1. Daniel Dadush Centrum Wiskunde & Informatica (CWI) Aussois 2019
On Approximating the Covering Radius and Finding Dense Lattice Subspaces
Daniel Dadush
Centrum Wiskunde & Informatica (CWI)
Aussois 2019

2. Lattices A lattice ℒ⊆ ℝ 𝑛 is 𝐵 ℤ 𝑛 for a basis 𝐵= 𝑏 1 ,…, 𝑏 𝑛 .
ℒ(𝐵) denotes the
lattice generated by 𝐵.
Note: a lattice has many equivalent bases.
𝑏 1
𝑏 2 ℒ

3. Lattices A lattice ℒ⊆ ℝ 𝑛 is 𝐵 ℤ 𝑛 for a basis 𝐵= 𝑏 1 ,…, 𝑏 𝑛 .
ℒ(𝐵) denotes the lattice generated by 𝐵.
The determinant of
ℒ is | det 𝐵 | .
𝑏 1
𝑏 2 ℒ

4. Lattices A lattice ℒ⊆ ℝ 𝑛 is 𝐵 ℤ 𝑛 for a basis 𝐵= 𝑏 1 ,…, 𝑏 𝑛 .
ℒ(𝐵) denotes the lattice generated by 𝐵.
The determinant of
ℒ is | det 𝐵 | .
Equal to volume of
any tiling set.
𝑏 1
𝑏 2 ℒ

5. Notation 𝑀⊆ℒ sublattice of dimension 𝑘 Normalized Determinant:
nd 𝑀 ≔ det 𝑀 1 𝑘
Projected Sublattice:
ℒ 𝑀 ≔ Π span 𝑀 ⊥ (ℒ)

6. ℓ 2 Covering Radius 𝜇 ℒ ≔𝜇 𝐵 2 𝑛 ,ℒ
𝜇 ℒ ≔𝜇 𝐵 2 𝑛 ,ℒ
Distance of farthest point to the lattice ℒ. 𝜇 𝒱 ℒ
Voronoi cell 𝒱≔ all points closest to 0

7. Distance of farthest point to the lattice ℒ.
ℓ 2 Covering Radius
𝜇 ℒ ≔𝜇 𝐵 2 𝑛 ,ℒ
Distance of farthest point to the lattice ℒ. 𝜇 𝒱 ℒ
Question: Does the covering radius admit a “good” dual characterization?

8. Volumetric Lower Bounds
vol 𝑛 𝐵 2 𝑛 𝜇 ℒ ≥ vol 𝑛 𝒱 = det (ℒ) 𝜇 𝒱 ℒ

9. Volumetric Lower Bounds
𝜇(ℒ) ≥ vol 𝑛 𝐵 2 𝑛 − 1 𝑛 det ℒ 1 𝑛 𝜇 𝒱 ℒ

10. Volumetric Lower Bounds
𝜇 ℒ ≿ 𝑛 det ℒ 1 𝑛 ≔ dim ℒ nd(ℒ) 𝜇 𝒱 ℒ

11. Volumetric Lower Bounds
𝜇 ℒ ≥𝜇 ℒ/𝑀 ≿ dim ℒ 𝑀 nd(ℒ/𝑀) ℒ
𝜇 ℒ/𝑀 𝑀

12. ℓ 2 Kannan-Lovász Constant
Define 𝐶 𝐾𝐿,2 (𝑛) to be smallest number such that
𝜇 ℒ ≤ 𝐶 𝐾𝐿,2 (𝑛) max 𝑀⊆ℒ dim ℒ 𝑀 nd(ℒ/𝑀)
for all lattices of dimension at most 𝑛.
𝐶 𝐾𝐿,2 𝑛 =Ω( log 𝑛 ) : Basis 𝑒 1 , 𝑒 2 ,…, 1 𝑛 𝑒 𝑛 .
Kannan−Lovász `88: 𝐶 𝐾𝐿,2 𝑛 =𝑂( 𝑛 )

13. ℓ 2 Kannan-Lovász Conjecture
Define 𝐶 𝐾𝐿,2 (𝑛) to be smallest number such that
𝜇 ℒ ≤ 𝐶 𝐾𝐿,2 (𝑛) max 𝑀⊆ℒ dim ℒ 𝑀 nd(ℒ/𝑀)
for all lattices of dimension at most 𝑛.
𝐶 𝐾𝐿,2 𝑛 =Ω( log 𝑛 ) : Basis 𝑒 1 , 𝑒 2 ,…, 1 𝑛 𝑒 𝑛 .
Kannan−Lovász `88: 𝐶 𝐾𝐿,2 𝑛 =𝑂( log 𝑛 )

14. ℓ 2 KL Conjecture Resolution
D. Regev 16: Reduction to Reverse Minkowski Conj.
If all sublattices of ℒ have determinant at least 1
then ∀𝑟>0 ℒ∩𝑟 𝐵 2 𝑛 ≤exp⁡(polylog 𝑛 𝑟 2 ).
Regev-S. Davidowitz 17:
Reverse Minkowski proved.
𝐶 𝐾𝐿,2 𝑛 =𝑂( log 𝑛 )
Questions: 1. Can we compute KL projections? Is there a better characterization?

15. Canonical Polygon [Stuhler 76]
𝑛 dimensional lattice ℒ
𝒫(ℒ)
Log det
(𝑛, log det ℒ )
(0,0)
dim. 1 2
𝑛-1 𝑛
{ 𝑘, log det 𝑀 :sublattice 𝑀⊆ℒ, dim 𝑀 =𝑘 }

16. Canonical Filtration [Stuhler 76]
𝑛 dimensional lattice ℒ ℒ
Log det
Vertices of 𝒫(ℒ)
ℒ 2
(𝑛, log det ℒ )
(0,0)
ℒ 1
{0}
dim. 1 2
𝑛-1 𝑛
Form Chain: 0 = ℒ 0 ⊂ ℒ 1 ⊂⋯⊂ ℒ 𝑘 =ℒ

17. Canonical Filtration [Stuhler 76]
𝑛 dimensional lattice ℒ
Slope: ln nd( ℒ 2 ℒ 1 ) ℒ
Log det
ℒ 2
(𝑛, log det ℒ )
(0,0)
ℒ 1
{0}
dim. 1 2
𝑛-1 𝑛
Form Chain: 0 = ℒ 0 ⊂ ℒ 1 ⊂⋯⊂ ℒ 𝑘 =ℒ

18. Stable Lattice [Stuhler 76]
𝑛 dimensional lattice ℒ is stable
I.e. no dense sublattices ℒ
Log det
(𝑛, log det ℒ )
(0,0)
{0}
dim. 1 2
𝑛-1 𝑛
If canonical filtration is trivial: 0 ⊂ℒ

19. ℓ 2 KL for stable lattices
𝐶 𝑛 := max ℒ 𝜇 ℒ dim ℒ , where ℒ ranges over all
stable lattices of det 1 and dimension ≤𝑛.
Conjecture [Shapira-Weiss 16]:
𝐶 𝑛 =𝑂(1) ( ℤ 𝑛 is worst-case)
Regev-S. Davidowitz 17: 𝐶 𝑛 =𝑂( log 𝑛 )

20. Canonical filtration and ℓ 2 KL
Regev-S. Davidowitz 17:
For the canonical filtration 0 = ℒ 0 ⊂ ℒ 1 ⊂⋯⊂ ℒ 𝑘 =ℒ
we have that
𝜇 ℒ 2 ≤ 𝐶 𝑛 2 𝑖∈ 𝑘 dim ( ℒ 𝑖 ℒ 𝑖−1 ) nd ℒ 𝑖 ℒ 𝑖−1 2
≤ log 𝑛 𝐶 𝑛 2 max i∈[𝑘] dim ( ℒ ℒ 𝑖−1 ) nd ℒ ℒ 𝑖−1 2

21. Above NP certificate tight up to 𝐶 𝑛 !
Main Results
Theorem [D. 18]:
1. Can compute 𝑂( log 𝑛 ) ``approximate’’ canonical filtration in 2 𝑂(𝑛) time.
2. For any chain 0 = ℒ 0 ⊂ ℒ 1 ⊂⋯⊂ ℒ 𝑘 =ℒ
𝜇 ℒ 2 ≿ 𝑖∈ 𝑘 dim ( ℒ 𝑖 ℒ 𝑖−1 ) nd ℒ ℒ 𝑖−1 2
Above NP certificate tight up to 𝐶 𝑛 !

22. Densest Sublattice Problem
𝜏 ℒ ≔ min 𝑀⊆ℒ 𝑀≠{0} nd(𝑀)
𝛼-DSP: Given ℒ find 𝑀⊆ℒ, 𝑀≠{0} such that nd 𝑀 ≤𝛼 𝜏(ℒ).
Remark: dimension of 𝑀 is not fixed!
Key primitive for finding sparse lattice projections.

23. Densest Sublattice Problem
𝑛 dimensional lattice ℒ
Want sublattice with slope
≤ log 𝛼 + log nd ( ℒ 1 ) ℒ
Log det
ℒ 2
(𝑛, log det ℒ )
(0,0)
ℒ 1
{0}
dim. 1 2
𝑛-1 𝑛

24. Densest Sublattice Problem
Theorem:
Can solve 𝑂( log 𝑛 )-DSP in 2 𝑂(𝑛) time with high probability.
High Level Approach:
If ℒ is not approximate minimizer:
find 𝑦≠0, orthogonal to actual minimizer, and recurse on ℒ∩ 𝑦 ⊥

25. Densest Sublattice Problem
High Level Approach:
If ℒ is not approximate minimizer:
find 𝑦≠0, orthogonal to actual minimizer, and recurse on ℒ∩ 𝑦 ⊥
Q: Where to find 𝑦?
A: The dual lattice ℒ ∗
Q: How to find it in ℒ ∗ ?
A: Discrete Gaussian sampling

26. Discrete Gaussian Distribution
6:30-8:30
Aharonov and Regev use the periodic Gaussian function. You can see how oracle access to this can be helpful.

27. Discrete Gaussian Distribution
Aggarwal-D.-Regev-S. Davidowitz: `15: Can sample in 2 𝑛+𝑜(𝑛) time for any parameter.
6:30-8:30
Aharonov and Regev use the periodic Gaussian function. You can see how oracle access to this can be helpful.

28. Conclusions Open Problem Algorithmic version of ℓ 2 KL conjecture.
Lower bound certificates for covering radius that are conjecturally tight within 𝑂(1).
Open Problem
Prove that ℓ 2 KL for stable lattices is 𝑂(1).
Prove KL conjecture for general convex bodies.

29. Kannan-Lovász (KL) Conjecture𝑃= 𝐾
If 𝐾∩ ℤ 𝑛 =∅ then ∃ 𝑘∈[𝑛], 𝑃∈ ℤ 𝑘×𝑛 , 𝑟𝑎𝑛𝑘 𝑃 =𝑘
such that vol 𝑃𝐾 1 𝑘 ≤𝑂( log 𝑛 ).
[Kannan `87, Kannan-Lovász `88]