On Approximating the Covering Radius and Finding Dense Lattice Subspaces

Transcription

1 On Approximating the Covering Radius and Finding Dense Lattice Subspaces Daniel Dadush Centrum Wiskunde & Informatica (CWI) ICERM April 2018

2 Outline 1. Integer Programming and the Kannan-Lovász (KL) Conjecture. 2. l 2 KL Conjecture & the Reverse Minkowski Conjecture. 3. Finding dense lattice subspaces.

3 Integer Programming (IP) min cx s.t. Ax b x ε Z n n variables, m constraints K c Open Question: Is there a 2 O(n) time algorithm? First result: 2 O(n2) [Lenstra `83] Best known complexity: n O(n) [Kannan `87]

4 Main Dichotomy μ K, Z n every shift smallest scaling s such that sk + t contains an integer point. K Z 2 Either covering radius μ K, Z n 1.

5 Main Dichotomy μ K, Z n every shift smallest scaling s such that sk + t contains an integer point. K Z 2 Either covering radius μ K, Z n 1.

6 Main Dichotomy μ K, Z n every shift smallest scaling s such that sk + t contains an integer point. K Z 2 Either covering radius μ K, Z n 1.

7 Main Dichotomy μ K, Z n every shift smallest scaling s such that sk + t contains an integer point. μ K, Z 2 = K Z 2 Either covering radius μ K, Z n 1.

8 Main Dichotomy Can find integer point in 2 O(n) time [D. 12] K Z 2 Either covering radius μ K, Z n 1.

9 Main Dichotomy Or K is flat : Projection on y-axis P = (0 1) K Z 2 There exists rank k 1 integer projection P Z n k such vol k PK 1 k is small.

10 Main Dichotomy Or K is flat : Projection on y-axis P = (0 1) K Z 2 There exists rank k 1 integer projection P Z n k such vol k PK 1 k is small.

11 Main Dichotomy Or K is flat : Recurse on vol k PK subproblems [D. 12] K Z 2 There exists rank k 1 integer projection P Z n k such vol k PK 1 k is small.

12 Duality Relation 1 μ K, Z n min P Z k n rk P =k 1 vol k PK 1 k? Easy side Hard side Either covering radius μ(k, Z n ) is small or K is flat.

13 Khinchine Flatness Theorem K 1 μ K, Z n 4 min vol P Z 1 n 1 (PK) O(n Τ3 ) rk P =1 [Khinchine `48, Babai `86, Hastad `86, Lenstra-Lagarias- Schnorr `87, Kannan-Lovasz `88, Banaszczyk `93-96, Banaszczyk-Litvak-Pajor-Szarek `99, Rudelson `00]

14 Kannan-Lovász Flatness Theorem K 1 μ K, Z n min P Z k n rk P =k 1 vol k PK 1 k n [Kannan `87, Kannan-Lovász `88]

15 Kannan-Lovász (KL) Conjecture K 1 μ K, Z n min vol k PK 1 k O log n!! P Z k n rk P =k 1

16 Faster Algorithm for IP? 1 μ K, Z n min P Z k n rk P =k 1 vol k PK 1 k O log n D. `12: Assuming KL conjecture + P computable in (log n) O(n) time then there is log n O(n) time algorithm for IP.

17 l 2 Kannan-Lovász Conjecture Does the conjecture hold for ellipsoids? E 0 Z n An ellipsoid is E = TB 2 n

18 l 2 Kannan-Lovász Conjecture Answer: YES* [Regev-S.Davidowitz 17] 0 E * up to polylogarithmic factors Z n

19 l 2 Kannan-Lovász Conjecture Can we compute the projection P? E 0 Z n THIS TALK: YES, in 2 O(n) time.

20 Outline 1. Integer Programming and the Kannan-Lovász (KL) Conjecture. 2. l 2 KL Conjecture & the Reverse Minkowski Conjecture. 3. Finding dense lattice subspaces.

21 l 2 Kannan-Lovász Conjecture Easier to think of Euclidean ball vs general lattice. TE = B 2 n 0 L = TZ n

22 Lattices A lattice L R n is BZ 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 1 b 2 b 2 b 2 L

23 Lattices A lattice L R n is BZ 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

24 Lattices A lattice L R n is BZ 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

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

26 Volumetric Lower Bounds vol n B 2 n μ L vol n V = det(l) μ V Voronoi cell V all points closer to 0 L

27 Volumetric Lower Bounds μ(l) vol n B 2 n 1 n det L 1 n μ V L Voronoi cell V all points closer to 0

28 Volumetric Lower Bounds μ(l) n det L 1 n μ V Voronoi cell V all points closer to 0 L

29 Volumetric Lower Bounds μ L μ L W μ W μ W L L W projection onto W

30 Volumetric Lower Bounds μ L μ L W k det L W 1 k μ W μ W L L W projection onto W dim(w) = k 1

31 Volumetric Lower Bounds μ L max dim W =k 1 k det L W 1 k μ W μ W L L W projection onto W dim(w) = k 1

32 l 2 Kannan-Lovász Conjecture Define C KL,2 (n) to be smallest number such that μ L C KL,2 (n) max dim W =k 1 k det L W 1 k for all lattices of dimension at most n. C KL,2 n = Ω( log n) Lower bound for L with basis e 1, 1 2 e 2,, 1 n e n.

33 KL Bounds μ L C KL,2 (n) max dim W =k 1 k det L W 1 k Kannan-Lovász `88: n D. Regev `16: log O(1) n Assuming Reverse Minkowski Conjecture. Regev, S.Davidowitz `17: log Τ 3 2 n Reverse Minkowski Conjecture is proved!

34 Our Results n dimensional lattice L L(B) 1. Can compute subspace W, dim W = k 1 μ L O(log n) k det L W k in 2 O(n) time with high probability. Prior work: Kannan Lovász `88: n in 2 O(n) time. D. Micciancio `13: best subspace in n O(n2) time.

35 Our Results n dimensional lattice L L(B) 2. Can combine lower bounds over different subspaces to certify μ L up to the slicing constant L n for stable Voronoi cells*. * If vol n V = 1 can find hyperplane H s.t. vol n 1 V H = Ω( 1 Ln ) H V

36 Our Results n dimensional lattice L L(B) 2. Can combine lower bounds over different subspaces to certify μ L up to the slicing constant L n for stable Voronoi cells*. Slicing Conjecture: L n = O(1) for all convex bodies! For stable Voronoi cells: L n = O(log n) [RS `17] H V

37 Notation M L sublattice of dimension k Convention: M = {0} then det M 1. Normalized Determinant: nd M det M Τ 1 k Projected Sublattice: LΤM L projected onto span M

38 Lower Bounds for Chains Theorem [D. 17]: For 0 = L 0 L 1 L k = L then μ L 2 σk i=1 dim( L i ΤL i 1 ) nd LΤL 2 i 1 Only missing ingredient : Combined with techniques from [R.S. `17] easily get tightness within slicing constant L n.

39 Lower Bounds for Chains Theorem [D. 17]: For 0 = L 0 L 1 L k = L then μ L 2 σk i=1 dim( L i ΤL i 1 ) nd LΤL 2 i 1 Proof Idea: 1. Establish SDP based lower bound: [D.R. `16] μ L 2 max σ i rk P i nd P i L 2 s.t. σ i P i P i I n 2. Build solution to above starting from any chain.

40 Lattice Density lim r L rb 2 n vol n (rb 2 n ) = 1 det L r L

41 Lattice Density lim r L rb 2 n vol n (rb 2 n ) = 1 det L r Global density of lattice points per unit volume L

42 Minkowski s First Theorem L rb 2 n 2 n vol n(rb 2 n ) det L 1889 Global density implies local density r L

43 Reverse Minkowski Theorem Regev-S.Davidowitz `17: L lattice dimension n. If all sublattices of L have determinant at least 1 then: L has at most 2 O(log2 n r 2) points at distance r. Almost tight: Z n has n Ω(k) points at distance r for k n.

44 Outline 1. Integer Programming and the Kannan-Lovász (KL) Conjecture. 2. l 2 KL Conjecture & the Reverse Minkowski Conjecture. 3. Finding dense lattice subspaces.

45 Notation M L sublattice of dimension k Convention: M = {0} then det M 1. Normalized Determinant: nd M det M Τ 1 k Projected Sublattice: LΤM L projected onto span M

46 Densest Subspace Problem nd L min M L M {0} nd(m) α-dsp: Given L find M L, M {0} such that nd M α nd (L). Remark: dimension of M is not fixed! Key primitive for finding sparse lattice projections. Will focus on this problem.

47 Densest Subspace 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

48 Canonical Polytope [Stuhler 76] n dimensional lattice L Log det P(L) (0,0) (n, log det L) dim n-1 n { k, log det M : sublattice M L, dim M = k}

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

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

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

52 Stable Lattice [Stuhler 76] Example: L = Z n Log det P(L) (0,0) {0} dim n-1 Z n has trivial filtration: 0 Z n n

53 Canonical Filtration [Stuhler 76] L {0} L 1 L n-1 n 1. Form Chain: 0 = L 0 L 1 L k = L. 2. Blocks L i ΤL i 1 are stable. 3. Slope increasing: nd L i ΤL i 1 < nd L i+1 ΤL i.

54 Densest Subspace Problem n dimensional lattice L Log det Want sublattice with approx. minimum slope. L L 2 (0,0) L 1 {0} dim (n, log det L) n-1 n

55 Densest Subspace 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

56 Dual Lattice A lattice L R n is BZ n for a basis B = b 1,, b n. The dual lattice is L = {y span L : y T x Z x L} L is generated by B T. Remark: Z n = Z n y L y T x = 0 y T x = 1 y T x = 2 y T x = 3 y T x = 4 L

57 Dual Lattice A lattice L R n is BZ n for a basis B = b 1,, b n. The dual lattice is L = {y span L : y T x Z x L} L is generated by B T. det L = 1/det(L) y L y T x = 0 y T x = 1 y T x = 2 y T x = 3 y T x = 4 L

58 Discrete Gaussian Distribution

59 Discrete Gaussian Distribution ADRS `15: Can sample in 2 n+o(n) time for any parameter.

60 Main Procedure Repeat until L = {0} s nd L Update M L if nd M > s Sample y D L,c/s until y 0 L L y Main Lemma: At any iteration, if L not O(log n) approximate minimizer, then L y contains minimizer w.p. Ω 1. Proc. finds apx minimizer with prob. 2 O(n).

61 Proof of Main Lemma wlog det L = det(l ) = 1 L 1 L densest sublattice sample y D L,c If nd L 1 1 log n must show that y 0 and y L 1 w.p. Ω(1).

62 det L = det L = 1 L 1 L densest sublattice, nd L 1 1 log n Sample y D L,c Want: 1. Pr y = 0 ε 2. Pr y L L 1 1 ε

63 det L = det L = 1 1. Pr y D L,c y = 0 = 1 ρ c (L ) (By Minkowski) 1 L nb n 2 e nτc n e Τ n c 2 = o(1) ρ c A σ x A e xτc 2

64 det L = det L = 1 L 1 L densest sublattice, nd L 1 1Τlog n W L 1 2. Pr y D L,c y W (ortho. is worst-case) = ρ c(l W) ρ c (L ) = ρ c (L W) ρ c L W ρ c ( L ΤW) 1 ρ c (L /W) ρ c A σ x A e xτc 2

65 det L = det L = 1 L 1 L densest sublattice, nd L 1 1Τlog n W L 1 2. Need to show ρ c L ΤW 1 + o(1) Key: nd Τ L W = Τ 1 nd(l 1 ) log n Reverse-Minkowski ( Τ L W) rb 2 n e o r2, r 0 ρ c A σ x A e xτc 2

66 Canonical Polytope of L? det L = 1 Log det {0} L (n, 0) Slope: ln nd(l 1 ) L 1 L 2 dim n-1 n Assumption: det L = 1

67 Canonical Polytope of L? det L = 1 Log det {0} L L 2 L L 1 L (n, 0) dim n-1 n Map M L M

68 Canonical Polytope of L? det L = 1 Log det {0} Slope: ln nd(l 1 ) L L 2 L L 1 L ΤW L (n, 0) dim n-1 n P(L ) is reflection of P(L)

69 Conclusions 1. Algorithmic version of l 2 Kannan-Lovász conjecture via discrete Gaussian sampling. 2. Lower bound certificates for covering radius that are tight within O(1) under slicing conjecture. Open Problem 1. Prove KL conjecture for general convex bodies. 2. Prove Slicing conjecture for Voronoi cells.