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

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

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

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

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

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

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?

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

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

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

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

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 , 1 2 𝑒 2 ,…, 1 𝑛 𝑒 𝑛 . Kannan−Lovász `88: 𝐶 𝐾𝐿,2 𝑛 =𝑂( 𝑛 )

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 , 1 2 𝑒 2 ,…, 1 𝑛 𝑒 𝑛 . Kannan−Lovász `88: 𝐶 𝐾𝐿,2 𝑛 =𝑂( log 𝑛 )

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 3 2 𝑛 ) Questions: 1. Can we compute KL projections? 2. Is there a better characterization?

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 }

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

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

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

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 )

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

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 !

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.

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

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 ⊥

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

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.

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.

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.

Kannan-Lovász (KL) Conjecture 𝑃= 1 0 0 1 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]