SDPs with rank constraint & Grothendieck inequalities Frank - - PowerPoint PPT Presentation

SDPs with rank constraint & Grothendieck inequalities

Frank Vallentin (TU Delft, CWI Amsterdam) Jop Briët (CWI Amsterdam) Fernando de Oliveira Filho (U Tilburg)

15th Workshop on Combinatorial optimization January 13, 2010

Overview

• 1. The problem
• 2. SDP relaxation
• 3. Approximation algorithm

http://arxiv.org/abs/1011.1754 http://arxiv.org/abs/0910.5765

• 1. The problem

image credit: Sturmfels, Uhler

interaction graph G = (V, E) symmetric matrix A: V × V → R

SDPr(G, A) = max

• {u,v}∈E

A(u, v)f(u) · f(v) : f : V → Sr−1

• Sr−1 = { x ∈ Rr : x · x = 1 }

graphical Grothendieck problem with rank-r constraint

linear optimization

• ver elliptope

with rank constraint

• X ∈ Sn

≥0 : X11 = . . . = Xnn = 1

• SDPr(G, A) = max
• {u,v}∈E

A(u, v)f(u) · f(v) : f : V → Sr−1

• = 1

2 max{A, X : X ∈ elliptope, rank X ≤ r}

MAX CUT(G) = SDP1

• Kn,n,

LG LG

• r = 1

S0 = {−1, +1}

LG Laplacian matrix of G

r-vector spin glass model

interaction graph G = (V, E)

atoms bonds

symmetric matrix A: V × V → R

A(u, v) > 0 for ferromagnetic interaction A(u, v) < 0 for antiferromagnetic interaction A(u, v) = 0 if {u, v} ∈ E, no interaction

minimize energy: −

• {u,v}∈E

A(u, v)f(u) · f(v)

spins f : V → Sr−1

Hardness results

NP-hard: Approximate MAX CUT within factor 16/17 = 0.94 . . . Unique games hard: Approximate MAX CUT within 0.87 . . .

Unique games hard: Approximate SDPr(Kn, A) within Θ(1 − 1

r)

(H˚ astad (2001)) (Kindler, Khot, Mossel, O’Donnell (2007)) (BOV (2009))

Natural SDP relaxation

Drop rank constraint! (set r = ∞)

Quality measured by dimensionality gap: Smallest constant K(r, G) with ∀A : SDP∞(G, A) ≤ K(r, G)SDPr(G, A)

Grothendieck inequality

• 2. SDP relaxation

Why Grothendieck?

Grothendieck (1953) showed: K(1, Kn,n) < C

(C independent of n, best possible C unknown until today) context: metric theory of tensor products

More propaganda

Grothendieck inequalities are unifying & have many applications:

• combinatorics: cut norms, graph limits, Szem´

eredi partitions (Alon-Naor (2006), Lov´ asz-Sz´ egey (2007)

• machine learning: kernel clustering

(Khot-Naor (2008))

• system theory: matrix cube theorem

(Ben-Tal, Nemirovski (2003))

• quantum information theory: XOR games

(Bri¨ et, Buhrman, Toner (2009))

• numerical linear algebra: column subset selection

(Tropp (2009))

• 3. Approximation

algorithm

• 1. Solve SDP∞(G, A),
• btaining optimal solution f : V → S|V |−1.
• 2. Use f to construct g: V → S|V |−1

according to Lemma 1.2.

• 3. Choose Z = (Zij) ∈ Rr×|V | at random, Zij ∼ N(0, 1).
• 4. Set h(u) = Zg(u)/Zg(u).

Approximation algorithm

Results: Upper bounds for K(r, G)

r bipartite G tripartite G 1 1.782213 . . . 3.264251 . . . 2 1.404909 . . . 2.621596 . . . 3 1.280812 . . . 2.412700 . . . 4 1.216786 . . . 2.309224 . . . 5 1.177179 . . . 2.247399 . . .

Krivine (1979): Bound for bipartite K(1, G) Haagerup (1987): Bound for bipartite K(2, G)

if r = 1 crucial in analysis of Goemans-Williamson

Crucial integral

E Zu Zu · Zv Zv

• = 2

π arcsin(u · v)

·

∞

• k=0

(1 · 3 · · · (2k − 1))2 (2 · 4 · · · 2k)((r + 2) · (r + 4) · · · (r + 2k))(u · v)2k+1

2 r Γ((r + 1)/2) Γ(r/2) 2

if r > 1:

E Zu Zu · Zv Zv

• =

Z ∈ Rr×|V |

set up notation

write down integral spherical coordinates substitutions, integration by part hypergeometric functions some tricks done

Conclusion

• 1. Grothendieck inequalities measure dimensionality gaps
• 2. Upper bounds for dimensionality gap give

Goemans-Williamson-style approximation algorithms

• 3. Exact analysis involved because of complicated integrals
• 4. Unique games conjecture =

Can we go beyound these SDP relaxations?

http://arxiv.org/abs/1011.1754 http://arxiv.org/abs/0910.5765

Details?