New Conjectures in the Geometry of Numbers Daniel Dadush Centrum - PowerPoint PPT Presentation

New Conjectures in the Geometry of Numbers Daniel Dadush Centrum Wiskunde & Informatica (CWI) Oded Regev New York University

Talk Outline 1. A Reverse Minkowski Inequality & its conjectured Strengthening. 2. Strong Reverse Minkowski implies the Kannan & Lovász conjecture for ℓ 2 . 3. From Decomposing Integer Programs to the general Kannan & Lovász conjecture.

Lattices The standard integer lattice ℤ 𝑜 . 𝑓 2 ℤ 𝑜 𝑓 1

Lattices A lattice ℒ ⊆ ℝ 𝑜 is 𝐶ℤ 𝑜 for a basis 𝐶 = 𝑐 1 , … , 𝑐 𝑜 . ℒ(𝐶) denotes the lattice generated by 𝐶 . 𝑐 2 𝑐 1 𝑐 1 Note: a lattice has many equivalent bases. 𝑐 2 𝑐 2 ℒ

Lattices A lattice ℒ ⊆ ℝ 𝑜 is 𝐶ℤ 𝑜 for a basis 𝐶 = 𝑐 1 , … , 𝑐 𝑜 . ℒ(𝐶) denotes the lattice generated by 𝐶 . 𝑐 1 The determinant of ℒ is | det 𝐶 | . 𝑐 2 ℒ

Lattices A lattice ℒ ⊆ ℝ 𝑜 is 𝐶ℤ 𝑜 for a basis 𝐶 = 𝑐 1 , … , 𝑐 𝑜 . ℒ(𝐶) denotes the lattice generated by 𝐶 . 𝑐 1 The determinant of ℒ is | det 𝐶 | . Equal to volume of 𝑐 2 any tiling set. ℒ

Lattices A lattice ℒ ⊆ ℝ 𝑜 is 𝐶ℤ 𝑜 for a basis 𝐶 = 𝑐 1 , … , 𝑐 𝑜 . The dual lattice is ℒ ∗ = {𝑧 ∈ span ℒ : 𝑧 T 𝑦 ∈ ℤ ∀𝑦 ∈ ℒ} ℒ ∗ is generated by 𝐶 −T . The identity ℒ det ℒ ∗ = Τ 1 det(ℒ) holds.

Lattices A lattice ℒ ⊆ ℝ 𝑜 is 𝐶ℤ 𝑜 for a basis 𝐶 = 𝑐 1 , … , 𝑐 𝑜 . The dual lattice is ℒ ∗ = {𝑧 ∈ span ℒ : 𝑧 T 𝑦 ∈ ℤ ∀𝑦 ∈ ℒ} ℒ ∗ is generated by 𝐶 −T . 𝑧 ∈ ℒ ∗ ℒ 𝑧 T 𝑦 = 0 𝑧 T 𝑦 = 1 𝑧 T 𝑦 = 2 𝑧 T 𝑦 = 3 𝑧 T 𝑦 = 4

Reversing Minkowski’s Convex Body Theorem

Minkowski’s Convex Body Theorem Theorem [Minkowski] : For an 𝑜 -dimensional symmetric convex body 𝐿 and lattice ℒ , we have that 2 −𝑜 vol 𝑜 𝐿 Τ 𝐿 ∩ ℒ ≥ ⌈ det ℒ ⌉ 𝐿 0 ℒ

Minkowski’s Convex Body Theorem Theorem [Minkowski] : For an 𝑜 -dimensional symmetric convex body 𝐿 and lattice ℒ , we have that 2 −𝑜 vol 𝑜 𝐿 Τ 𝐿 ∩ ℒ ≥ ⌈ det ℒ ⌉ Question: Is the above lower bound also “close” to being an upper bound?

Minkowski’s Convex Body Theorem Theorem [Minkowski] : For an 𝑜 -dimensional symmetric convex body 𝐿 and lattice ℒ , we have that 2 −𝑜 vol 𝑜 𝐿 Τ 𝐿 ∩ ℒ ≥ ⌈ det ℒ ⌉ Clearly NO! 𝐿 0 ℒ

Minkowski’s Convex Body Theorem Theorem [Minkowski] : For an 𝑜 -dimensional symmetric convex body 𝐿 and lattice ℒ , we have that 2 −𝑜 vol 𝑜 𝐿 Τ 𝐿 ∩ ℒ ≥ ⌈ det ℒ ⌉ Can we strengthen the lower bound? 𝐿 0 ℒ Points all lie in a lattice subspace.

Reverse Minkowski Inequality For a symmetric convex body 𝐿 and lattice ℒ , let 2 −𝑒 vol d (𝐿 ∩ 𝑋) MB 𝐿, ℒ = max max det(ℒ ∩ 𝑋) 𝑋 𝑚𝑏𝑢. 𝑡𝑣𝑐. 𝑒≥0 dim 𝑋 =𝑒 𝐿 0 𝑋 ℒ 𝑋 is a lattice subspace of ℒ if dim 𝑋 ∩ ℒ = dim(𝑋) .

Reverse Minkowski Inequality For a symmetric convex body 𝐿 and lattice ℒ , let 2 −𝑒 vol d (𝐿 ∩ 𝑋) MB 𝐿, ℒ = max max det(ℒ ∩ 𝑋) 𝑋 𝑚𝑏𝑢. 𝑡𝑣𝑐. 𝑒≥0 dim 𝑋 =𝑒 𝐿 0 𝑋 ℒ Is this bound any better?

Weak Reverse Minkowski For a symmetric convex body 𝐿 and lattice ℒ , let 2 −𝑒 vol d (𝐿 ∩ 𝑋) MB 𝐿, ℒ = max max det(ℒ ∩ 𝑋) 𝑋 𝑚𝑏𝑢. 𝑡𝑣𝑐. 𝑒≥0 dim 𝑋 =𝑒 𝑜 denote the unit Euclidean ball. Let 𝐶 2 Theorem [D.,Regev `16] : For an 𝑜 -dimensional lattice ℒ 𝑜 ∩ 𝑀 ≤ MB(6 𝑜𝐶 2 𝑜 , ℒ) . 𝐶 2 Furthermore, for any symmetric convex body 𝐿 𝐿 ∩ 𝑀 ≤ MB(6𝑜𝐿, ℒ) .

Strong Reverse Minkowski For a symmetric convex body 𝐿 and lattice ℒ , let 2 −𝑒 vol d (𝐿 ∩ 𝑋) MB 𝐿, ℒ = max max det(ℒ ∩ 𝑋) 𝑋 𝑚𝑏𝑢. 𝑡𝑣𝑐. 𝑒≥0 dim 𝑋 =𝑒 𝑜 denote the unit Euclidean ball. Let 𝐶 2 Conjecture [D.,Regev `16] : For an 𝑜 -dimensional lattice ℒ 𝑜 ∩ 𝑀 ≤ MB(𝑃( log 𝑜)𝐶 2 𝑜 , ℒ) . 𝐶 2 Furthermore, for any symmetric convex body 𝐿 𝐿 ∩ 𝑀 ≤ MB(𝑃(log 𝑜)𝐿, ℒ) . 𝑜 and ℒ = ℤ 𝑜 . Tight example: 𝐿 = 𝐶 1

Successive Minima Symmetric convex body 𝐿 and lattice ℒ in ℝ 𝑜 . 𝜇 𝑗 𝐿, ℒ = inf 𝑡 ≥ 0: dim ℒ ∩ 𝑡𝐿 ≥ 𝑗 , 𝑗 𝜗[𝑜] 𝑧 2 𝜇 2 𝐿 𝜇 1 𝐿 - 𝑧 1 𝑧 1 0 - 𝑧 2

Minkowski’s Second Theorem Symmetric convex body 𝐿 and lattice ℒ in ℝ 𝑜 . 𝑜 𝜇 𝑗 𝐿, ℒ ≤ 2 𝑜 det ℒ 𝑜 𝜇 𝑗 (𝐿, ℒ) Π 𝑗=1 ≤ 𝑜! Π 𝑗=1 vol 𝑜 𝐿 𝑧 2 𝜇 2 𝐿 𝜇 1 𝐿 - 𝑧 1 𝑧 1 0 - 𝑧 2

Lattice points bounds via Minima Theorem [Henk `02] : 𝑜 ⌊1 + 2 𝐿 ∩ ℒ ≤ 2 𝑜−1 ς 𝑗=1 𝜇 𝑗 𝐿,ℒ ⌋ 𝐿 0 ℒ

Proof of Weak Minkowski 2 −𝑒 vol d (6𝑜𝐿∩𝑋) Must show 𝐿 ∩ ℒ ≤ max max det(ℒ∩𝑋) 𝑋 𝑚𝑏𝑢. 𝑡𝑣𝑐. 𝑒≥0 dim 𝑋 =𝑒 If max 𝑗∈[𝑜] 𝜇 𝑗 𝐿, ℒ > 1 , then 𝐿 ∩ ℒ is lower dimensional and we can induct. If max 𝑗∈[𝑜] 𝜇 𝑗 𝐿, ℒ ≤ 1 , then we have that 𝐿 ∩ ℒ ≤ 2 𝑜−1 ς 𝑗=1 2 𝑜 1 + (by Henk) 𝜇 𝑗 𝐿,ℒ ≤ 2 𝑜−1 ς 𝑗=1 𝜇 𝑗 𝐿,ℒ ≤ 6 𝑜 ς 𝑗=1 3 1 𝑜 𝑜 𝜇 𝑗 𝐿,ℒ ≤ 𝑜! 3 𝑜 Τ vol n (K) det ℒ (by Minkowski) Τ ≤ vol n 3nK det ℒ ≤ MB(6𝑜𝐿, ℒ)

The Kannan & Lovász conjecture for ℓ 2

ℓ 2 covering radius 𝑜 , ℒ = max Let 𝜈 𝐶 2 𝑢∈ℝ 𝑜 min 𝑧∈ℒ 𝑢 − 𝑧 2 , i.e. the farthest distance from the lattice. 𝜈 ℒ Main question: How to get good lower bounds on the covering radius?

Lower bounds for covering radius 1 𝑜 , ℒ ≥ Lemma: 𝜈 𝐶 2 𝑜 ,ℒ ∗ ) 2𝜇 1 (𝐶 2 𝑢 𝑧 ∈ ℒ ∗ ℒ 𝑧 T 𝑦 = 1 𝑧 T 𝑦 = 2 𝑧 T 𝑦 = 3 𝑧 T 𝑦 = 4

Lower bounds for covering radius 1 det ℒ 𝑜 𝑜 , ℒ ≥ = Ω( 𝑜) det ℒ 1/𝑜 Lemma: 𝜈 𝐶 2 𝑜 1 vol n 𝐶 2 𝑜 𝜈 𝒲 ℒ 𝑜 , Since 𝒲 ⊆ 𝜈𝐶 2 𝑜 = 𝜈 𝑜 vol n 𝐶 2 𝑜 vol n 𝒲 = det(ℒ) ≤ vol n 𝜈B 2

Lower bounds for covering radius Lemma: Let 𝜌 𝑋 denote the orthogonal projection onto a 𝑒 -dimensional subspace 𝑋 (*). Then 1 𝑒 det 𝜌 𝑋 ℒ 𝑜 , ℒ ≥ 𝜈 𝐶 2 1 𝑒 𝑒 vol d 𝐶 2 1 𝑒 = = Ω 1 . 1 1 1 det ℒ ∗ ∩𝑋 𝑒 𝑒 𝑒 det ℒ ∗ ∩ 𝑋 𝑒 vol d 𝐶 2 Pf: For the first inequality, since orthogonal projections shrink distances we get 𝑜 , 𝜌 𝑋 ℒ 𝑜 , ℒ ≥ 𝜈 𝜌 𝑋 𝐶 2 𝜈 𝐶 2 . The second equality follows from the identity 𝜌 𝑋 ℒ ∗ = ℒ ∗ ∩ 𝑋 .

Kannan & Lovász for ℓ 2 Theorem [Kannan-Lovasz `88] : 1 det ℒ ∗ ∩𝑋 𝑒 𝑜 , ℒ Ω(1) ≤ 𝜈 𝐶 2 min ≤ 𝑃( 𝑜) 𝑒 𝑚𝑏𝑢.subspace W 1≤dim 𝑋 =𝑒≤𝑜

Kannan & Lovász for ℓ 2 Conjecture [Kannan-Lovász `88] : det ℒ ∗ ∩𝑋 1/𝑒 𝑜 , ℒ Ω 1 ≤ 𝜈 𝐶 2 min = O( log 𝑜) 𝑒 𝑚𝑏𝑢.subspace W 1≤dim 𝑋 =𝑒≤𝑜 Remark: implies that there are very good NP-certificates for showing that the covering radius is large.

Reverse Minkowski vs Kannan & Lovász 𝑜 ∩ 𝑋) 2 −𝑒 vol d (𝐶 2 𝑜 , ℒ = max MB 𝐶 2 max det(ℒ ∩ 𝑋) 𝑋 𝑚𝑏𝑢. 𝑡𝑣𝑐. 𝑒≥0 dim 𝑋 =𝑒 Theorem [D. Regev `16] : If for any 𝑜 -dimensional lattice ℒ 𝑜 ∩ ℒ ≤ MB(𝑔 𝑜 𝐶 2 𝑜 , ℒ) 𝐶 2 for a non-decreasing function 𝑔(𝑜) , then the Kannan & Lovász conjecture for ℓ 2 holds with bound 𝑃 log 𝑜 𝑔 𝑜 .

Main Approach 𝑜 , ℒ 2 : Use convex programming relaxation for 𝜈 𝐶 2 1. 𝑜 , ℒ 2 ≤ 𝑃 1 𝜈 𝐶 2 min trace 𝐵 s.t. σ 𝑧∈ℒ ∗ ∖{0} 𝑓 −𝑧 T 𝐵𝑧 ≤ 1, A psd. 2. Use Reverse Minkowski to formulate an approximate dual to the above program. 3. Round / massage optimal dual solution to get the subspace 𝑋 .

A Sufficient Conjecture 𝑜 , ℒ 2 ≤ 𝑃 1 𝜈 𝐶 2 min trace 𝐵 s.t. σ 𝑧∈ℒ ∗ ∖{0} 𝑓 −𝑧 T 𝐵𝑧 ≤ 1, A psd. To formulate the required dual for the above program, we can rely on the following weaker conjecture: Conjecture [D. Regev `16] : 𝑜 ∩ ℒ log 𝑠𝐶 2 1 Ω 1 ≤ max min det ℒ ∩ 𝑋 𝑒 𝑠 𝑚𝑏𝑢.𝑡𝑣𝑐. 𝑋 𝑠>0 1≤dim 𝑋 =𝑒 ≤ 𝑃 log 𝑜

The Relaxed Program min trace 𝐵 s.t. σ 𝑧∈ℒ ∗ ∖{0} 𝑓 −𝑧 T 𝐵𝑧 ≤ 1, A psd. We relax the above using the following weaker constraints 1 det 𝑋 𝐵 ≥ det ℒ ∗ ∩ 𝑋 2 ∀ 𝑚𝑏𝑢. 𝑡𝑣𝑐𝑡𝑞𝑏𝑑𝑓 𝑋 𝑈 𝐵𝑃 𝑋 ) , where 𝑃 𝑋 is any where det 𝑋 (𝐵) ≔ det(𝑃 𝑋 orthonormal basis of 𝑋 . This relaxation makes the value of the program drop by at most an 𝑔 𝑜 2 factor (the Reverse Minkowski bound).