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On Approximating the Covering Radius and Finding Dense Lattice Subspaces Daniel Dadush Centrum Wiskunde & Informatica (CWI) ICERM April 2018 Outline 1. Integer Programming and the Kannan-Lovsz (KL) Conjecture. 2. 2 KL Conjecture

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SLIDE 1

On Approximating the Covering Radius and Finding Dense Lattice Subspaces

Daniel Dadush

Centrum Wiskunde & Informatica (CWI) ICERM April 2018

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SLIDE 2

Outline

  • 1. Integer Programming and

the Kannan-LovΓ‘sz (KL) Conjecture.

  • 2. β„“2 KL Conjecture &

the Reverse Minkowski Conjecture.

  • 3. Finding dense lattice subspaces.
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SLIDE 3

Integer Programming (IP)

min𝑑𝑦 s.t. 𝐡𝑦 ≀ 𝑐 𝑦 πœ— β„€π‘œ π‘œ variables, 𝑛 constraints Open Question: Is there a 2𝑃(π‘œ) time algorithm? First result: 2𝑃(π‘œ2) [Lenstra `83] Best known complexity: π‘œπ‘ƒ(π‘œ) [Kannan `87]

𝐿 𝑑

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SLIDE 4

𝜈 𝐿, β„€π‘œ ≔ smallest scaling 𝑑 such that every shift 𝑑𝐿 + 𝑒 contains an integer point.

β„€2

𝐿

Main Dichotomy

Either covering radius 𝜈 𝐿, β„€π‘œ ≀ 1.

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SLIDE 5

𝐿

𝜈 𝐿, β„€π‘œ ≔ smallest scaling 𝑑 such that every shift 𝑑𝐿 + 𝑒 contains an integer point.

β„€2

Main Dichotomy

Either covering radius 𝜈 𝐿, β„€π‘œ ≀ 1.

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SLIDE 6

𝐿

𝜈 𝐿, β„€π‘œ ≔ smallest scaling 𝑑 such that every shift 𝑑𝐿 + 𝑒 contains an integer point.

β„€2

Main Dichotomy

Either covering radius 𝜈 𝐿, β„€π‘œ ≀ 1.

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SLIDE 7

𝜈 𝐿, β„€π‘œ ≔ smallest scaling 𝑑 such that every shift 𝑑𝐿 + 𝑒 contains an integer point.

β„€2

Main Dichotomy

Either covering radius 𝜈 𝐿, β„€π‘œ ≀ 1.

2 3𝐿

𝜈 𝐿, β„€2 = 2

3

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SLIDE 8

𝐿

β„€2

Main Dichotomy

Either covering radius 𝜈 𝐿, β„€π‘œ ≀ 1. Can find integer point in 2𝑃(π‘œ) time [D. 12]

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SLIDE 9

β„€2

Main Dichotomy

There exists rank 𝑙 β‰₯ 1 integer projection 𝑄 ∈ β„€π‘œΓ—π‘™ such vol𝑙 𝑄𝐿

1 𝑙 is small.

𝐿

Projection on 𝑧-axis 𝑄 = (0 1)

Or 𝐿 is β€œflat”:

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SLIDE 10

β„€2

Main Dichotomy

There exists rank 𝑙 β‰₯ 1 integer projection 𝑄 ∈ β„€π‘œΓ—π‘™ such vol𝑙 𝑄𝐿

1 𝑙 is small.

𝐿

Or 𝐿 is β€œflat”:

Projection on 𝑧-axis 𝑄 = (0 1)

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SLIDE 11

β„€2

Main Dichotomy

There exists rank 𝑙 β‰₯ 1 integer projection 𝑄 ∈ β„€π‘œΓ—π‘™ such vol𝑙 𝑄𝐿

1 𝑙 is small.

𝐿

Recurse on β‰ˆ volk PK subproblems [D. 12]

Or 𝐿 is β€œflat”:

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SLIDE 12

Duality Relation

1 ≀ 𝜈 𝐿, β„€π‘œ min

π‘„βˆˆβ„€π‘™Γ—π‘œ 𝑠𝑙 𝑄 =𝑙β‰₯1

vol𝑙 𝑄𝐿

1 𝑙 ≀ ?

β€œEasy” side β€œHard” side

Either covering radius 𝜈(𝐿, β„€π‘œ) is small

  • r 𝐿 is β€œflat”.
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SLIDE 13

Khinchine Flatness Theorem

1 ≀ 𝜈 𝐿, β„€π‘œ min

π‘„βˆˆβ„€1Γ—π‘œ 𝑠𝑙 𝑄 =1

vol1(𝑄𝐿) ≀ ΰ·© O(π‘œ Ξ€

4 3)

𝐿

[Khinchine `48, Babai `86, Hastad `86, Lenstra-Lagarias-

Schnorr `87, Kannan-Lovasz `88, Banaszczyk `93-96, Banaszczyk-Litvak-Pajor-Szarek `99, Rudelson `00]

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SLIDE 14

Kannan-LovΓ‘sz Flatness Theorem

1 ≀ 𝜈 𝐿, β„€π‘œ min

π‘„βˆˆβ„€π‘™Γ—π‘œ 𝑠𝑙 𝑄 =𝑙β‰₯1

vol𝑙 𝑄𝐿

1 𝑙 ≀ π‘œ

[Kannan `87, Kannan-LovÑsz `88] 𝐿

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SLIDE 15

Kannan-LovΓ‘sz (KL) Conjecture

1 ≀ 𝜈 𝐿, β„€π‘œ min

π‘„βˆˆβ„€π‘™Γ—π‘œ 𝑠𝑙 𝑄 =𝑙β‰₯1

vol𝑙 𝑄𝐿

1 𝑙 ≀ 𝑃 log π‘œ β€Ό

𝐿

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SLIDE 16

Faster Algorithm for IP?

1 ≀ 𝜈 𝐿, β„€π‘œ min

π‘„βˆˆβ„€π‘™Γ—π‘œ 𝑠𝑙 𝑄 =𝑙β‰₯1

vol𝑙 𝑄𝐿

1 𝑙 ≀ 𝑃 log π‘œ

  • D. `12: Assuming KL conjecture

+ 𝑄 computable in (log π‘œ)𝑃(π‘œ) time then there is log π‘œ 𝑃(π‘œ) time algorithm for IP.

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SLIDE 17

β„“2 Kannan-LovΓ‘sz Conjecture

Does the conjecture hold for ellipsoids? β„€π‘œ 𝐹 An ellipsoid is 𝐹 = π‘ˆπΆ2

π‘œ

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SLIDE 18

β„“2 Kannan-LovΓ‘sz Conjecture

Answer: YES* [Regev-S.Davidowitz 17] β„€π‘œ 𝐹

* up to polylogarithmic factors

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SLIDE 19

β„“2 Kannan-LovΓ‘sz Conjecture

Can we compute the projection P? β„€π‘œ 𝐹

THIS TALK: YES, in 2𝑃(π‘œ) time.

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SLIDE 20

Outline

  • 1. Integer Programming and

the Kannan-LovΓ‘sz (KL) Conjecture.

  • 2. β„“2 KL Conjecture &

the Reverse Minkowski Conjecture.

  • 3. Finding dense lattice subspaces.
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SLIDE 21

π‘ˆπΉ = 𝐢2

π‘œ

Easier to think of Euclidean ball vs general lattice. β„’ = π‘ˆβ„€π‘œ

β„“2 Kannan-LovΓ‘sz Conjecture

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SLIDE 22

A lattice β„’ βŠ† β„π‘œ is πΆβ„€π‘œ for a basis 𝐢 = 𝑐1, … , π‘π‘œ . β„’(𝐢) denotes the lattice generated by 𝐢. Note: a lattice has many equivalent bases.

𝑐2 𝑐2 𝑐1 𝑐2 𝑐1

Lattices

β„’

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SLIDE 23

A lattice β„’ βŠ† β„π‘œ is πΆβ„€π‘œ for a basis 𝐢 = 𝑐1, … , π‘π‘œ . β„’(𝐢) denotes the lattice generated by 𝐢. The determinant of β„’ is | det 𝐢 |.

Lattices

𝑐1 𝑐2 β„’

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SLIDE 24

Lattices

𝑐1 𝑐2

A lattice β„’ βŠ† β„π‘œ is πΆβ„€π‘œ for a basis 𝐢 = 𝑐1, … , π‘π‘œ . β„’(𝐢) denotes the lattice generated by 𝐢. The determinant of β„’ is | det 𝐢 |. Equal to volume of any tiling set.

β„’

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SLIDE 25

β„“2 Covering Radius

β„’

𝜈

𝜈 β„’ ≔ 𝜈 𝐢2

π‘œ, β„’

Distance of farthest point to the lattice β„’.

𝒲

Voronoi cell 𝒲 ≔ all points closer to 0

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SLIDE 26

Volumetric Lower Bounds

β„’

𝜈

volπ‘œ 𝐢2

π‘œπœˆ β„’

β‰₯ voln 𝒲 = det(β„’)

𝒲

Voronoi cell 𝒲 ≔ all points closer to 0

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SLIDE 27

Volumetric Lower Bounds

β„’

𝜈

𝜈(β„’) β‰₯ voln 𝐢2

π‘œ βˆ’1 π‘œ det β„’ 1 π‘œ

𝒲

Voronoi cell 𝒲 ≔ all points closer to 0

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SLIDE 28

Volumetric Lower Bounds

β„’

𝜈

𝜈(β„’) β‰Ώ π‘œ det β„’

1 π‘œ

𝒲

Voronoi cell 𝒲 ≔ all points closer to 0

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SLIDE 29

Volumetric Lower Bounds

β„’

𝜈

𝜈 β„’ β‰₯ 𝜈 ℒ↓𝑋 ℒ↓𝑋 projection onto 𝑋

𝑋

πœˆβ†“π‘‹

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SLIDE 30

Volumetric Lower Bounds

β„’

𝜈

𝜈 β„’ β‰₯ 𝜈 ℒ↓𝑋 ≳ 𝑙 det ℒ↓𝑋

1 𝑙

ℒ↓𝑋 projection onto 𝑋 dim(𝑋) = 𝑙 β‰₯ 1

𝑋

πœˆβ†“π‘‹

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SLIDE 31

Volumetric Lower Bounds

β„’

𝜈

𝜈 β„’ ≳ max

dim 𝑋 =𝑙β‰₯1

𝑙 det ℒ↓𝑋

1 𝑙

ℒ↓𝑋 projection onto 𝑋 dim(𝑋) = 𝑙 β‰₯ 1

𝑋

πœˆβ†“π‘‹

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SLIDE 32

β„“2 Kannan-LovΓ‘sz Conjecture

Define 𝐷𝐿𝑀,2(π‘œ) to be smallest number such that 𝜈 β„’ ≀ 𝐷𝐿𝑀,2(π‘œ) max

dim 𝑋 =𝑙β‰₯1

𝑙 det ℒ↓𝑋

1 𝑙

for all lattices of dimension at most π‘œ. 𝐷𝐿𝑀,2 π‘œ = Ξ©( log π‘œ) Lower bound for β„’ with basis 𝑓1,

1 2 𝑓2, … , 1 π‘œ π‘“π‘œ.

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SLIDE 33

KL Bounds

𝜈 β„’ ≀ 𝐷𝐿𝑀,2(π‘œ) max

dim 𝑋 =𝑙β‰₯1

𝑙 det ℒ↓𝑋

1 𝑙

Kannan-LovΓ‘sz `88: π‘œ

  • D. Regev `16: log𝑃(1) π‘œ

Assuming Reverse Minkowski Conjecture. Regev, S.Davidowitz `17: log Ξ€

3 2 π‘œ

Reverse Minkowski Conjecture is proved!

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SLIDE 34

Our Results

π‘œ dimensional lattice β„’ ≔ β„’(𝐢)

  • 1. Can compute subspace 𝑋, dim 𝑋 = 𝑙 β‰₯ 1

𝜈 β„’ ≀ 𝑃(log2.5 π‘œ) 𝑙 det ℒ↓𝑋

1 𝑙

in 2𝑃(π‘œ) time with high probability. Prior work: Kannan LovΓ‘sz `88: π‘œ in 2𝑃(π‘œ) time.

  • D. Micciancio `13: best subspace in π‘œπ‘ƒ(π‘œ2) time.
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SLIDE 35

Our Results

π‘œ dimensional lattice β„’ ≔ β„’(𝐢)

  • 2. Can combine lower bounds over different

subspaces to certify 𝜈 𝑀 up to the slicing constant π‘€π‘œ for β€œstable” Voronoi cells*. 𝒲 * If voln 𝒲 = 1 can find hyperplane 𝐼 s.t. volnβˆ’1 𝒲 ∩ 𝐼 = Ξ©( 1

π‘€π‘œ)

𝐼

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SLIDE 36

Our Results

π‘œ dimensional lattice β„’ ≔ β„’(𝐢)

  • 2. Can combine lower bounds over different

subspaces to certify 𝜈 𝑀 up to the slicing constant π‘€π‘œ for β€œstable” Voronoi cells*. 𝒲 Slicing Conjecture: π‘€π‘œ = 𝑃(1) for all convex bodies! For β€œstable” Voronoi cells: π‘€π‘œ = 𝑃(log π‘œ) [RS `17] 𝐼

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SLIDE 37

Notation

𝑁 βŠ† β„’ sublattice of dimension 𝑙 Convention: 𝑁 = {0} then det 𝑁 ≔ 1. Normalized Determinant: nd 𝑁 ≔ det 𝑁

Ξ€ 1 𝑙

Projected Sublattice: Ξ€ β„’ 𝑁 ≔ β„’ projected onto span 𝑁 βŠ₯

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SLIDE 38

Lower Bounds for Chains

Theorem [D. 17]: For 0 = β„’0 βŠ‚ β„’1 βŠ‚ β‹― βŠ‚ ℒ𝑙 = β„’ then 𝜈 β„’ 2 β‰Ώ σ𝑗=1

𝑙

dim( Ξ€ ℒ𝑗 β„’π‘—βˆ’1) nd Ξ€ β„’ β„’π‘—βˆ’1 2 Only β€œmissing ingredient”: Combined with techniques from [R.S. `17] easily get tightness within slicing constant π‘€π‘œ.

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SLIDE 39

Lower Bounds for Chains

Theorem [D. 17]: For 0 = β„’0 βŠ‚ β„’1 βŠ‚ β‹― βŠ‚ ℒ𝑙 = β„’ then 𝜈 β„’ 2 β‰Ώ σ𝑗=1

𝑙

dim( Ξ€ ℒ𝑗 β„’π‘—βˆ’1) nd Ξ€ β„’ β„’π‘—βˆ’1 2 Proof Idea:

  • 1. Establish SDP based lower bound: [D.R. `16]

𝜈 β„’ 2 β‰Ώ max σ𝑗 rk 𝑄𝑗 nd 𝑄𝑗ℒ 2 s.t. σ𝑗 𝑄𝑗

βˆ—π‘„π‘— β‰Ό π½π‘œ

  • 2. Build solution to above starting from any chain.
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SLIDE 40

Lattice Density

β„’ 𝑠

lim

π‘ β†’βˆž

β„’ ∩ 𝑠𝐢2

π‘œ

volπ‘œ(𝑠𝐢2

π‘œ) =

1 det β„’

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SLIDE 41

Lattice Density

β„’

lim

π‘ β†’βˆž

β„’ ∩ 𝑠𝐢2

π‘œ

volπ‘œ(𝑠𝐢2

π‘œ) =

1 det β„’

𝑠

Global density of lattice points per unit volume

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SLIDE 42

Minkowski’s First Theorem

1889

β„’

β„’ ∩ 𝑠𝐢2

π‘œ β‰₯ 2βˆ’π‘œ volπ‘œ(𝑠𝐢2 π‘œ)

det β„’

𝑠

Global density implies β€œlocal density”

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SLIDE 43

Reverse Minkowski Theorem

Regev-S.Davidowitz `17: β„’ lattice dimension π‘œ. If all sublattices of β„’ have determinant at least 1 then: β„’ has at most 2𝑃(log2 π‘œ 𝑠2) points at distance 𝑠. Almost tight: β„€π‘œ has π‘œΞ©(𝑙) points at distance 𝑠 for 𝑙 β‰ͺ π‘œ.

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SLIDE 44

Outline

  • 1. Integer Programming and

the Kannan-LovΓ‘sz (KL) Conjecture.

  • 2. β„“2 KL Conjecture &

the Reverse Minkowski Conjecture.

  • 3. Finding dense lattice subspaces.
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SLIDE 45

Notation

𝑁 βŠ† β„’ sublattice of dimension 𝑙 Convention: 𝑁 = {0} then det 𝑁 ≔ 1. Normalized Determinant: nd 𝑁 ≔ det 𝑁

Ξ€ 1 𝑙

Projected Sublattice: Ξ€ β„’ 𝑁 ≔ β„’ projected onto span 𝑁 βŠ₯

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SLIDE 46

Densest Subspace Problem

ndβˆ— β„’ ≔ min

π‘βŠ†β„’ 𝑁≠{0}

nd(𝑁) 𝛽-DSP: Given β„’ find 𝑁 βŠ† β„’, 𝑁 β‰  {0} such that nd 𝑁 ≀ 𝛽 ndβˆ—(β„’). Remark: dimension of 𝑁 is not fixed! Key primitive for finding sparse lattice

  • projections. Will focus on this problem.
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SLIDE 47

Densest Subspace Problem

Theorem: Can solve 𝑃(log π‘œ)-DSP in 2𝑃(π‘œ) time with high probability. High Level Approach: If β„’ is not approximate minimizer: find 𝑧 β‰  0, orthogonal to actual minimizer, and recurse on β„’ ∩ 𝑧βŠ₯

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SLIDE 48

Canonical Polytope [Stuhler 76]

π‘œ dimensional lattice β„’

1 dim. π‘œ π‘œ-1 2 { 𝑙, log det 𝑁 : sublattice 𝑁 βŠ† β„’, dim 𝑁 = 𝑙} (π‘œ, log det β„’) Log det (0,0) 𝒬(β„’)

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SLIDE 49

Canonical Filtration [Stuhler 76]

π‘œ dimensional lattice β„’

1 dim. π‘œ π‘œ-1 2 Form Chain: 0 = β„’0 βŠ‚ β„’1 βŠ‚ β‹― βŠ‚ ℒ𝑙 = β„’ (π‘œ, log det β„’) Log det (0,0) β„’1 β„’2 Vertices of 𝒬(β„’) β„’ {0}

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SLIDE 50

Canonical Filtration [Stuhler 76]

π‘œ dimensional lattice β„’

1 dim. π‘œ π‘œ-1 2 Form Chain: 0 = β„’0 βŠ‚ β„’1 βŠ‚ β‹― βŠ‚ ℒ𝑙 = β„’ (π‘œ, log det β„’) Log det (0,0) β„’1 β„’2 Slope: ln nd( Ξ€ β„’2 β„’1) β„’ {0}

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SLIDE 51

Stable Lattice [Stuhler 76]

π‘œ dimensional lattice β„’ is stable

1 dim. π‘œ π‘œ-1 2 If canonical filtration is trivial: 0 βŠ‚ β„’ (π‘œ, log det β„’) Log det (0,0) β„’ {0} I.e. no dense sublattices

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SLIDE 52

Stable Lattice [Stuhler 76]

Example: β„’ = β„€π‘œ

1 dim. π‘œ π‘œ-1 2 β„€π‘œ has trivial filtration: 0 βŠ‚ β„€π‘œ Log det (0,0) 𝒬(β„’) {0}

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SLIDE 53

Canonical Filtration [Stuhler 76]

  • 1. Form Chain: 0 = β„’0 βŠ‚ β„’1 βŠ‚ β‹― βŠ‚ ℒ𝑙 = β„’ .
  • 2. Blocks

Ξ€ ℒ𝑗 β„’π‘—βˆ’1 are stable.

  • 3. Slope increasing: nd

Ξ€ ℒ𝑗 β„’π‘—βˆ’1 < nd Ξ€ ℒ𝑗+1 ℒ𝑗 . 1 π‘œ π‘œ-1 2 β„’1 β„’2 β„’ {0}

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SLIDE 54

Densest Subspace Problem

π‘œ dimensional lattice β„’

1 dim. π‘œ π‘œ-1 2 (π‘œ, log det β„’) Log det (0,0) β„’1 β„’2 Want sublattice with approx. minimum slope. β„’ {0}

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SLIDE 55

Densest Subspace Problem

High Level Approach: If β„’ is not approximate minimizer: find 𝑧 β‰  0, orthogonal to actual minimizer, and recurse on β„’ ∩ 𝑧βŠ₯ Q: Where to find 𝑧? A: The dual lattice β„’βˆ— Q: How to find it in β„’βˆ—? A: Discrete Gaussian sampling

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SLIDE 56

𝑧 ∈ β„’βˆ—

A lattice β„’ βŠ† β„π‘œ is πΆβ„€π‘œ for a basis 𝐢 = 𝑐1, … , π‘π‘œ . The dual lattice is β„’βˆ— = {𝑧 ∈ span β„’ : 𝑧T𝑦 ∈ β„€ βˆ€π‘¦ ∈ β„’} β„’βˆ— is generated by πΆβˆ’T. Remark: β„€π‘œ βˆ— = β„€π‘œ

Dual Lattice

𝑧T𝑦 = 0 𝑧T𝑦 = 1 𝑧T𝑦 = 2 𝑧T𝑦 = 3 𝑧T𝑦 = 4

β„’

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SLIDE 57

𝑧 ∈ β„’βˆ—

A lattice β„’ βŠ† β„π‘œ is πΆβ„€π‘œ for a basis 𝐢 = 𝑐1, … , π‘π‘œ . The dual lattice is β„’βˆ— = {𝑧 ∈ span β„’ : 𝑧T𝑦 ∈ β„€ βˆ€π‘¦ ∈ β„’} β„’βˆ— is generated by πΆβˆ’T. det β„’βˆ— = 1/det(β„’)

Dual Lattice

𝑧T𝑦 = 0 𝑧T𝑦 = 1 𝑧T𝑦 = 2 𝑧T𝑦 = 3 𝑧T𝑦 = 4

β„’

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SLIDE 58

Discrete Gaussian Distribution

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SLIDE 59

Discrete Gaussian Distribution

ADRS `15: Can sample in 2π‘œ+𝑝(π‘œ) time for any parameter.

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SLIDE 60

Main Procedure

Repeat until β„’ = {0} 𝑑 ← nd β„’ Update 𝑁 ← β„’ if nd 𝑁 > 𝑑 Sample 𝑧 ∼ πΈβ„’βˆ—,𝑑/𝑑 until 𝑧 β‰  0 β„’ ← β„’ ∩ 𝑧βŠ₯ Main Lemma: At any iteration, if β„’ not 𝑃(log π‘œ) approximate minimizer, then β„’ ∩ 𝑧βŠ₯ contains minimizer w.p. Ξ© 1 .

  • Proc. finds apx minimizer with prob. 2βˆ’π‘ƒ(π‘œ).
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SLIDE 61

Proof of Main Lemma

wlog det β„’ = det(β„’βˆ—) = 1 β„’1 βŠ† β„’ densest sublattice sample 𝑧 ∼ πΈβ„’βˆ—,𝑑 If nd β„’1 β‰ͺ

1 log π‘œ

must show that 𝑧 β‰  0 and 𝑧 βŠ₯ β„’1 w.p. Ξ©(1).

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SLIDE 62

det β„’ = det β„’βˆ— = 1 β„’1 βŠ† β„’ densest sublattice, nd β„’1 β‰ͺ

1 log π‘œ

Sample 𝑧 ∼ πΈβ„’βˆ—,𝑑 Want:

  • 1. Pr 𝑧 = 0 ≀ πœ—
  • 2. Pr 𝑧 ∈ β„’βˆ— ∩ β„’1

βŠ₯ β‰₯ 1 βˆ’ πœ—

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SLIDE 63

det β„’ = det β„’βˆ— = 1 1. Pr

π‘§βˆΌπΈβ„’βˆ—,𝑑

𝑧 = 0 =

1 πœπ‘‘(β„’βˆ—)

≀

1 β„’βˆ—βˆ© π‘œπΆ2

π‘œ π‘“βˆ’ Ξ€ π‘œ 𝑑2

(By Minkowski) ≀

1 2π‘œπ‘“βˆ’ Ξ€

π‘œ 𝑑2 = 𝑝(1)

πœπ‘‘ 𝐡 ≔ Οƒπ‘¦βˆˆπ΅ π‘“βˆ’

Ξ€ 𝑦 𝑑 2

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SLIDE 64

det β„’ = det β„’βˆ— = 1 β„’1 βŠ† β„’ densest sublattice, nd β„’1 β‰ͺ Ξ€ 1 log π‘œ 𝑋 ≔ β„’1

βŠ₯

2. Pr

π‘§βˆΌπΈβ„’βˆ—,𝑑

𝑧 ∈ 𝑋 =

πœπ‘‘(β„’βˆ—βˆ© 𝑋) πœπ‘‘(β„’βˆ—)

(ortho. is worst-case) β‰₯

πœπ‘‘(β„’βˆ—βˆ©π‘‹) πœπ‘‘ β„’βˆ—βˆ©π‘‹ πœπ‘‘( Ξ€ β„’βˆ— 𝑋)

=

1 πœπ‘‘(β„’βˆ—/𝑋)

πœπ‘‘ 𝐡 ≔ Οƒπ‘¦βˆˆπ΅ π‘“βˆ’

Ξ€ 𝑦 𝑑 2

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SLIDE 65

det β„’ = det β„’βˆ— = 1 β„’1 βŠ† β„’ densest sublattice, nd β„’1 β‰ͺ Ξ€ 1 log π‘œ 𝑋 ≔ β„’1

βŠ₯

  • 2. Need to show πœπ‘‘

Ξ€ β„’βˆ— 𝑋 ≀ 1 + 𝑝(1) Key: ndβˆ— Ξ€ β„’βˆ— 𝑋 = Ξ€ 1 nd(β„’1) ≫ log π‘œ Reverse-Minkowski β‡’ ( Ξ€ β„’βˆ— 𝑋) ∩ 𝑠𝐢2

π‘œ β‰ͺ 𝑓𝑝 𝑠2 , βˆ€π‘  β‰₯ 0

πœπ‘‘ 𝐡 ≔ Οƒπ‘¦βˆˆπ΅ π‘“βˆ’

Ξ€ 𝑦 𝑑 2

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SLIDE 66

Canonical Polytope of β„’βˆ—?

Assumption: det β„’ = 1

1 dim. π‘œ π‘œ-1 2 (π‘œ, 0) Log det β„’1 β„’2 β„’ {0}

det β„’ = 1

Slope: ln nd(β„’1)

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SLIDE 67

Canonical Polytope of β„’βˆ—?

1 dim. π‘œ π‘œ-1 2 Log det β„’βˆ— ∩ β„’1

βŠ₯

β„’βˆ— ∩ β„’2

βŠ₯

(π‘œ, 0) β„’βˆ— {0}

det β„’βˆ— = 1 Map 𝑁 β†’ β„’βˆ— ∩ 𝑁βŠ₯

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SLIDE 68

Canonical Polytope of β„’βˆ—?

1 dim. π‘œ π‘œ-1 2 Log det β„’βˆ— ∩ β„’1

βŠ₯

β„’βˆ— ∩ β„’2

βŠ₯

(π‘œ, 0) β„’βˆ— {0}

𝒬(β„’βˆ—) is β€œreflection” of 𝒬(β„’)

Slope: βˆ’ln nd(β„’1) Ξ€ β„’βˆ— 𝑋

det β„’βˆ— = 1

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SLIDE 69

Conclusions

  • 1. Algorithmic version of β„“2 Kannan-LovΓ‘sz

conjecture via discrete Gaussian sampling.

  • 2. Lower bound certificates for covering radius that

are tight within 𝑃(1) under slicing conjecture.

Open Problem

  • 1. Prove KL conjecture for general convex bodies.
  • 2. Prove Slicing conjecture for Voronoi cells.