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On the existence of 0/1 polytopes with high semidefinite extension complexity

Daniel Dadush

Centrum Wiskunde & Informatica (CWI)

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 1

Introduction

Joint Work with. . . My coauthors:

• Sebastian Pokutta (Georgia Tech)
• Jop Bri¨

et (CWI)

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 2

Introduction

What is the expressive power of linear / semidefinite programming?

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 3

Introduction

What is the expressive power of linear / semidefinite programming? For convex hulls such as Matchings, Hamiltonian cycles, Graph Cuts, ... what is the smallest linear / semidefinite program whose feasible region captures them (even approximately)?

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 3

Introduction

What is the expressive power of linear / semidefinite programming? For convex hulls such as Matchings, Hamiltonian cycles, Graph Cuts, ... what is the smallest linear / semidefinite program whose feasible region captures them (even approximately)? Alternative measure of complexity independent from P vs NP.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 3

Introduction

Lower bounds on the size of Linear Programs:

1 Any symmetric LP that captures the TSP or Matching

polytope must have size 2Ω(n) [Yannakakis ’91]

2 There exists a convex hull of 0/1 points that cannot be

captured by an LP of size less than 2Ω(n) [ Rothvoss ’11]

3 Any LP that captures the TSP polytope must have size

2Ω(n1/2) [Fiorini, Massar, Pokutta, Tiwary, de Wolf ’12]

4 Any LP that ρ-approximates the Correlation polytope must

have size 2Ω(n/ρ) [Braun, Fiornini, Pokutta, Steurer ’12,

Braverman, Moitra ’13, Pokutta, Braun ’13]

5 Any LP of relaxation of size nr for the Correlation polytope

has integrality gap at least as large as O(r) levels of Sherali-Adams [Chan, Lee, Raghavendra, Steurer ’13]

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 4

Introduction

Lower bounds on the size of Linear Programs:

1 Any symmetric LP that captures the TSP or Matching

polytope must have size 2Ω(n) [Yannakakis ’91]

2 There exists a convex hull of 0/1 points that cannot be

captured by an LP of size less than 2Ω(n) [ Rothvoss ’11]

3 Any LP that captures the TSP polytope must have size

2Ω(n1/2) [Fiorini, Massar, Pokutta, Tiwary, de Wolf ’12]

4 Any LP that ρ-approximates the Correlation polytope must

have size 2Ω(n/ρ) [Braun, Fiornini, Pokutta, Steurer ’12,

Braverman, Moitra ’13, Pokutta, Braun ’13]

5 Any LP of relaxation of size nr for the Correlation polytope

has integrality gap at least as large as O(r) levels of Sherali-Adams [Chan, Lee, Raghavendra, Steurer ’13]

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 4

Introduction

Lower bounds on the size of Linear Programs:

1 Any symmetric LP that captures the TSP or Matching

polytope must have size 2Ω(n) [Yannakakis ’91]

2 There exists a convex hull of 0/1 points that cannot be

captured by an LP of size less than 2Ω(n) [ Rothvoss ’11]

3 Any LP that captures the TSP polytope must have size

2Ω(n1/2) [Fiorini, Massar, Pokutta, Tiwary, de Wolf ’12]

4 Any LP that ρ-approximates the Correlation polytope must

have size 2Ω(n/ρ) [Braun, Fiornini, Pokutta, Steurer ’12,

Braverman, Moitra ’13, Pokutta, Braun ’13]

5 Any LP of relaxation of size nr for the Correlation polytope

has integrality gap at least as large as O(r) levels of Sherali-Adams [Chan, Lee, Raghavendra, Steurer ’13]

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 4

Introduction

Lower bounds on the size of Linear Programs:

1 Any symmetric LP that captures the TSP or Matching

polytope must have size 2Ω(n) [Yannakakis ’91]

2 There exists a convex hull of 0/1 points that cannot be

captured by an LP of size less than 2Ω(n) [ Rothvoss ’11]

3 Any LP that captures the TSP polytope must have size

2Ω(n1/2) [Fiorini, Massar, Pokutta, Tiwary, de Wolf ’12]

4 Any LP that ρ-approximates the Correlation polytope must

have size 2Ω(n/ρ) [Braun, Fiornini, Pokutta, Steurer ’12,

Braverman, Moitra ’13, Pokutta, Braun ’13]

5 Any LP of relaxation of size nr for the Correlation polytope

has integrality gap at least as large as O(r) levels of Sherali-Adams [Chan, Lee, Raghavendra, Steurer ’13]

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 4

Introduction

Lower bounds on the size of Linear Programs:

1 Any symmetric LP that captures the TSP or Matching

polytope must have size 2Ω(n) [Yannakakis ’91]

2 There exists a convex hull of 0/1 points that cannot be

captured by an LP of size less than 2Ω(n) [ Rothvoss ’11]

3 Any LP that captures the TSP polytope must have size

2Ω(n1/2) [Fiorini, Massar, Pokutta, Tiwary, de Wolf ’12]

4 Any LP that ρ-approximates the Correlation polytope must

have size 2Ω(n/ρ) [Braun, Fiornini, Pokutta, Steurer ’12,

Braverman, Moitra ’13, Pokutta, Braun ’13]

5 Any LP of relaxation of size nr for the Correlation polytope

has integrality gap at least as large as O(r) levels of Sherali-Adams [Chan, Lee, Raghavendra, Steurer ’13]

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 4

Introduction

What about Semidefinite Programs? Theorem ([Bri¨

et, D., Pokutta ’13])

There exists a convex hull of 0/1 points that cannot be captured by an SDP of size less than 2Ω(n). Extend framework of Rothvoss to SDP setting. Baby step towards understanding lower bounds for SDPs.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 5

Introduction

What about Semidefinite Programs? Theorem ([Bri¨

et, D., Pokutta ’13])

There exists a convex hull of 0/1 points that cannot be captured by an SDP of size less than 2Ω(n). Extend framework of Rothvoss to SDP setting. Baby step towards understanding lower bounds for SDPs.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 5

Introduction

What about Semidefinite Programs? Theorem ([Bri¨

et, D., Pokutta ’13])

There exists a convex hull of 0/1 points that cannot be captured by an SDP of size less than 2Ω(n). Extend framework of Rothvoss to SDP setting. Baby step towards understanding lower bounds for SDPs.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 5

Introduction

What about Semidefinite Programs? Theorem ([Bri¨

et, D., Pokutta ’13])

There exists a convex hull of 0/1 points that cannot be captured by an SDP of size less than 2Ω(n). Extend framework of Rothvoss to SDP setting. Baby step towards understanding lower bounds for SDPs.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 5

Linear Extensions

Polytope P = {x ∈ Rn : Ax ≤ b} with m facets. Question Can we reduce the number of inequalities needed to define P by adding auxilliary variables? Definition (Linear Extension) Q = {(z, y) : Cz + Dy = d, y ≥ 0, y ∈ Rr, z ∈ Rl}, is a linear extension of P of size r if ∃ π : Rl+r → Rn such that P = π(Q). Definition (Linear Extension Complexity) xc(P) := minimum size of any linear extension of P.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 6

Linear Extensions

Polytope P = {x ∈ Rn : Ax ≤ b} with m facets. Question Can we reduce the number of inequalities needed to define P by adding auxilliary variables? Definition (Linear Extension) Q = {(z, y) : Cz + Dy = d, y ≥ 0, y ∈ Rr, z ∈ Rl}, is a linear extension of P of size r if ∃ π : Rl+r → Rn such that P = π(Q). Definition (Linear Extension Complexity) xc(P) := minimum size of any linear extension of P.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 6

Linear Extensions

Polytope P = {x ∈ Rn : Ax ≤ b} with m facets. Question Can we reduce the number of inequalities needed to define P by adding auxilliary variables? Definition (Linear Extension) Q = {(z, y) : Cz + Dy = d, y ≥ 0, y ∈ Rr, z ∈ Rl}, is a linear extension of P of size r if ∃ π : Rl+r → Rn such that P = π(Q). Definition (Linear Extension Complexity) xc(P) := minimum size of any linear extension of P.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 6

Linear Extensions

Polytope P = {x ∈ Rn : Ax ≤ b} with m facets. Question Can we reduce the number of inequalities needed to define P by adding auxilliary variables? Definition (Linear Extension) Q = {(z, y) : Cz + Dy = d, y ≥ 0, y ∈ Rr, z ∈ Rl}, is a linear extension of P of size r if ∃ π : Rl+r → Rn such that P = π(Q). Definition (Linear Extension Complexity) xc(P) := minimum size of any linear extension of P.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 6

Linear Extensions

Extension Complexity. Some known results (constructions & lower bounds):

• xc(regular n-gon) = Θ(log n)

[Ben-Tal, Nemirovski’01]

• xc(generic n-gon) = Ω(√n)

[Fironi, Rothvoss, Tiwary’11]

• xc(n-permutahedron) = Θ(n log n)

[Goemans’09]

• xc(spanning tree polytope of Kn) = O(n3)

[Kipp-Martin’87]

• xc(spanning tree polytope of planar graph G) = Θ(n)

[Williams’01]

• xc(stable set polytope of perfect graph G) = nO(log n)

[Yannakakis’91]

• . . .

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 7

Linear Extensions

Slack Matrices. Let A ∈ Rm×n, b ∈ Rm, V = {v1, . . . , vN} ⊆ Rn s.t. P = {x ∈ Rn | Ax b} = conv(V)

Aix = bi Sij vj

Definition (slack matrix) Slack matrix S ∈ Rm×N

+

• f P (w.r.t. Ax b and V):

Sij := bi − Aivj, ∀ i ∈ [m], j ∈ [N]

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 8

Linear Extensions

Slack Matrices. Let A ∈ Rm×n, b ∈ Rm, V = {v1, . . . , vN} ⊆ Rn s.t. P = {x ∈ Rn | Ax b} = conv(V)

Aix = bi Sij vj

Definition (slack matrix) Slack matrix S ∈ Rm×N

+

• f P (w.r.t. Ax b and V):

Sij := bi − Aivj, ∀ i ∈ [m], j ∈ [N]

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 8

Linear Extensions

Nonnegative Factorizations and Extensions. Definition (slack matrix) Slack matrix S ∈ Rm×N

+

• f P (w.r.t. Ax b and V):

Sij := bi − Aivj, ∀ i ∈ [m], j ∈ [N] Definition A rank-r nonnegative factorization of S ∈ Rm×n

+

is S = UV where U ∈ Rm×r

+

and V ∈ Rr×n

+

Proposition (Extensions from Factorizations) Q = {(x, y) : Ax + Uy = b, y ≥ 0} is a linear extension of P of size r.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 9

Linear Extensions

Nonnegative Factorizations and Extensions. Definition (slack matrix) Slack matrix S ∈ Rm×N

+

• f P (w.r.t. Ax b and V):

Sij := bi − Aivj, ∀ i ∈ [m], j ∈ [N] Definition A rank-r nonnegative factorization of S ∈ Rm×n

+

is S = UV where U ∈ Rm×r

+

and V ∈ Rr×n

+

Proposition (Extensions from Factorizations) Q = {(x, y) : Ax + Uy = b, y ≥ 0} is a linear extension of P of size r.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 9

Linear Extensions

Nonnegative Factorizations and Extensions. Definition (Nonnegative Rank) rk+(S) := min{r | ∃ rank-r nonnegative factorization of S} Theorem (Factorization Theorem [Yannakakis’91]) For every slack matrix S of P: xc(P) = rk+(S)

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 10

PSD Extensions

PSD Matrices. Definition (PSD matrix) A matrix U ∈ Rr×r is PSD if U is symmetric and xTUx ≥ 0 ∀x ∈ Rr. Let Sr

+ denote the set of r × r PSD matrices.

Definition (Spectral Decomposition) U is r × r PSD iff U admits a Spectral Decomposition U =

r

• i=1

λiuiuT

i ,

λ1, . . . , λr ≥ 0, u1, . . . , ur an orthonormal basis.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 11

PSD Extensions

PSD Matrices. Definition (PSD matrix) A matrix U ∈ Rr×r is PSD if U is symmetric and xTUx ≥ 0 ∀x ∈ Rr. Let Sr

+ denote the set of r × r PSD matrices.

Definition (Spectral Decomposition) U is r × r PSD iff U admits a Spectral Decomposition U =

r

• i=1

λiuiuT

i ,

λ1, . . . , λr ≥ 0, u1, . . . , ur an orthonormal basis.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 11

PSD Extensions

PSD Matrices. Definition (Operator norm) For a matrix T ∈ Rr×r the operator norm of T is Top = max

x2=1 Tx2

For a PSD matrix U ∈ Rr×r Uop = max

x2=1 xTUx = largest eigenvalue of U.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 12

PSD Extensions

PSD Matrices. Definition (Trace) For a matrix T ∈ Rr×r, we define Tr [T] = r

i=1 Tii.

Remark (Trace Inner Product) For A, B ∈ Rr×r symmetric, Tr [AB] =

i,j∈[r] AijBij.

Proposition For PSD matrices U, V ∈ Sr

+,

Tr [UV ] =

• i,j∈[r]

λiγj ui, vj2 ≥ 0 , where U = r

i=1 λiuiuT i and V = r j=1 γjvjvT j are the respective

spectral decompositions.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 13

PSD Extensions

PSD Matrices. Definition (Trace) For a matrix T ∈ Rr×r, we define Tr [T] = r

i=1 Tii.

Remark (Trace Inner Product) For A, B ∈ Rr×r symmetric, Tr [AB] =

i,j∈[r] AijBij.

Proposition For PSD matrices U, V ∈ Sr

+,

Tr [UV ] =

• i,j∈[r]

λiγj ui, vj2 ≥ 0 , where U = r

i=1 λiuiuT i and V = r j=1 γjvjvT j are the respective

spectral decompositions.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 13

PSD Extensions

PSD Matrices. Definition (Trace) For a matrix T ∈ Rr×r, we define Tr [T] = r

i=1 Tii.

Remark (Trace Inner Product) For A, B ∈ Rr×r symmetric, Tr [AB] =

i,j∈[r] AijBij.

Proposition For PSD matrices U, V ∈ Sr

+,

Tr [UV ] =

• i,j∈[r]

λiγj ui, vj2 ≥ 0 , where U = r

i=1 λiuiuT i and V = r j=1 γjvjvT j are the respective

spectral decompositions.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 13

PSD Extensions

Polytope P = {x ∈ Rn : Ax ≤ b} with m facets. Definition (PSD Extension) Q = {(z, Y ) : Ciz + Tr [DiY ] = di, ∀ i ∈ [l], Y ∈ Sr

+, z ∈ Rl}, is a

PSD extension of P of size r if ∃ π : Rl × Sr

+ → Rn such that

P = π(Q). Definition (PSD Extension Complexity) xcpsd(P) := minimum size of any PSD extension of P.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 14

PSD Extensions

Polytope P = {x ∈ Rn : Ax ≤ b} with m facets. Definition (PSD Extension) Q = {(z, Y ) : Ciz + Tr [DiY ] = di, ∀ i ∈ [l], Y ∈ Sr

+, z ∈ Rl}, is a

PSD extension of P of size r if ∃ π : Rl × Sr

+ → Rn such that

P = π(Q). Definition (PSD Extension Complexity) xcpsd(P) := minimum size of any PSD extension of P.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 14

PSD Extensions

Polytope P = {x ∈ Rn : Ax ≤ b} with m facets. Definition (PSD Extension) Q = {(z, Y ) : Ciz + Tr [DiY ] = di, ∀ i ∈ [l], Y ∈ Sr

+, z ∈ Rl}, is a

PSD extension of P of size r if ∃ π : Rl × Sr

+ → Rn such that

P = π(Q). Definition (PSD Extension Complexity) xcpsd(P) := minimum size of any PSD extension of P.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 14

PSD Extensions

PSD Factorizations and Extensions Definition (Slack Matrix) Slack matrix S ∈ Rm×N

+

• f P (w.r.t. Ax b and vertices V):

Sij := bi − Aivj, ∀ i ∈ [m], j ∈ [N] Definition A rank-r PSD factorization of S ∈ Rm×N

+

is Sij = Tr [UiVj] where Ui, Vj are r × r PSD ∀ i ∈ [m], j ∈ [N]. Proposition (Extensions from Factorizations) Q = {(x, Y ) : Aix + Tr [UiY ] = bi, ∀i ∈ [m], Y ∈ Sr

+}

is a PSD extension of P of size r.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 15

PSD Extensions

PSD Factorizations and Extensions Definition (Slack Matrix) Slack matrix S ∈ Rm×N

+

• f P (w.r.t. Ax b and vertices V):

Sij := bi − Aivj, ∀ i ∈ [m], j ∈ [N] Definition A rank-r PSD factorization of S ∈ Rm×N

+

is Sij = Tr [UiVj] where Ui, Vj are r × r PSD ∀ i ∈ [m], j ∈ [N]. Proposition (Extensions from Factorizations) Q = {(x, Y ) : Aix + Tr [UiY ] = bi, ∀i ∈ [m], Y ∈ Sr

+}

is a PSD extension of P of size r.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 15

PSD Extensions

PSD Factorizations and Extensions. Definition (PSD Rank) rkpsd(S) := min{r | ∃ rank-r PSD factorization of S} Theorem (Factorization Theorem [Gouviea, Thomas, Parillo ’11,

Fiorini, Massar, Pokutta, Tiwary, de Wolf ’12])

For every slack matrix S of P: xcpsd(P) = rkpsd(S)

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 16

Counting Argument

Rothvoss’s Counting Argument

Goal: Show existence of X ⊆ {0, 1}n with xc(conv(X)) = 2Ω(n). Let R = maxX⊆{0,1}n xc(conv(X)). High level:

1 “Discretize” an optimal a linear EF for conv(X) to compress

the description of X.

2 Show that the number of discretized linear EFs of size R is

bounded by 2poly(R,n). Conclusion: 22n subsets of {0, 1}n means that R ≥ 2Ω(n).

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 17

Counting Argument

Rothvoss’s Counting Argument

Goal: Show existence of X ⊆ {0, 1}n with xc(conv(X)) = 2Ω(n). Let R = maxX⊆{0,1}n xc(conv(X)). High level:

1 “Discretize” an optimal a linear EF for conv(X) to compress

the description of X.

2 Show that the number of discretized linear EFs of size R is

bounded by 2poly(R,n). Conclusion: 22n subsets of {0, 1}n means that R ≥ 2Ω(n).

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 17

Counting Argument

Rothvoss’s Counting Argument

Goal: Show existence of X ⊆ {0, 1}n with xc(conv(X)) = 2Ω(n). Let R = maxX⊆{0,1}n xc(conv(X)). High level:

1 “Discretize” an optimal a linear EF for conv(X) to compress

the description of X.

2 Show that the number of discretized linear EFs of size R is

bounded by 2poly(R,n). Conclusion: 22n subsets of {0, 1}n means that R ≥ 2Ω(n).

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 17

Counting Argument

Rothvoss’s Counting Argument

Goal: Show existence of X ⊆ {0, 1}n with xc(conv(X)) = 2Ω(n). Let R = maxX⊆{0,1}n xc(conv(X)). High level:

1 “Discretize” an optimal a linear EF for conv(X) to compress

the description of X.

2 Show that the number of discretized linear EFs of size R is

bounded by 2poly(R,n). Conclusion: 22n subsets of {0, 1}n means that R ≥ 2Ω(n).

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 17

Counting Argument

Rothvoss’s Counting Argument

Goal: Show existence of X ⊆ {0, 1}n with xc(conv(X)) = 2Ω(n). Let R = maxX⊆{0,1}n xc(conv(X)). High level:

1 “Discretize” an optimal a linear EF for conv(X) to compress

the description of X.

2 Show that the number of discretized linear EFs of size R is

bounded by 2poly(R,n). Conclusion: 22n subsets of {0, 1}n means that R ≥ 2Ω(n).

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 17

Counting Argument

Rothvoss’s Counting Argument

Goal: Show existence of X ⊆ {0, 1}n with xcpsd(conv(X)) = 2Ω(n). Let R = maxX⊆{0,1}n xcpsd(conv(X)). High level:

1 “Discretize” an optimal a PSD EF for conv(X) to compress

the description of X.

2 Show that the number of discretized PSD EFs of size R is

bounded by 2poly(R,n). Conclusion: 22n subsets of {0, 1}n means that R ≥ 2Ω(n).

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 18

Discretizing EFs

Discretizing Linear EFs

X = {v1, . . . , vN} ⊆ {0, 1}n. conv(X) = {x ∈ Rn : Ax ≤ b}. By Hadamard can assume A ∈ Zm×n, b ∈ Zm and max |Aij|, max |bi| ≤ ∆ ≈ nn/2. Let Sij = bi − Aivj, ∀i ∈ [m], j ∈ [N], be the slack matrix.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 19

Discretizing EFs

Discretizing Linear EFs

X = {v1, . . . , vN} ⊆ {0, 1}n. conv(X) = {x ∈ Rn : Ax ≤ b}. By Hadamard can assume A ∈ Zm×n, b ∈ Zm and max |Aij|, max |bi| ≤ ∆ ≈ nn/2. Let Sij = bi − Aivj, ∀i ∈ [m], j ∈ [N], be the slack matrix.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 19

Discretizing EFs

Discretizing Linear EFs

X = {v1, . . . , vN} ⊆ {0, 1}n. conv(X) = {x ∈ Rn : Ax ≤ b}. By Hadamard can assume A ∈ Zm×n, b ∈ Zm and max |Aij|, max |bi| ≤ ∆ ≈ nn/2. Let Sij = bi − Aivj, ∀i ∈ [m], j ∈ [N], be the slack matrix.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 19

Discretizing EFs

Discretizing Linear EFs

Let S = UV be rank r nonnegative factorization. Examine linear EF Q = {(x, y) : Ax + Uy = b, y ≥ 0}. Let (AS, US), S ⊆ [m], be any maximal set of linearly independent set of rows of (A, U). Note that Q = {(x, y) : ASx + USy = bS, y ≥ 0}. Can represent conv(X) using only at most (n + r) × (n + r + 1) numbers!

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 20

Discretizing EFs

Discretizing Linear EFs

Let S = UV be rank r nonnegative factorization. Examine linear EF Q = {(x, y) : Ax + Uy = b, y ≥ 0}. Let (AS, US), S ⊆ [m], be any maximal set of linearly independent set of rows of (A, U). Note that Q = {(x, y) : ASx + USy = bS, y ≥ 0}. Can represent conv(X) using only at most (n + r) × (n + r + 1) numbers!

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 20

Discretizing EFs

Discretizing Linear EFs

Let S = UV be rank r nonnegative factorization. Examine linear EF Q = {(x, y) : Ax + Uy = b, y ≥ 0}. Let (AS, US), S ⊆ [m], be any maximal set of linearly independent set of rows of (A, U). Note that Q = {(x, y) : ASx + USy = bS, y ≥ 0}. Can represent conv(X) using only at most (n + r) × (n + r + 1) numbers!

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 20

Discretizing EFs

Discretizing Linear EFs

Let S = UV be rank r nonnegative factorization. Examine linear EF Q = {(x, y) : Ax + Uy = b, y ≥ 0}. Let (AS, US), S ⊆ [m], be any maximal set of linearly independent set of rows of (A, U). Note that Q = {(x, y) : ASx + USy = bS, y ≥ 0}. Can represent conv(X) using only at most (n + r) × (n + r + 1) numbers!

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 20

Discretizing EFs

Discretizing Linear EFs

Let S = UV be rank r nonnegative factorization. Examine linear EF Q = {(x, y) : Ax + Uy = b, y ≥ 0}. Let (AS, US), S ⊆ [m], be any maximal set of linearly independent set of rows of (A, U). Note that Q = {(x, y) : ASx + USy = bS, y ≥ 0}. Can represent conv(X) using only at most (n + r) × (n + r + 1) numbers!

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 20

Discretizing EFs

Discretizing Linear EFs

Q = {(x, y) : ASx + USy = bS, y ≥ 0}. Can represent conv(X) using only at most (n + r) × (n + r + 1) numbers! Issues with counting:

1 Numbers in US may be unbounded. 2 Numbers in US don’t fall in a discrete set.

Fix idea:

1 Round numbers in U to a grid. 2 Build “rounded” EF ¯

Q that contains SAME 0/1 points as Q.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 21

Discretizing EFs

Discretizing Linear EFs

Q = {(x, y) : ASx + USy = bS, y ≥ 0}. Can represent conv(X) using only at most (n + r) × (n + r + 1) numbers! Issues with counting:

1 Numbers in US may be unbounded. 2 Numbers in US don’t fall in a discrete set.

Fix idea:

1 Round numbers in U to a grid. 2 Build “rounded” EF ¯

Q that contains SAME 0/1 points as Q.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 21

Discretizing EFs

Discretizing Linear EFs

Rounded Linear EF: ¯ Q = {(x, y) : ASx + ¯ USy − bS∞ ≤ 1/poly(n, r), y ≥ 0, y∞ ≤ √ ∆}. High Level Fix:

1 Rescale factorization S = UV so that max entry U, V ≤

√ ∆.

2 Round U → ¯

U to nearest multiple of 1/poly(n, r, ∆).

3 Choose rows S “carefully”. 4 Show that πx( ¯

Q) ∩ {0, 1}n = X. Can represent X using only at most (n + r) × (n + r + 1) numbers in a bounded set!

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 22

Discretizing EFs

Discretizing Linear EFs

Rounded Linear EF: ¯ Q = {(x, y) : ASx + ¯ USy − bS∞ ≤ 1/poly(n, r), y ≥ 0, y∞ ≤ √ ∆}. High Level Fix:

1 Rescale factorization S = UV so that max entry U, V ≤

√ ∆.

2 Round U → ¯

U to nearest multiple of 1/poly(n, r, ∆).

3 Choose rows S “carefully”. 4 Show that πx( ¯

Q) ∩ {0, 1}n = X. Can represent X using only at most (n + r) × (n + r + 1) numbers in a bounded set!

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 22

Discretizing EFs

Discretizing Linear EFs

Rounded Linear EF: ¯ Q = {(x, y) : ASx + ¯ USy − bS∞ ≤ 1/poly(n, r), y ≥ 0, y∞ ≤ √ ∆}. High Level Fix:

1 Rescale factorization S = UV so that max entry U, V ≤

√ ∆.

2 Round U → ¯

U to nearest multiple of 1/poly(n, r, ∆).

3 Choose rows S “carefully”. 4 Show that πx( ¯

Q) ∩ {0, 1}n = X. Can represent X using only at most (n + r) × (n + r + 1) numbers in a bounded set!

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 22

Discretizing EFs

Discretizing Linear EFs

Rounded Linear EF: ¯ Q = {(x, y) : ASx + ¯ USy − bS∞ ≤ 1/poly(n, r), y ≥ 0, y∞ ≤ √ ∆}. High Level Fix:

1 Rescale factorization S = UV so that max entry U, V ≤

√ ∆.

2 Round U → ¯

U to nearest multiple of 1/poly(n, r, ∆).

3 Choose rows S “carefully”. 4 Show that πx( ¯

Q) ∩ {0, 1}n = X. Can represent X using only at most (n + r) × (n + r + 1) numbers in a bounded set!

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 22

Discretizing EFs

Discretizing Linear EFs

Rounded Linear EF: ¯ Q = {(x, y) : ASx + ¯ USy − bS∞ ≤ 1/poly(n, r), y ≥ 0, y∞ ≤ √ ∆}. High Level Fix:

1 Rescale factorization S = UV so that max entry U, V ≤

√ ∆.

2 Round U → ¯

U to nearest multiple of 1/poly(n, r, ∆).

3 Choose rows S “carefully”. 4 Show that πx( ¯

Q) ∩ {0, 1}n = X. Can represent X using only at most (n + r) × (n + r + 1) numbers in a bounded set!

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 22

Discretizing EFs

Discretizing PSD EFs

Question: What changes for PSD factorizations? Rounded PSD EF: ¯ Q = {(x, y) : Aix + Tr ¯ UiY

• − bi∞ ≤ 1/poly(n, r), ∀i ∈ S,

Y op ≤???, Y ∈ Sr

+}.

Solution steps:

1 Rescale factorization Sij = Tr [UiVj] so that

maxi Uiop, maxj Vjop ≤???.

2 Round each Ui ⇒ ¯

Ui to nearest multiple of 1/poly(n, r, ???).

3 Choose rows of S “carefully”. 4 Show that πx( ¯

Q) ∩ {0, 1}n = X. Can represent X using only at most (n + r2) × (n + r2 + 1) numbers in a bounded set!

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 23

Discretizing EFs

Discretizing PSD EFs

Question: What changes for PSD factorizations? Rounded PSD EF: ¯ Q = {(x, y) : Aix + Tr ¯ UiY

• − bi∞ ≤ 1/poly(n, r), ∀i ∈ S,

Y op ≤???, Y ∈ Sr

+}.

Solution steps:

1 Rescale factorization Sij = Tr [UiVj] so that

maxi Uiop, maxj Vjop ≤???.

2 Round each Ui ⇒ ¯

Ui to nearest multiple of 1/poly(n, r, ???).

3 Choose rows of S “carefully”. 4 Show that πx( ¯

Q) ∩ {0, 1}n = X. Can represent X using only at most (n + r2) × (n + r2 + 1) numbers in a bounded set!

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 23

Discretizing EFs

Discretizing PSD EFs

Question: What changes for PSD factorizations? Rounded PSD EF: ¯ Q = {(x, y) : Aix + Tr ¯ UiY

• − bi∞ ≤ 1/poly(n, r), ∀i ∈ S,

Y op ≤???, Y ∈ Sr

+}.

Solution steps:

1 Rescale factorization Sij = Tr [UiVj] so that

maxi Uiop, maxj Vjop ≤???.

2 Round each Ui ⇒ ¯

Ui to nearest multiple of 1/poly(n, r, ???).

3 Choose rows of S “carefully”. 4 Show that πx( ¯

Q) ∩ {0, 1}n = X. Can represent X using only at most (n + r2) × (n + r2 + 1) numbers in a bounded set!

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 23

Discretizing EFs

Discretizing PSD EFs

Question: What changes for PSD factorizations? Rounded PSD EF: ¯ Q = {(x, y) : Aix + Tr ¯ UiY

• − bi∞ ≤ 1/poly(n, r), ∀i ∈ S,

Y op ≤???, Y ∈ Sr

+}.

Solution steps:

1 Rescale factorization Sij = Tr [UiVj] so that

maxi Uiop, maxj Vjop ≤???.

2 Round each Ui ⇒ ¯

Ui to nearest multiple of 1/poly(n, r, ???).

3 Choose rows of S “carefully”. 4 Show that πx( ¯

Q) ∩ {0, 1}n = X. Can represent X using only at most (n + r2) × (n + r2 + 1) numbers in a bounded set!

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 23

Discretizing EFs

Discretizing PSD EFs

Question: What changes for PSD factorizations? Rounded PSD EF: ¯ Q = {(x, y) : Aix + Tr ¯ UiY

• − bi∞ ≤ 1/poly(n, r), ∀i ∈ S,

Y op ≤???, Y ∈ Sr

+}.

Solution steps:

1 Rescale factorization Sij = Tr [UiVj] so that

maxi Uiop, maxj Vjop ≤???.

2 Round each Ui ⇒ ¯

Ui to nearest multiple of 1/poly(n, r, ???).

3 Choose rows of S “carefully”. 4 Show that πx( ¯

Q) ∩ {0, 1}n = X. Can represent X using only at most (n + r2) × (n + r2 + 1) numbers in a bounded set!

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 23

Discretizing EFs

Discretizing PSD EFs

Question: What changes for PSD factorizations? Rounded PSD EF: ¯ Q = {(x, y) : Aix + Tr ¯ UiY

• − bi∞ ≤ 1/poly(n, r), ∀i ∈ S,

Y op ≤???, Y ∈ Sr

+}.

Solution steps:

1 Rescale factorization Sij = Tr [UiVj] so that

maxi Uiop, maxj Vjop ≤???.

2 Round each Ui ⇒ ¯

Ui to nearest multiple of 1/poly(n, r, ???).

3 Choose rows of S “carefully”. 4 Show that πx( ¯

Q) ∩ {0, 1}n = X. Can represent X using only at most (n + r2) × (n + r2 + 1) numbers in a bounded set!

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 23

Rescaling Factorizations

Rescaling PSD Factorizations

Theorem ([Bri¨

et, D., Pokutta])

Let S ∈ Rm×N

+

with rkpsd(S) = r and S∞ = ∆. Then ∃ rank r PSD factorization Sij = Tr [UiVj], ∀ i, j such that max

i

Uiop, max

j

Vjop ≤ √ r∆

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 24

Rescaling Factorizations

Rescaling Nonnegative Factorizations

Lemma Let S ∈ Rm×N

+

with rk+(S) = r and S∞ = ∆. Then ∃ rank r nonnegative factorization S = UV such that U∞, V ∞ ≤ √ ∆

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 25

Rescaling Factorizations

Rescaling Nonnegative Factorizations

Proof: U =

• u1

. . . ur

• , V =

   vT

1

. . . vT

r

   S = r

i=1 uivT i .

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 26

Rescaling Factorizations

Rescaling Nonnegative Factorizations

Proof: U =

• u1

. . . ur

• , V =

   vT

1

. . . vT

r

   S = r

i=1 uivT i .

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 26

Rescaling Factorizations

Rescaling Nonnegative Factorizations

Choose λ1, . . . , λr > 0 such that λiui∞ = 1/λivi∞. Let U ′ =

• λ1u1

. . . λrur

• , V ′ =

  1/λ1vT

1

. . . 1/λrvT

r

 . Note that U ′V ′ = r

i=1(λiui)(1/λivT i ) = r i=1 uivT i = S.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 27

Rescaling Factorizations

Rescaling Nonnegative Factorizations

Choose λ1, . . . , λr > 0 such that λiui∞ = 1/λivi∞. Let U ′ =

• λ1u1

. . . λrur

• , V ′ =

  1/λ1vT

1

. . . 1/λrvT

r

 . Note that U ′V ′ = r

i=1(λiui)(1/λivT i ) = r i=1 uivT i = S.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 27

Rescaling Factorizations

Rescaling Nonnegative Factorizations

Choose λ1, . . . , λr > 0 such that λiui∞ = 1/λivi∞. Let U ′ =

• λ1u1

. . . λrur

• , V ′ =

  1/λ1vT

1

. . . 1/λrvT

r

 . Note that U ′V ′ = r

i=1(λiui)(1/λivT i ) = r i=1 uivT i = S.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 27

Rescaling Factorizations

Rescaling Nonnegative Factorizations

M def = U ′∞ = maxi λiui∞ = maxi 1/λvi∞ = V ′∞. Pick j ∈ [r] such that M = λjuj∞ = 1/λjvj∞. ∆ = S∞ = U ′V ′∞ ≥ (λjuj)(1/λjvT

j )∞

= λjuj∞1/λjvj∞ = M2. Therefore U ′∞ = V ′∞ = M ≤ √ ∆ as needed.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 28

Rescaling Factorizations

Rescaling Nonnegative Factorizations

M def = U ′∞ = maxi λiui∞ = maxi 1/λvi∞ = V ′∞. Pick j ∈ [r] such that M = λjuj∞ = 1/λjvj∞. ∆ = S∞ = U ′V ′∞ ≥ (λjuj)(1/λjvT

j )∞

= λjuj∞1/λjvj∞ = M2. Therefore U ′∞ = V ′∞ = M ≤ √ ∆ as needed.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 28

Rescaling Factorizations

Rescaling Nonnegative Factorizations

M def = U ′∞ = maxi λiui∞ = maxi 1/λvi∞ = V ′∞. Pick j ∈ [r] such that M = λjuj∞ = 1/λjvj∞. ∆ = S∞ = U ′V ′∞ ≥ (λjuj)(1/λjvT

j )∞

= λjuj∞1/λjvj∞ = M2. Therefore U ′∞ = V ′∞ = M ≤ √ ∆ as needed.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 28

Rescaling Factorizations

Rescaling Nonnegative Factorizations

M def = U ′∞ = maxi λiui∞ = maxi 1/λvi∞ = V ′∞. Pick j ∈ [r] such that M = λjuj∞ = 1/λjvj∞. ∆ = S∞ = U ′V ′∞ ≥ (λjuj)(1/λjvT

j )∞

= λjuj∞1/λjvj∞ = M2. Therefore U ′∞ = V ′∞ = M ≤ √ ∆ as needed.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 28

Rescaling Factorizations

Rescaling PSD Factorizations

Theorem ([Bri¨

et, D., Pokutta])

Let S ∈ Rm×N

+

with rkpsd(S) = r and S∞ = ∆. Then ∃ rank r PSD factorization Sij = Tr [UiVj], ∀ i, j such that max

i

Uiop, max

j

Vjop ≤ √ r∆ Comparison with Nonnegative Factorizations:

• Nonnegative setting: can rescale entries of non-negative

vector independently and maintain non-negativity.

• PSD setting: entries can be NEGATIVE.

CANNOT rescale entries of PSD matrix independently while maintaining PSD property.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 29

Rescaling Factorizations

Rescaling PSD Factorizations

Theorem ([Bri¨

et, D., Pokutta])

Let S ∈ Rm×N

+

with rkpsd(S) = r and S∞ = ∆. Then ∃ rank r PSD factorization Sij = Tr [UiVj], ∀ i, j such that max

i

Uiop, max

j

Vjop ≤ √ r∆ Comparison with Nonnegative Factorizations:

• Nonnegative setting: can rescale entries of non-negative

vector independently and maintain non-negativity.

• PSD setting: entries can be NEGATIVE.

CANNOT rescale entries of PSD matrix independently while maintaining PSD property.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 29

Rescaling Factorizations

Rescaling PSD Factorizations

Theorem ([Bri¨

et, D., Pokutta])

Let S ∈ Rm×N

+

with rkpsd(S) = r and S∞ = ∆. Then ∃ rank r PSD factorization Sij = Tr [UiVj], ∀ i, j such that max

i

Uiop, max

j

Vjop ≤ √ r∆ Comparison with Nonnegative Factorizations:

• Nonnegative setting: can rescale entries of non-negative

vector independently and maintain non-negativity.

• PSD setting: entries can be NEGATIVE.

CANNOT rescale entries of PSD matrix independently while maintaining PSD property.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 29

Rescaling Factorizations

Rescaling PSD Factorizations

Theorem ([Bri¨

et, D., Pokutta])

Let S ∈ Rm×N

+

with rkpsd(S) = r and S∞ = ∆. Then ∃ rank r PSD factorization Sij = Tr [UiVj], ∀ i, j such that max

i

Uiop, max

j

Vjop ≤ √ r∆ Admissible PSD Rescalings: For A ∈ Rr×r invertible, send Ui → ATUiA and Vj → A−1ViA−T. Preserves inner product: Tr

• ATUiAA−1VjA−T

= Tr

• UiVjA−TAT

= Tr [UiVj] = Sij. Map U → ATUA is a symmetry of PSD cone.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 30

Rescaling Factorizations

Rescaling PSD Factorizations

Theorem ([Bri¨

et, D., Pokutta])

Let S ∈ Rm×N

+

with rkpsd(S) = r and S∞ = ∆. Then ∃ rank r PSD factorization Sij = Tr [UiVj], ∀ i, j such that max

i

Uiop, max

j

Vjop ≤ √ r∆ Admissible PSD Rescalings: For A ∈ Rr×r invertible, send Ui → ATUiA and Vj → A−1ViA−T. Preserves inner product: Tr

• ATUiAA−1VjA−T

= Tr

• UiVjA−TAT

= Tr [UiVj] = Sij. Map U → ATUA is a symmetry of PSD cone.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 30

Rescaling Factorizations

Rescaling PSD Factorizations

Theorem ([Bri¨

et, D., Pokutta])

Let S ∈ Rm×N

+

with rkpsd(S) = r and S∞ = ∆. Then ∃ rank r PSD factorization Sij = Tr [UiVj], ∀ i, j such that max

i

Uiop, max

j

Vjop ≤ √ r∆ Admissible PSD Rescalings: For A ∈ Rr×r invertible, send Ui → ATUiA and Vj → A−1ViA−T. Preserves inner product: Tr

• ATUiAA−1VjA−T

= Tr

• UiVjA−TAT

= Tr [UiVj] = Sij. Map U → ATUA is a symmetry of PSD cone.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 30

Rescaling Factorizations

Rescaling PSD Factorizations

Proof Idea: Variational Argument

1 Choose rescaling A ∈ Sr + such that potential

maxi AUiAop × maxj A−1VjA−1op is minimized.

2 Show that if potential > r∆, can find infinitessimal

pertubation A → (I + ǫP)A(I + ǫP) which decreases potential. Require convex geometric tools (John’s decomposition of the identity) to build perturbation P.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 31

Rescaling Factorizations

Rescaling PSD Factorizations

Proof Idea: Variational Argument

1 Choose rescaling A ∈ Sr + such that potential

maxi AUiAop × maxj A−1VjA−1op is minimized.

2 Show that if potential > r∆, can find infinitessimal

pertubation A → (I + ǫP)A(I + ǫP) which decreases potential. Require convex geometric tools (John’s decomposition of the identity) to build perturbation P.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 31

Rescaling Factorizations

Rescaling PSD Factorizations

Proof Idea: Variational Argument

1 Choose rescaling A ∈ Sr + such that potential

maxi AUiAop × maxj A−1VjA−1op is minimized.

2 Show that if potential > r∆, can find infinitessimal

pertubation A → (I + ǫP)A(I + ǫP) which decreases potential. Require convex geometric tools (John’s decomposition of the identity) to build perturbation P.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 31

Open problems and future work.

1 Is the rescaling theorem tight? Do we need dependance on r? 2 Show that any nr PSD extended formulation for Correlation

polytope has integrality gap as large as O(r) levels of Lasserre.

3 Extension complexity of the matching polytope.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 32

Open problems and future work.

1 Is the rescaling theorem tight? Do we need dependance on r? 2 Show that any nr PSD extended formulation for Correlation

polytope has integrality gap as large as O(r) levels of Lasserre.

3 Extension complexity of the matching polytope.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 32

Open problems and future work.

1 Is the rescaling theorem tight? Do we need dependance on r? 2 Show that any nr PSD extended formulation for Correlation

polytope has integrality gap as large as O(r) levels of Lasserre.

3 Extension complexity of the matching polytope.

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 32

Thank you!

Daniel Dadush On the existence of 0/1 polytopes with high SDP rank 33