Ik−2

4 Results

a = 1/11, 1/9, 1/7, 1/5

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

1 Problem: equiangular lines and spherical codes

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Equiangular lines

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Equiangular lines

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Equiangular lines

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Equiangular lines

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Equiangular lines

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Equiangular lines

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Equiangular lines

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Equiangular lines

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Equiangular lines

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Equiangular lines

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Equiangular lines

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Equiangular lines

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Equiangular lines

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Spherical codes

x · y := n

i=1 xiyi

Sn−1 := {x ∈ Rn : x · x = 1} D ⊆ [−1, 1) - set of allowable inner products Examples: D = {−a, a} equiangular lines with com- mon angle arccos a |D| = s spherical s-distance sets D = [−1, a] packing of spherical caps

A(n, D) := max

  • |C| : C ⊆ Sn−1, x · y ∈ D for all distinct x, y ∈ C
  • Fabr´

ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 19

6/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Spherical codes

x · y := n

i=1 xiyi

Sn−1 := {x ∈ Rn : x · x = 1} D ⊆ [−1, 1) - set of allowable inner products Examples: D = {−a, a} equiangular lines with com- mon angle arccos a |D| = s spherical s-distance sets D = [−1, a] packing of spherical caps

A(n, D) := max

  • |C| : C ⊆ Sn−1, x · y ∈ D for all distinct x, y ∈ C
  • Fabr´

ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 20

6/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Spherical codes

x · y := n

i=1 xiyi

Sn−1 := {x ∈ Rn : x · x = 1} D ⊆ [−1, 1) - set of allowable inner products Examples: D = {−a, a} equiangular lines with com- mon angle arccos a |D| = s spherical s-distance sets D = [−1, a] packing of spherical caps

A(n, D) := max

  • |C| : C ⊆ Sn−1, x · y ∈ D for all distinct x, y ∈ C
  • Fabr´

ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 21

6/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Spherical codes

x · y := n

i=1 xiyi

Sn−1 := {x ∈ Rn : x · x = 1} D ⊆ [−1, 1) - set of allowable inner products Examples: D = {−a, a} equiangular lines with com- mon angle arccos a |D| = s spherical s-distance sets D = [−1, a] packing of spherical caps

A(n, D) := max

  • |C| : C ⊆ Sn−1, x · y ∈ D for all distinct x, y ∈ C
  • Fabr´

ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 22

6/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Spherical codes

x · y := n

i=1 xiyi

Sn−1 := {x ∈ Rn : x · x = 1} D ⊆ [−1, 1) - set of allowable inner products Examples: D = {−a, a} equiangular lines with com- mon angle arccos a |D| = s spherical s-distance sets D = [−1, a] packing of spherical caps

A(n, D) := max

  • |C| : C ⊆ Sn−1, x · y ∈ D for all distinct x, y ∈ C
  • Fabr´

ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 23

6/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Spherical codes

x · y := n

i=1 xiyi

Sn−1 := {x ∈ Rn : x · x = 1} D ⊆ [−1, 1) - set of allowable inner products Examples: D = {−a, a} equiangular lines with com- mon angle arccos a |D| = s spherical s-distance sets D = [−1, a] packing of spherical caps

A(n, D) := max

  • |C| : C ⊆ Sn−1, x · y ∈ D for all distinct x, y ∈ C
  • Fabr´

ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 24

6/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results

Spherical codes

x · y := n

i=1 xiyi

Sn−1 := {x ∈ Rn : x · x = 1} D ⊆ [−1, 1) - set of allowable inner products Examples: D = {−a, a} equiangular lines with com- mon angle arccos a |D| = s spherical s-distance sets D = [−1, a] packing of spherical caps

A(n, D) := max

  • |C| : C ⊆ Sn−1, x · y ∈ D for all distinct x, y ∈ C
  • Fabr´

ıcio C. Machado k-point SDP bounds for equiangular lines

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7/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results Independent sets The bound ∆k(G)∗

2 A hierarchy of k-point bounds for packing problems

Independent sets The bound ∆k(G)∗

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results Independent sets The bound ∆k(G)∗

Independent sets

Let G = (V, E) be a graph. α(G) := max{|S| : S is a independent set} Let Ik be the set of independent sets in G of size at most k. (I=k for size exactly k.) ∅ ∈ Ik, I=1 ≃ V

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 27

8/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results Independent sets The bound ∆k(G)∗

Independent sets

Let G = (V, E) be a graph. α(G) := max{|S| : S is a independent set} Let Ik be the set of independent sets in G of size at most k. (I=k for size exactly k.) ∅ ∈ Ik, I=1 ≃ V

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 28

8/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results Independent sets The bound ∆k(G)∗

Independent sets

Let G = (V, E) be a graph. α(G) := max{|S| : S is a independent set} Let Ik be the set of independent sets in G of size at most k. (I=k for size exactly k.) ∅ ∈ Ik, I=1 ≃ V

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 29

8/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results Independent sets The bound ∆k(G)∗

Independent sets

Let G = (V, E) be a graph. α(G) := max{|S| : S is a independent set} Let Ik be the set of independent sets in G of size at most k. (I=k for size exactly k.) ∅ ∈ Ik, I=1 ≃ V

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 30

8/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results Independent sets The bound ∆k(G)∗

Independent sets

Let G = (V, E) be a graph. α(G) := max{|S| : S is a independent set} Let Ik be the set of independent sets in G of size at most k. (I=k for size exactly k.) ∅ ∈ Ik, I=1 ≃ V

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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8/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results Independent sets The bound ∆k(G)∗

Independent sets

α(G) := max{|S| : S is a independent set} Let Ik be the set of independent sets in G of size at most k. (I=k for size exactly k.) ∅ ∈ Ik, I=1 ≃ V Upper bounds for α(G) can be computed with semidefinite programming (the Lov´ asz theta number, Lasserre hierarchy)

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results Independent sets The bound ∆k(G)∗

A graph in the sphere

V = Sn−1 E = {{x, y} ⊆ V : x · y / ∈ D} Independent sets = spherical codes α(G) = A(n, D) Assuming D is closed, it is a “topological packing graph” and α(G) is finite. Automorphism group: O(n)

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 33

9/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results Independent sets The bound ∆k(G)∗

A graph in the sphere

V = Sn−1 E = {{x, y} ⊆ V : x · y / ∈ D} Independent sets = spherical codes α(G) = A(n, D) Assuming D is closed, it is a “topological packing graph” and α(G) is finite. Automorphism group: O(n)

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 34

9/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results Independent sets The bound ∆k(G)∗

A graph in the sphere

V = Sn−1 E = {{x, y} ⊆ V : x · y / ∈ D} Independent sets = spherical codes α(G) = A(n, D) Assuming D is closed, it is a “topological packing graph” and α(G) is finite. Automorphism group: O(n)

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 35

9/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results Independent sets The bound ∆k(G)∗

A graph in the sphere

V = Sn−1 E = {{x, y} ⊆ V : x · y / ∈ D} Independent sets = spherical codes α(G) = A(n, D) Assuming D is closed, it is a “topological packing graph” and α(G) is finite. Automorphism group: O(n)

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 36

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results Independent sets The bound ∆k(G)∗

∆k(G)∗

α(G) ≤ ∆k(G)∗ ∆k(G)∗ := inf{ 1 + λ : λ ∈ R, T ∈ C(V 2 × Ik−2)0, and BkT ≤ λχI=1 − 2χI=2 }

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 37

10/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results Independent sets The bound ∆k(G)∗

∆k(G)∗

α(G) ≤ ∆k(G)∗ ∆k(G)∗ := inf{ 1 + λ : λ ∈ R, T ∈ C(V 2 × Ik−2)0, and BkT ≤ λχI=1 − 2χI=2 } T ∈ C(V 2 × Ik−2)0 A function T : V 2 × Ik−2 → R is in C(V 2 × Ik−2)0 if it is continuous and for every Q ∈ Ik−2 and finite U ⊆ V the ma- trix

  • T(x, y, Q)
  • x,y∈U is positive semidefinite.

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results Independent sets The bound ∆k(G)∗

∆k(G)∗

α(G) ≤ ∆k(G)∗ ∆k(G)∗ := inf{ 1 + λ : λ ∈ R, T ∈ C(V 2 × Ik−2)0, and BkT ≤ λχI=1 − 2χI=2 } Bk : C(V 2 × Ik−2) → C(Ik \ {∅}) BkT(S) :=

  • Q⊆S

|Q|≤k−2

  • x,y∈S

Q∪{x,y}=S

T(x, y, Q).

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results Independent sets The bound ∆k(G)∗

∆k(G)∗

Bk : C(V 2 × Ik−2) → C(Ik \ {∅}) BkT(S) :=

  • Q⊆S

|Q|≤k−2

  • x,y∈S

Q∪{x,y}=S

T(x, y, Q). Example: If S = {a, b}, then B3T({a,b}) = T(a, b, ∅) + T(b, a, ∅) + T(a, b, {a}) + T(b, a, {a}) + T(b, b, {a}) + T(a, b, {b}) + T(b, a, {b}) + T(a, a, {b}).

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 40

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results Independent sets The bound ∆k(G)∗

∆k(G)∗

α(G) ≤ ∆k(G)∗ ∆k(G)∗ := inf{ 1 + λ : λ ∈ R, T ∈ C(V 2 × Ik−2)0, and BkT ≤ λχI=1 − 2χI=2 } BkT(S) ≤      λ if |S| = 1, −2 if |S| = 2, if 3 ≤ |S| ≤ k.

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 41

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results Independent sets The bound ∆k(G)∗

∆k(G)∗

α(G) ≤ ∆k(G)∗ ∆k(G)∗ := inf{ 1 + λ : λ ∈ R, T ∈ C(V 2 × Ik−2)0, and BkT ≤ λχI=1 − 2χI=2 }

Let C ⊆ V be a nonempty independent set and let (λ, T) be a feasible solution of ∆k(G)∗.

  • S⊆C

|S|≤k, S=∅

BkT(S) ≤ |C| 1

  • λ +

|C| 2

  • (−2) = |C|(1 + λ − |C|).
  • S⊆C

|S|≤k, S=∅

BkT(S) =

  • S⊆C

|S|≤k, S=∅

  • Q⊆S

|Q|≤k−2

  • x,y∈S

Q∪{x,y}=S

T(x, y, Q) =

  • Q⊆C

|Q|≤k−2

  • x,y∈C

T(x, y, Q) ≥ 0.

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 42

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results T (x, y, Q) and O(n)-invariance Gegenbauer polynomials P n

l

Fixing a set Q ∈ Ik−2

3 Parameterizing invariant kernels on the sphere

T(x, y, Q) and O(n)-invariance Gegenbauer polynomials P n

l

Fixing a set Q ∈ Ik−2

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 43

12/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results T (x, y, Q) and O(n)-invariance Gegenbauer polynomials P n

l

Fixing a set Q ∈ Ik−2

T(x, y, Q) and O(n)-invariance

Due the symmetries in the problem, we consider functions T : Sn−1 × Sn−1 × Ik−2 → R such that T(γx, γy, γQ) = T(x, y, Q) ∀ x, y ∈ Sn−1, Q ∈ Ik−2, γ ∈ O(n). These functions depend only on the inner products between x, y, and Q. ⇒ The complexity of the computations depend only on k and not on n.

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 44

12/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results T (x, y, Q) and O(n)-invariance Gegenbauer polynomials P n

l

Fixing a set Q ∈ Ik−2

T(x, y, Q) and O(n)-invariance

Due the symmetries in the problem, we consider functions T : Sn−1 × Sn−1 × Ik−2 → R such that T(γx, γy, γQ) = T(x, y, Q) ∀ x, y ∈ Sn−1, Q ∈ Ik−2, γ ∈ O(n). These functions depend only on the inner products between x, y, and Q. ⇒ The complexity of the computations depend only on k and not on n.

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 45

12/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results T (x, y, Q) and O(n)-invariance Gegenbauer polynomials P n

l

Fixing a set Q ∈ Ik−2

T(x, y, Q) and O(n)-invariance

Due the symmetries in the problem, we consider functions T : Sn−1 × Sn−1 × Ik−2 → R such that T(γx, γy, γQ) = T(x, y, Q) ∀ x, y ∈ Sn−1, Q ∈ Ik−2, γ ∈ O(n). These functions depend only on the inner products between x, y, and Q. ⇒ The complexity of the computations depend only on k and not on n.

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

slide-46
SLIDE 46

13/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results T (x, y, Q) and O(n)-invariance Gegenbauer polynomials P n

l

Fixing a set Q ∈ Ik−2

T(x, y, Q) and O(n)-invariance

Fix Q ∈ Ik−2 and define KQ : Sn−1 × Sn−1 → R, KQ(x, y) := T(x, y, Q). T(γx, γy, γQ) = T(x, y, Q) ∀γ ∈ O(n) ⇒ KQ(γx, γy) = KQ(x, y) ∀γ ∈ StabO(n)(Q) StabO(n)(Q) := {γ ∈ O(n) : γQ = Q}

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 47

13/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results T (x, y, Q) and O(n)-invariance Gegenbauer polynomials P n

l

Fixing a set Q ∈ Ik−2

T(x, y, Q) and O(n)-invariance

Fix Q ∈ Ik−2 and define KQ : Sn−1 × Sn−1 → R, KQ(x, y) := T(x, y, Q). T(γx, γy, γQ) = T(x, y, Q) ∀γ ∈ O(n) ⇒ KQ(γx, γy) = KQ(x, y) ∀γ ∈ StabO(n)(Q) StabO(n)(Q) := {γ ∈ O(n) : γQ = Q}

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 48

13/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results T (x, y, Q) and O(n)-invariance Gegenbauer polynomials P n

l

Fixing a set Q ∈ Ik−2

T(x, y, Q) and O(n)-invariance

Fix Q ∈ Ik−2 and define KQ : Sn−1 × Sn−1 → R, KQ(x, y) := T(x, y, Q). T(γx, γy, γQ) = T(x, y, Q) ∀γ ∈ O(n) ⇒ KQ(γx, γy) = KQ(x, y) ∀γ ∈ StabO(n)(Q) StabO(n)(Q) := {γ ∈ O(n) : γQ = Q}

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 49

13/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results T (x, y, Q) and O(n)-invariance Gegenbauer polynomials P n

l

Fixing a set Q ∈ Ik−2

T(x, y, Q) and O(n)-invariance

Fix Q ∈ Ik−2 and define KQ : Sn−1 × Sn−1 → R, KQ(x, y) := T(x, y, Q). T(γx, γy, γQ) = T(x, y, Q) ∀γ ∈ O(n) ⇒ KQ(γx, γy) = KQ(x, y) ∀γ ∈ StabO(n)(Q) StabO(n)(Q) := {γ ∈ O(n) : γQ = Q}

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 50

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Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results T (x, y, Q) and O(n)-invariance Gegenbauer polynomials P n

l

Fixing a set Q ∈ Ik−2

Gegenbauer polynomials P n

l

The Gegenbauer polynomials P n

l can be recursively defined as

P n

0 (t) := 1, P n 1 (t) := t, and

P n

l (t) := (n + 2l − 4)tP n l−1(t) − (l − 1)P n l−2(t)

n + l − 3 . Positive property For any finite U ⊆ Sn−1, the matrix

  • P n

l (x · y)

  • x,y∈U is positive

semidefinite. For a0 . . . , ad ≥ 0, (x, y) → d

l=0 alP n l (x · y) is O(n)-invariant and

has the positive property (Schoenberg’s theorem)

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

slide-51
SLIDE 51

14/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results T (x, y, Q) and O(n)-invariance Gegenbauer polynomials P n

l

Fixing a set Q ∈ Ik−2

Gegenbauer polynomials P n

l

The Gegenbauer polynomials P n

l can be recursively defined as

P n

0 (t) := 1, P n 1 (t) := t, and

P n

l (t) := (n + 2l − 4)tP n l−1(t) − (l − 1)P n l−2(t)

n + l − 3 . Positive property For any finite U ⊆ Sn−1, the matrix

  • P n

l (x · y)

  • x,y∈U is positive

semidefinite. For a0 . . . , ad ≥ 0, (x, y) → d

l=0 alP n l (x · y) is O(n)-invariant and

has the positive property (Schoenberg’s theorem)

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

slide-52
SLIDE 52

14/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results T (x, y, Q) and O(n)-invariance Gegenbauer polynomials P n

l

Fixing a set Q ∈ Ik−2

Gegenbauer polynomials P n

l

The Gegenbauer polynomials P n

l can be recursively defined as

P n

0 (t) := 1, P n 1 (t) := t, and

P n

l (t) := (n + 2l − 4)tP n l−1(t) − (l − 1)P n l−2(t)

n + l − 3 . Positive property For any finite U ⊆ Sn−1, the matrix

  • P n

l (x · y)

  • x,y∈U is positive

semidefinite. For a0 . . . , ad ≥ 0, (x, y) → d

l=0 alP n l (x · y) is O(n)-invariant and

has the positive property (Schoenberg’s theorem) Q = ∅

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

slide-53
SLIDE 53

15/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results T (x, y, Q) and O(n)-invariance Gegenbauer polynomials P n

l

Fixing a set Q ∈ Ik−2

Q = ∅?

|Q| = 1 — Bachoc and Vallentin (2008) |Q| > 1 — Musin (2014)

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

slide-54
SLIDE 54

16/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results T (x, y, Q) and O(n)-invariance Gegenbauer polynomials P n

l

Fixing a set Q ∈ Ik−2

L L⊥ Q x y x′ y′

L := span(Q), m := dim(L). Let C′ ⊆ Sn−m−1 be the projection of the spherical code C

  • nto L⊥.

x′ · y′ can be computed in terms of the inner products between x, y, and Q. Positive and StabO(n)(Q)-invariant function can be defined with C′ and the polynomials P n−m

l

.

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

slide-55
SLIDE 55

16/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results T (x, y, Q) and O(n)-invariance Gegenbauer polynomials P n

l

Fixing a set Q ∈ Ik−2

L L⊥ Q x y x′ y′

L := span(Q), m := dim(L). Let C′ ⊆ Sn−m−1 be the projection of the spherical code C

  • nto L⊥.

x′ · y′ can be computed in terms of the inner products between x, y, and Q. Positive and StabO(n)(Q)-invariant function can be defined with C′ and the polynomials P n−m

l

.

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

slide-56
SLIDE 56

16/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results T (x, y, Q) and O(n)-invariance Gegenbauer polynomials P n

l

Fixing a set Q ∈ Ik−2

L L⊥ Q x y x′ y′

L := span(Q), m := dim(L). Let C′ ⊆ Sn−m−1 be the projection of the spherical code C

  • nto L⊥.

x′ · y′ can be computed in terms of the inner products between x, y, and Q. Positive and StabO(n)(Q)-invariant function can be defined with C′ and the polynomials P n−m

l

.

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

slide-57
SLIDE 57

16/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results T (x, y, Q) and O(n)-invariance Gegenbauer polynomials P n

l

Fixing a set Q ∈ Ik−2

L L⊥ Q x y x′ y′

L := span(Q), m := dim(L). Let C′ ⊆ Sn−m−1 be the projection of the spherical code C

  • nto L⊥.

x′ · y′ can be computed in terms of the inner products between x, y, and Q. Positive and StabO(n)(Q)-invariant function can be defined with C′ and the polynomials P n−m

l

.

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

slide-58
SLIDE 58

17/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results a = 1/11, 1/9, 1/7, 1/5

4 Results

a = 1/11, 1/9, 1/7, 1/5

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

slide-59
SLIDE 59

18/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results a = 1/11, 1/9, 1/7, 1/5

Results D = {−1/11, 1/11}

100 200 300 400 500 dimension 10000 20000 30000 40000 bound

a = 1/11 Thm 5.1 [Gerzon] ∆3(G) ∗ [Barg, Yu - King, Tang] ∆4(G) ∗ ∆5(G) ∗ ∆6(G) ∗ Thm 5.11 [Glazyrin, Yu]

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

slide-60
SLIDE 60

19/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results a = 1/11, 1/9, 1/7, 1/5

Results D = {−1/9, 1/9}

50 100 150 200 250 300 350 400 dimension 5000 10000 15000 20000 25000 30000 bound

a = 1/9 Thm 5.1 [Gerzon] ∆3(G) ∗ [Barg, Yu - King, Tang] ∆4(G) ∗ ∆5(G) ∗ ∆6(G) ∗ Thm 5.11 [Glazyrin, Yu]

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

slide-61
SLIDE 61

20/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results a = 1/11, 1/9, 1/7, 1/5

Results D = {−1/7, 1/7}

50 100 150 200 250 300 350 400 dimension 5000 10000 15000 20000 25000 bound

a = 1/7 Thm 5.1 [Gerzon] ∆3(G) ∗ [Barg, Yu - King, Tang] ∆4(G) ∗ ∆5(G) ∗ ∆6(G) ∗ Thm 5.11 [Glazyrin, Yu]

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

slide-62
SLIDE 62

21/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results a = 1/11, 1/9, 1/7, 1/5

Results D = {−1/5, 1/5}

50 100 150 200 250 300 dimension 1000 2000 3000 4000 5000 6000 7000 bound

a = 1/5 Thm 5.1 [Gerzon] ∆3 (G) ∗ [Barg, Yu - King, Tang] ∆4 (G) ∗ ∆5 (G) ∗ ∆6 (G) ∗ Thm 5.12 + ∆3 (G) ∗ [King, Tang] Thm 5.12 + ∆5 (G) ∗ Thm 5.12 + ∆6 (G) ∗ Thm 5.12 + Thm 5.10 [Glazyrin, Yu - Lin, Yu]

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

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SLIDE 63

22/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results a = 1/11, 1/9, 1/7, 1/5

Questions

What is the smallest n such that Ma(n) = (1/a2 − 2)(1/a2 − 1)/2? (the “stable range” in the plots) What is the smallest n such that Ma(n) > (1/a2 − 2)(1/a2 − 1)/2? When a = 1/5, Lemmens and Seidel (1973) conjectures that n > 185, the SDP bound shows n > 60 and we show n > 70.

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

slide-64
SLIDE 64

22/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results a = 1/11, 1/9, 1/7, 1/5

Questions

What is the smallest n such that Ma(n) = (1/a2 − 2)(1/a2 − 1)/2? (the “stable range” in the plots) What is the smallest n such that Ma(n) > (1/a2 − 2)(1/a2 − 1)/2? When a = 1/5, Lemmens and Seidel (1973) conjectures that n > 185, the SDP bound shows n > 60 and we show n > 70.

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines

slide-65
SLIDE 65

23/23

Problem: equiangular lines and spherical codes A hierarchy of k-point bounds for packing problems Parameterizing invariant kernels on the sphere Results a = 1/11, 1/9, 1/7, 1/5

Thank you for your attention!

Fabr´ ıcio Caluza Machado fabcm1@gmail.com Mathematics and Statistics Institute, University of S~ ao Paulo, Brazil

Fabr´ ıcio C. Machado k-point SDP bounds for equiangular lines