Extension complexity bounds for polygons Numerical factorizations - - PowerPoint PPT Presentation

Extension complexity bounds for polygons Numerical factorizations and conjectures

François Glineur

Université catholique de Louvain (UCLouvain)

Center for Operations Research and Econometrics and Information and Communication Technologies, Electronics and Applied Mathematics Institute

joint work with Nicolas Gillis (UMons), Arnaud Vandaele (UMons) and Julien Dewez (UCLouvain) Limitations of convex programming: lower bounds on extended formulations and factorization ranks Dagstuhl, February 20 2015

1

Main message

Nonnegative factorizations of slack matrices computed numerically can be useful to prove upper bounds and infer exact values for the extension complexity of (small) polytopes

2

Outline

Numerical factorization of nonnegative matrices Extension complexity of the correlation polytope Cartesian products and extension complexity Worst-case extension complexity of polygons An explicit factorization for regular n-gons Lower bounds on extension complexity Geometric lower bound on extension complexity Computable bounds for regular n-gons Positive semidefinite rank This work/talk has mostly a computational focus

3

Link between extension and nonnegative factorization

Given a polytope P and its slack matrix S Extension complexity of P = rank+(S) [Yannanakis, 1991]

Link between extension and nonnegative factorization

Given a polytope P and its slack matrix S Extension complexity of P = rank+(S) [Yannanakis, 1991] P extension with r facets S = UV T U ∈ Rm×r

+

V ∈ Rn×r

+

≡ S ≥ 0

4

Link between extension and nonnegative factorization

Given a polytope P and its slack matrix S Extension complexity of P = rank+(S) [Yannanakis, 1991] P extension with r facets S = UV T U ∈ Rm×r

+

V ∈ Rn×r

+

≡ S ≥ 0 Moreover any slack matrix factorization provides an explicit extended formulation (with some redundant equalities) P = {x | Ax ≤ b} = {x | Ax + Uy = b and y ≥ 0} i.e. columns of U generate the cone of slack vectors

4

Link between extension and nonnegative factorization

Given a polytope P and its slack matrix S Extension complexity of P = rank+(S) [Yannanakis, 1991] P extension with r facets S = UV T U ∈ Rm×r

+

V ∈ Rn×r

+

≡ S ≥ 0 Moreover any slack matrix factorization provides an explicit extended formulation (with some redundant equalities) P = {x | Ax ≤ b} = {x | Ax + Uy = b and y ≥ 0} i.e. columns of U generate the cone of slack vectors Our goal: compute (bounds on) the nonnegative rank of (small) matrices and deduce the extension complexity of (small) polytopes

4

Example for the hexagon

6 facets 5 facets π         1 1 1 1 1 2 1 1 1 1 2 1               1 2 1 1 2 1 1 1 1 1 1 1       =         1 2 2 1 1 2 2 1 1 1 2 2 2 1 1 2 2 2 1 1 1 2 2 1        

5

Outline

Numerical factorization of nonnegative matrices Extension complexity of the correlation polytope Cartesian products and extension complexity Worst-case extension complexity of polygons An explicit factorization for regular n-gons Lower bounds on extension complexity Geometric lower bound on extension complexity Computable bounds for regular n-gons Positive semidefinite rank

6

Numerical factorization of nonnegative matrices

(cf. Nicolas’ talk yesterday) Simulated annealing-based heuristic factorization code (improved multi-start with accelerated coordinate descent) Seems to perform well in practice (e.g. for a few dozen rows and columns) Matlab code (user-friendly) available at http://sites.google.com/site/exactnmf/ Also maintain a list of benchmark matrices and best known lower and upper bounds on rank+

7

Benchmarked matrices

m n rank(X) rank+(X) Code 6 6 3 5 LEDM6 Linear EDM’s 8 8 3 6 LEDM8 X(i, j) = (i − j)2, 12 12 3 7 LEDM12 for 1 ≤ i ≤ m, 1 ≤ j ≤ n 16 16 3 8 LEDM16 32 32 3 10∗ LEDM32 Slack Matrix of the Hexagon 6 6 3 5 6-G Slack Matrix of the Heptagon 7 7 3 6 7-G Slack Matrix of the Octagon 8 8 3 6 8-G Slack Matrix of the Nonagon 9 9 3 7 9-G Slack Matrix of the Hexadecagon 16 16 3 8 12-G Slack Matrix of the 32-gon 32 32 3 10 32-G Slack Matrix of the dodecahedron 20 12 4 9 20-D Slack Matrix of the 24-cell 24 24 5 12∗ 24-C UDISJ (n = 4) 16 16 9 9 UDISJ4 UDISJ (n = 5) 32 32 18 18 UDISJ5 UDISJ (n = 6) 64 64 27 27 UDISJ6 Randomly generated: X = WH density = 0.1 50 50 10 10 RND1 density = 0.3 50 50 10 10 RND3

8

Benchmark results

MS2 SA RBR Hybrid LEDM6 112/150 (3.1) 100/100 (19.6) 100/100 (1.4) 100/100 (19) LEDM8 107/600 (27.1) 100/100 (60.9) 100/100 (16.7) 148/150 (63.6) LEDM12 0/1000 (∼) 119/200 (42.9) 107/650 (15.1) 103/150 (36.9) LEDM16 0/1000 (∼) 100/250 (118) 100/550 (29.1) 121/250 (104) LEDM32 0/1000 (∼) 14/1000 (2592) 0/1000 (∼) 28/1000 (1371) 6-G 100/100 (2.1) 100/100 (1.2) 100/100 (1.4) 100/100 (1.5) 7-G 100/100 (2.2) 100/100 (4.2) 100/100 (1.5) 100/100 (4.4) 8-G 129/200 (3.8) 100/100 (15.4) 100/100 (1.5) 100/100 (15.3) 9-G 117/200 (4.6) 100/100 (22.9) 100/100 (1.6) 100/100 (23.2) 16-G 0/1000 (∼) 102/350 (91.6) 143/150 (1.9) 118/150 (34.2) 32-G 0/1000 (∼) 31/1000 (1086) 107/250 (6.6) 105/300 (97) 20-D 21/1000 (161) 100/100 (7.8) 129/150 (2.3) 100/100 (5.6) 24-C 0/1000 (∼) 100/100 (3.1) 119/200 (4.1) 100/100 (4.4) UDISJ4 100/100 (2.4) 100/100 (1.2) 100/100 (1.9) 100/100 (1.9) UDISJ5 102/500 (38) 100/100 (2.8) 100/100 (4.9) 100/100 (5.2) UDISJ6 8/1000 (1594) 100/100 (7.8) 112/450 (66.4) 100/100 (18.5) RND1 100/100 (2.8) 100/100 (1.1) 100/100 (2.2) 100/100 (2.2) RND3 100/100 (2.8) 100/100 (1.1) 100/100 (2.2) 100/100 (2.2)

9

Outline

Numerical factorization of nonnegative matrices Extension complexity of the correlation polytope Cartesian products and extension complexity Worst-case extension complexity of polygons An explicit factorization for regular n-gons Lower bounds on extension complexity Geometric lower bound on extension complexity Computable bounds for regular n-gons Positive semidefinite rank

10

Correlation polytope

Correlation polytope COR(n) is COR(n) = conv{xxT such that x ∈ {0, 1}n} xc(COR(n)) ≥ 2Ω(n) (Fiorini et al., 2012) xc(COR(n)) ≥ 1.5n (Kaibel and Weltge, 2013)

11

A submatrix of the slack matrix of COR(n)

Define the following 2n × 2n nonnegative matrix Mn(a, b) = (1 − |aTb|)2 for any a ∈ {0, 1}n, b ∈ {0, 1}n Mn appears as a submatrix of the slack matrix of COR(n) Mn has rank 1

2n(n + 1) + 1

Numerical tests suggest that for 3 ≤ n ≤ 6 we have rank+ Mn = 2n (1000 attempts for each n) which would imply xc(COR(n)) ≥ 2n

12

Outline

Numerical factorization of nonnegative matrices Extension complexity of the correlation polytope Cartesian products and extension complexity Worst-case extension complexity of polygons An explicit factorization for regular n-gons Lower bounds on extension complexity Geometric lower bound on extension complexity Computable bounds for regular n-gons Positive semidefinite rank

13

Cartesian product of polyhedral cones

Let C1, C2 be two polyhedral cones (with fi faces, ri extreme rays) and S1 ∈ Rf1×r1

+

, S2 ∈ Rf2×r2

+

their slack matrices Their Cartesian product C1 × C2 = {(x, y) such that x ∈ C1, y ∈ C2} features f1 + f2 faces and v1 + v2 extreme rays and its slack matrix S12 satisfies rank+ S12 = rank+ S1 + rank+ S2

• r

xc(C1 × C2) = xc(C1) + xc(C2) Is the same true for polytopes ?

14

Cartesian product for polytopes and nonnegative matrices

Let P1, P2 be two polytopes (with fi faces, vi vertices) and S1 ∈ Rf1×v1

+

, S2 ∈ Rf2×v2

+

their slack matrices We have that P1 × P2 features f1 + f2 faces and v1v2 vertices and xc(P1 × P2) ≤ xc(P1) + xc(P2) and its slack matrix S12 = S1 ⊙ S2 (size (f1 + f2) × v1v2) satisfies rank+ S1 ⊙ S2 ≤ rank+ S1 + rank+ S2 (⊙ can be defined for arbitrary nonnegative matrices S1 and S2) Does equality always hold ? Intuitively obvious but ... No one knows ! (except when P2 is a simplex or S2 identity) (for standard rank we have rank S1 ⊙ S2 = rank S1 + rank S2 unless 1 ∈ rowspanS1 and 1 ∈ rowspanS2, in which case rank S1 ⊙ S2 = rank S1 + rank S2 − 1)

15

Numerical tests for Cartesian product

Numerical tests for products of two (or three) of the following

• 1. 5-gon, 6-gon, 7-gon, 8-gon

(with respective xc equal to 5,5,6,6)

• 2. dodecahedron (xc=9)
• 3. cuboctahedron (xc=8)
• 4. polar of any of the above
• 5. generalized slack matrix obtained by relaxing 5-gon

(adding a constant δ ≈ 0.499 to all entries of its slack matrix) 0 ≤ δ < 1 2 ⇒ rank+(S5 + δE) = 5 and rank+(S5 + 1 2E) = 4 All tests strongly suggest that xc(P1 × P2) = xc(P1) + xc(P2) holds as well as rank+ S1 ⊙ S2 = rank+ S1 + rank+ S2

(with the same final error observed in all tests for each pair)

16

Outline

Numerical factorization of nonnegative matrices Extension complexity of the correlation polytope Cartesian products and extension complexity Worst-case extension complexity of polygons An explicit factorization for regular n-gons Lower bounds on extension complexity Geometric lower bound on extension complexity Computable bounds for regular n-gons Positive semidefinite rank

17

Worst-case extension complexity of polygons

What is the worst-case extension complexity of an n-gon ? Lower (LB) and upper (UB) bounds:

• UB: wcc(n) ≤ ⌈ 6n

7 ⌉ (Shitov 2013)

• UB: wcc(n) is sublinear (Shitov 2014)
• LB: wcc(n) ≥ ⌈

√ 2n⌉ (Fiorini et al. 2011)

• btained for generic polygons (using algebraic independence)
• LB: wcc(n) ≥ ⌈2√2n − 2 − 1⌉ (Padrol 2015)
• btained for generic polygons (using dimension counting)
• UB: from numerical tests, conjectured wcc(n) ≤ ⌈ n+5

2 ⌉

and tight for 5 ≤ n ≤ 15 ; would imply e.g. wcc(12) ≤ 9

• Conjecture (Padrol, 2015): wcc(n) = ⌈2√2n − 2 − 1⌉

18

Numerical tests for arbitrary n-gons

• Min. and max. number of exact factorizations computed out of

1000 runs (10 n-gons tested for each value of 6 ≤ n ≤ 17)

exact NMF’s found by Hybrid out of 1000 runs on ten such n-gons.

r/n 6 7 8 9 10 11 4 5 [0,0] [0,0] 6 [1000,1000] [463,1000] [0,0] [0,0] 7 [754,897] [160,353] [0,0] [0,0] 8 [743,873] [351,443] [25,48] 9 [787,858] [401,546] 10 [692,862] 11 12 13 14 15

19

Numerical tests for arbitrary n-gons (cont.)

• Min. and max. number of exact factorizations computed out of

1000 runs (10 n-gons tested for each value of 6 ≤ n ≤ 17)

• gons.

11 12 13 14 15 16 17 [0,0] [25,48] [0,0] [0,0] [401,546] [148,190] [10,19] [0,0] [0,0] [0,0] [692,862] [580,665] [242,389] [63,111] [5,19] [0,1] [0,0] [833,902] [533,726] [385,540] [150,247] [9,82] [5,14] [734,874] [643,766] [442,631] [138,365] [107,204] [671,824] [375,674] [405,517] [610,830] [583,734] [721,829]

20

5 10 15 20 25 30 35 40 −5 5 10 15 20 25 30 35 Bounds on the worst case extension complexity of n−gons 6n/7 (upper bound proved by Shitov) n/2 + 3 (upper bound conjectured by Vandaele et al.) 2 sqrt(2n−2) − 1 (lower bound for generic proved by Padrol, priv. comm.) sqrt(2n) (lower bound for generic proved by Fiorini et al.)

21

5 10 15 20 25 30 35 40 −5 5 10 15 20 25 30 35 Bounds on the worst case extension complexity of n−gons 6n/7 (upper bound proved by Shitov) n/2 + 3 (upper bound conjectured by Vandaele et al.) 2 sqrt(2n−2) − 1 (lower bound for generic proved by Padrol, priv. comm.) sqrt(2n) (lower bound for generic proved by Fiorini et al.)

22

5 10 15 20 25 30 35 40 −5 5 10 15 20 25 30 35 Bounds on the worst case extension complexity of n−gons 6n/7 (upper bound proved by Shitov) n/2 + 3 (upper bound conjectured by Vandaele et al.) 2 sqrt(2n−2) − 1 (lower bound for generic proved by Padrol, priv. comm.) sqrt(2n) (lower bound for generic proved by Fiorini et al.)

23

5 10 15 20 25 30 35 40 −5 5 10 15 20 25 30 35 Bounds on the worst case extension complexity of n−gons 6n/7 (upper bound proved by Shitov) n/2 + 3 (upper bound conjectured by Vandaele et al.) 2 sqrt(2n−2) − 1 (lower bound for generic proved by Padrol, priv. comm.) sqrt(2n) (lower bound for generic proved by Fiorini et al.)

24

Extension complexity of regular n-gons

Testing the slack matrices Xn of regular n-gons, a simple regular pattern for the extension complexity was found, corresponding to rank+(Xn) = 2k − 1 for 2k−1 < n ≤ 2k−1 + 2k−2, 2k for 2k−1 + 2k−2 < n ≤ 2k.

25

Extension complexity of regular n-gons

Testing the slack matrices Xn of regular n-gons, a simple regular pattern for the extension complexity was found, corresponding to rank+(Xn) = 2k − 1 for 2k−1 < n ≤ 2k−1 + 2k−2, 2k for 2k−1 + 2k−2 < n ≤ 2k. This was "verified" numerically up to n = 78, and matches the best-known Ben-Tal Nemirovsky formulation for powers of 2 (complexity 2k for n = 2k).

25

10 20 30 40 50 60 70 100 200 300 400 500 600 700 800 900 1000

n # exact NMF’s found out of 1000 runs of Hybrid

26

It turns out that, staring long enough at the numerical factors, a pattern was also found, leading after some additional thinking to a rigorously proved upper bound for all n

27

Outline

Numerical factorization of nonnegative matrices Extension complexity of the correlation polytope Cartesian products and extension complexity Worst-case extension complexity of polygons An explicit factorization for regular n-gons Lower bounds on extension complexity Geometric lower bound on extension complexity Computable bounds for regular n-gons Positive semidefinite rank

28

Slack matrix of a regular n-gon

With the radius of the circumscribed circle r = 1, we have: s = 2 sin π n

• .

The distance between a face and the kth vertex is: ck = cos π n

• − cos

π n (2k + 1)

• .

29

Slack matrix of a regular n-gon

Slack matrices of regular n-gons are circulant matrices. ck = cos π n

• − cos

π n + 2π n k

• .

For k = 0, ..., n − 1, we have c0 = cn−1 = 0, c1 = cn−2, ... With n = 9: S9 =               c1 c2 c3 c4 c3 c2 c1 c1 c2 c3 c4 c3 c2 c1 c1 c1 c2 c3 c4 c3 c2 c2 c1 c1 c2 c3 c4 c3 c3 c2 c1 c1 c2 c3 c4 c4 c3 c2 c1 c1 c2 c3 c3 c4 c3 c2 c1 c1 c2 c2 c3 c4 c3 c2 c1 c1 c1 c2 c3 c4 c3 c2 c1               .

30

Removing two rank-1 factors

S9 =               c1 c2 c3 c4 c3 c2 c1 c1 c2 c3 c4 c3 c2 c1 c1 c1 c2 c3 c4 c3 c2 c2 c1 c1 c2 c3 c4 c3 c3 c2 c1 c1 c2 c3 c4 c4 c3 c2 c1 c1 c2 c3 c3 c4 c3 c2 c1 c1 c2 c2 c3 c4 c3 c2 c1 c1 c1 c2 c3 c4 c3 c2 c1               . Goal : zeros on the underlined entries.

31

Removing two rank-1 factors

    c4 c3 c2 c1 c3 c4 c3 c2 c2 c3 c4 c3 c1 c2 c3 c4    −     c4 − c3 c3 − c2 c2 − c1 c1 c3 − c2 c4 − c1 c3 c2 c2 − c1 c3 c4 c3 − c1 c1 c2 c3 − c1 c4 − c2     =     c3 c2 c1 c2 c1 c1 c1 c1 c2     .     c4 − c3 c3 − c2 c2 − c1 c1 c3 − c2 c4 − c1 c3 c2 c2 − c1 c3 c4 c3 − c1 c1 c2 c3 − c1 c4 − c2     is a rank-1 matrix !

32

Removing two rank-1 factors

    c4 c3 c2 c1 c3 c4 c3 c2 c2 c3 c4 c3 c1 c2 c3 c4    −     c4 − c2 c3 − c1 c2 c1 c3 − c1 c4 c3 c2 − c1 c2 c3 c4 − c1 c3 − c2 c1 c2 − c1 c3 − c2 c4 − c3     =     c2 c1 c1 c1 c1 c2 c1 c2 c3     .     c4 − c2 c3 − c1 c2 c1 c3 − c1 c4 c3 c2 − c1 c2 c3 c4 − c1 c3 − c2 c1 c2 − c1 c3 − c2 c4 − c3     is a rank-1 matrix !

33

Rank-1 factors

With ck = cos( π

n ) − cos( π n (2k + 1)) = 2 sin( π n k) sin( π n (k + 1))

we have that (infinite) matrix

• cα+i−j − cβ−i−j
• i∈Z,j∈Z

is rank-1 for any fixed α ∈ Z, β ∈ Z and n ∈ N>0 ! Proof: It is straightforward to check that ci − cj = 2 sin(i − j) sin(i + j + 1) which implies the result

34

Result: a matrix with symmetry

S9 − u1v1 − u2v2 =               c1 c2 c3 c4 c3 c2 c1 c1 c2 c3 c2 c1 c1 c1 c2 c1 c1 c2 c1 c1 c1 c2 c3 c2 c1 c1 c2 c3 c3 c2 c1 c1 c2 c3 c2 c1 c1 c1 c2 c1 c1 c2 c1 c1 c1 c2 c3 c2 c1               . A cross matrix where each of the four parts is symmetric.

35

Result: a matrix with symmetry

              c1 c2 c3 c4 c3 c2 c1 c1 c2 c3 c2 c1 c1 c1 c2 c1 c1 c2 c1 c1 c1 c2 c3 c2 c1 c1 c2 c3 c3 c2 c1 c1 c2 c3 c2 c1 c1 c1 c2 c1 c1 c2 c1 c1 c1 c2 c3 c2 c1               . By symmetry, if we have the nonnegative factorization of the green submatrix, then we have the nonnegative factorization of the entire matrix with the same nonnegative rank.

36

Dealing with submatrices: removing again rank-1 factors

      c1 c2 c3 c4 c1 c2 c3 c1 c1 c2 c2 c1 c1 c3 c2 c1       →       c1 c2 c1 c1 c1 c1 c1 c1 c1       . Again, in the submatrix, we can remove two rank-1 factors to

• btain a cross matrix with symmetry.

      c1 c2 c3 c1 c1 c1 c1 c1 c1       .

37

Smallest submatrix where r+ is known

As we have: rank   c1 c2 c1 c1   = 3, the nonnegative matrix factorization of that submatrix with r+ = 3 is for example:   c1 c2 c1 c1   =   c1 c2 c1 c1     1 1 1   .

38

Factorization of the original slack matrix

Following back the same way with the help of the NMF of the small submatrix, we can obtain the NMF of the original slack matrix. size

• peration

r+ (3 × 3) 3 (5 × 5) adding two rank-1 factors 5 (9 × 9) adding two rank-1 factors 7

39

Factorization of the original slack matrix

Following back the same way with the help of the NMF of the small submatrix, we can obtain the NMF of the original slack matrix. size

• peration

r+ (3 × 3) 3 (5 × 5) adding two rank-1 factors 5 (9 × 9) adding two rank-1 factors 7 Bigger example with n = 293: size

• peration

r+ (3 × 3) 3 (6 × 5) adding two rank-1 factors 5 (10 × 10) adding two rank-1 factors 7 (19 × 19) adding two rank-1 factors 9 (38 × 37) adding two rank-1 factors 11 (74 × 74) adding two rank-1 factors 13 (147 × 147) adding two rank-1 factors 15 (293 × 293) adding two rank-1 factors 17

39

Upper bound on the extension complexity of regular n-gons

r+ = 2k − 1 for 2k−1 < n ≤ 2k−1 + 2k−2 r+ = 2k for 2k−1 + 2k−2 < n ≤ 2k Is this tight ?

40

Outline

Numerical factorization of nonnegative matrices Extension complexity of the correlation polytope Cartesian products and extension complexity Worst-case extension complexity of polygons An explicit factorization for regular n-gons Lower bounds on extension complexity Geometric lower bound on extension complexity Computable bounds for regular n-gons Positive semidefinite rank

41

Geometric f -vector bound

Given a n-dimensional polytope P, its f -vector is the vector (f0, f1, . . . , fn), where fk denotes the number of k-faces of P

42

Geometric f -vector bound

Given a n-dimensional polytope P, its f -vector is the vector (f0, f1, . . . , fn), where fk denotes the number of k-faces of P Example: the f -vector of the hexagon is (6, 6, 1)

42

Geometric f -vector bound

Given a n-dimensional polytope P, its f -vector is the vector (f0, f1, . . . , fn), where fk denotes the number of k-faces of P Example: the f -vector of the hexagon is (6, 6, 1) Goal : A lower bound depending on the f -vector only

42

Geometric f -vector bound

Given a n-dimensional polytope P, its f -vector is the vector (f0, f1, . . . , fn), where fk denotes the number of k-faces of P Example: the f -vector of the hexagon is (6, 6, 1) Goal : A lower bound depending on the f -vector only Key property : f -vectors decreases monotonically under projections

42

Geometric f -vector bound

Given a n-dimensional polytope P, its f -vector is the vector (f0, f1, . . . , fn), where fk denotes the number of k-faces of P Example: the f -vector of the hexagon is (6, 6, 1) Goal : A lower bound depending on the f -vector only Key property : f -vectors decreases monotonically under projections Let P be an n-dimensional polytope and let T be a projection, then fk(T(P)) ≤ fk(P), ∀k

42

Geometric f -vector bound

Given a n-dimensional polytope P, its f -vector is the vector (f0, f1, . . . , fn), where fk denotes the number of k-faces of P Example: the f -vector of the hexagon is (6, 6, 1) Goal : A lower bound depending on the f -vector only Key property : f -vectors decreases monotonically under projections Let P be an n-dimensional polytope and let T be a projection, then fk(T(P)) ≤ fk(P), ∀k Note: property also true if T is a linear fractional transformation

42

Illustration of the monotonicity under projection

Example: Q P # 0-faces # 1-faces # 2-faces # 3-faces     8 12 6 1     ≥ ≥ ≥ ≥     8 8 1    

43

Consequence of the property

Let P be an n-dimensional polytope and let Q be an extension for P (Q is d-dimensional with d ≥ n), then fk(Q) ≥ fk(P), ∀k

44

Consequence of the property

Let P be an n-dimensional polytope and let Q be an extension for P (Q is d-dimensional with d ≥ n), then fk(Q) ≥ fk(P), ∀k Maximal f -vector : In a given dimension d, for a given number of facets fd−1, we know an upper bound on the number of k-faces, f max

k

(d, fd−1) [McMullen, 1970]

44

Consequence of the property

Let P be an n-dimensional polytope and let Q be an extension for P (Q is d-dimensional with d ≥ n), then fk(Q) ≥ fk(P), ∀k Maximal f -vector : In a given dimension d, for a given number of facets fd−1, we know an upper bound on the number of k-faces, f max

k

(d, fd−1) [McMullen, 1970] A geometric lower bound for the extension complexity : Consider all dimensions d > n, min # facets of a polytope with maximal f -vector which satisfies the projection property

44

Geometric lower bound for the extension complexity

Lower bound : Extension complexity of P is greater than the optimal solution of min

d such that n<d<fn−1

• min

f

f such that f max

k

(d, f ) ≥ fk, k = 0, . . . , n−1

• (can be further strengthened by simulatenously considering the

polar polytope, using the fact that extension complexity is invariant when taking polars)

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Numerical example

24-cell : (24, 96, 96, 24, 1) (n = 4)

46

Numerical example

24-cell : (24, 96, 96, 24, 1) (n = 4) d = 5 : (30, 75, 74, 36, 9, 1) (42, 105, 100, 45, 10, 1)

46

Numerical example

24-cell : (24, 96, 96, 24, 1) (n = 4) d = 5 : (30, 75, 74, 36, 9, 1) (42, 105, 100, 45, 10, 1) d = 6 : (30, 90, 117, 84, 36, 9, 1) (50, 150, 185, 120, 45, 10, 1)

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Numerical example

24-cell : (24, 96, 96, 24, 1) (n = 4) d = 5 : (30, 75, 74, 36, 9, 1) (42, 105, 100, 45, 10, 1) d = 6 : (30, 90, 117, 84, 36, 9, 1) (50, 150, 185, 120, 45, 10, 1) d = 7 : (20, 70, 120, 125, 84, 36, 9, 1) (40, 140, 222, 205, 120, 45, 10, 1)

46

Numerical example

24-cell : (24, 96, 96, 24, 1) (n = 4) d = 5 : (30, 75, 74, 36, 9, 1) (42, 105, 100, 45, 10, 1) d = 6 : (30, 90, 117, 84, 36, 9, 1) (50, 150, 185, 120, 45, 10, 1) d = 7 : (20, 70, 120, 125, 84, 36, 9, 1) (40, 140, 222, 205, 120, 45, 10, 1) d = 8 : (9, 36, 84, 126, 126, 84, 36, 9, 1) (25, 100, 200, 250, 210, 120, 45, 10, 1)

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Numerical example

24-cell : (24, 96, 96, 24, 1) (n = 4) d = 5 : (30, 75, 74, 36, 9, 1) (42, 105, 100, 45, 10, 1) d = 6 : (30, 90, 117, 84, 36, 9, 1) (50, 150, 185, 120, 45, 10, 1) d = 7 : (20, 70, 120, 125, 84, 36, 9, 1) (40, 140, 222, 205, 120, 45, 10, 1) d = 8 : (9, 36, 84, 126, 126, 84, 36, 9, 1) (25, 100, 200, 250, 210, 120, 45, 10, 1) and for d ≥ 9 we must have at least d + 1 ≥ 10 facets

46

Numerical example

24-cell : (24, 96, 96, 24, 1) (n = 4) d = 5 : (30, 75, 74, 36, 9, 1) (42, 105, 100, 45, 10, 1) d = 6 : (30, 90, 117, 84, 36, 9, 1) (50, 150, 185, 120, 45, 10, 1) d = 7 : (20, 70, 120, 125, 84, 36, 9, 1) (40, 140, 222, 205, 120, 45, 10, 1) d = 8 : (9, 36, 84, 126, 126, 84, 36, 9, 1) (25, 100, 200, 250, 210, 120, 45, 10, 1) and for d ≥ 9 we must have at least d + 1 ≥ 10 facets Lower bound = 10

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Comparison of geometric lower bounds

Some results for low-dimensional regular polytopes.

Lower Bounds Best known (Higher is better) extension dimension, Bound from # vertices, Bound from [Gillis and Our bound Best upper # facets [Goemans, 2009] G., 2012] bound Square 2, 4, 4 2 4 4 4 Pentagon 2, 5, 5 3 5 5 5 Hexagon 2, 6, 6 3 5 5 5 Octagon 2, 8, 8 3 6 6 6 16-gon 2, 16, 16 4 8 8 8 210-gon 2, 210, 210 10 17 17 20 Cube 3, 8, 6 3 6 6 6 Dodecahedron 3, 20, 12 5 8 8 9 Cuboctahedron 3, 12, 14 4 7 7 8 24-cell 4, 24, 24 5 9 10 12 6-cube 6, 64, 12 6 11 11 12

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Comparison of geometric lower bounds

Some results for the family of uniform k21 polytopes.

Lower Bounds (Higher is better) dimension, Bound from # vertices, Bound from [Gillis and Our bound # facets [Goemans, 2009] G., 2012] −121 3 6 5 3 5 5 021 4 10 10 4 7 7 121 5 16 26 4 9 10 221 6 27 99 5 12 13 321 7 56 702 6 17 18 421 8 240 19440 8 24 25

48

Comparison of geometric lower bounds

Some results for the family of uniform k21 polytopes.

Lower Bounds (Higher is better) dimension, Bound from # vertices, Bound from [Gillis and Our bound # facets [Goemans, 2009] G., 2012] −121 3 6 5 3 5 5 021 4 10 10 4 7 7 121 5 16 26 4 9 10 221 6 27 99 5 12 13 321 7 56 702 6 17 18 421 8 240 19440 8 24 25

Bound better than previous bounds Optimality guarantee in some cases Could potentially be further improved by considering flag vectors

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Outline

Numerical factorization of nonnegative matrices Extension complexity of the correlation polytope Cartesian products and extension complexity Worst-case extension complexity of polygons An explicit factorization for regular n-gons Lower bounds on extension complexity Geometric lower bound on extension complexity Computable bounds for regular n-gons Positive semidefinite rank

49

Numerical comparison on the n-gon

Upper bounds

• Rank=3, Matrix dimension=n, Shitov=

6n

7

• Reflection-based extended formulation (Kaibel & Pashkovich,

Fiorini, Rothvoss & Tiwary)=2 ⌈log2 n⌉

• Factorization found numerically + recursive generalization

Lower bounds

• Fooling set bound ≤ rectangle covering bound
• (first-level-SOS) copositive bound ν+

0 ≤ atomic bound τ + SOS

(Fawzi & Parrilo)

• Goemans bound log2(2n + 2) ≤ geometric f -vector bound

(Dewez, Gillis & G.)

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Numerical comparison on the n-gon

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Outline

Numerical factorization of nonnegative matrices Extension complexity of the correlation polytope Cartesian products and extension complexity Worst-case extension complexity of polygons An explicit factorization for regular n-gons Lower bounds on extension complexity Geometric lower bound on extension complexity Computable bounds for regular n-gons Positive semidefinite rank

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Best known bounds on lifts of regular polygons

For a regular 2n-gon:

Equivariant Non-equivariant LP Lower bound: 2n [GPT13] Upper bound: 2n (trivial) Lower bound: n [Goe14] Upper bound: 2n + 1[BTN01] SDP Lower bound: (ln 2)(n − 1) (Theorem 9) Upper bound: 2n − 1 (Section 3) Lower bound: Ω q

n log n

• [GPT13, GRT13b]

Upper bound: 2n − 1 (Section 3)

Table 1. Bounds on the size of the smallest LP/PSD lifts for the regular 2n-gon in both the equivariant and non-equivariant cases. The main contributions of this paper are highlighted in bold.

from Fawzi, Saunderson and Parrilo (2014) Are the (non-equivariant) upper bounds tight ? Prelminary results from psd-factorization code

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Psd-rank of regular polygons

It is known that 4-gon has rankpsd = 3 6-gon has rankpsd = 4 We found numerically that 5-gon has rankpsd = 4 8-gon has rankpsd = 4 (which is less than 2 log2 8 − 1) Conjecture: regular 2n-gons have psd rank n + 1 Furthermore these factorization involve only rank-one psd matrices for one side of the factorization (and both for the square) Easier to analyze ?

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Thanks

Thank you for your attention and Many thanks to the organizers Rekha, Dirk, Hartmut and Troy!

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