1. Proving Analytic Inequalities
Avi Wigderson
IAS, Princeton
Math and Computation
New book on my website

2. Past 2 lectures
Alternate minimization & Symbolic matrices: analytic algorithms for algebraic problems.
Polynomial identities: algebraic tools for understanding analytic algorithms.
Today
Applications: Analysis & Optimization

3. Ubiquity of matrix tuples [GGOW’15’16]
Computational Complexity
(A1,A2,…,Am): symbolic matrix iAixi
Quantum Information Theory
(A1,A2,…,Am): completely positive operator
Non-commutative Algebra
(A1,A2,…,Am): rational expression
Invariant Theory
(A1,A2,…,Am): orbit under Left-Right action
Analysis
(A1,A2,…,Am): projectors in
Brascamp-Lieb inequalities
Optimization
(A1,A2,…,Am): exp size linear programs
In P
In P
In P
In P
In P
In P

4. Brascamp-Lieb Inequalities [BL’76,Lieb’90]
∫ ∏j fj ≤ C ∏j |fj|pj
Propaganda: special cases & extensions
Cauchy-Schwarz,Holder Precopa-Leindler
Loomis-Whitney Nelson Hypercontractive
Young’s convolution Brunn-Minkowski
Lieb’s Non-commutative BL Barthe Reverse BL
Bennett-Bez Nonlinear BL Quantitative Helly
Analysis, Geometry, Probability, Information Theory,…

5. Brascamp-Lieb Inequalities [BL’76,Lieb’90]
Input B = (B1,B2,…,Bm) Bj:RnRnj linear
(BL data) p = (p1,p2,…,pm) pj ≥0
∫xRn ∏j fj(Bj(x)) ≤ C ∏j |fj|pj
f = (f1,f2,…,fm) ( fj:Rnj R+ integrable )
[Garg-Gurvits-Oliveira-W’16]
Feasibility & Optimal C in P
(through Operator Scaling & Alternate Minimization)
Optimization: solving (some) exponential size LPs

6. Plan Examples Notation General statement Structural theory f:Rd R+
Algorithm
Consequences:
Structure
Optimization (?)
Notation
f:Rd R+
|f|1/p = (∫xRd f(x)1/p)p

7. Examples

8. Cauchy-Schwarz, Holder
d=1 f1,f2:R R+
[CS] ∫xR f1(x)f2(x) ≤ |f1|2|f2| p1=p2=1/2
any other norms?
[H] ∫xR f1(x)f2(x) ≤ |f1|1/p1|f2|1/p p1+p2=1
pi≥0

9. Loomis-Whitney I d=2, x=(x1,x2) f1,f2:R R+
[Trivial] ∫xR2 f1(x1)f2(x2) = |f1|1|f2| p1=p2=1 x1 x2 a1 a2 A
area(A) ≤len(a1)len(a2)
H(Z1Z2) ≤ H(Z1)+H(Z2)

10. Loomis-Whitney II d=3, x=(x1,x2,x3) f1,f2,f3:R2 R+
[LW] ∫xR3 f1(x2x3)f2(x1x3)f3(x1x2) ≤ |f1|2|f2|2|f3|2
pi=½ x1 x2
A12
vol(S) ≤ [area(A12)area(A13)area(A23)]1/2
A13
A23 x3 S
any other norms?
H(Z1Z2Z2) ≤
½[H(Z1Z2)+H(Z2Z3)+H(Z1Z3)]

11. Young I d=2, x=(x1,x2) f1,f2,f3:R R+
[Young] ∫xR2 f1(x1)f2(x2)f3(x1+2)
≤ (√3)/2 |f1|3/2|f2|3/2|f3|3/2 pi=2/3 x1 x2 a1 a2 A
x1+x2 a3
Any other norms?
area(A) ≤
(√3)/2 [len(a1)len(a2)len(a3)]2/3

12. Young II C = d=2, x=(x1,x2) f1,f2,f3:R2 R+
[Young] ∫xR2 f1(x1)f2(x2)f3(x1+2)
≤ C |f1|1/p1|f2|1/p2|f3|1/p p1+p2+p3=2
1≥pi≥0
q1q1q2q2q3q3
p1p1p2p2p3p qi=1-pi
C =

13. Brascamp-Lieb Inequalities [BL’76,Lieb’90]
Input B = (B1,B2,…,Bm) Bj:RnRnj
(BL data) p = (p1,p2,…,pm) pj ≥0
∫xRn ∏j fj(Bj(x)) ≤ C ∏j |fj|1/pj
f = (f1,f2,…,fm) ( fj:Rnj R+ integrable )
Given BL data (B,p): Is there a finite C?
What is the smallest C? [ BL(B,p) ]
[GGOW’16] Feasibility & Optimal C in P

14. (look for similarities to lecture 1)
Structure
(look for similarities to lecture 1)

15. Feasibility: C<∞ [Bennett-Carbery-Christ-Tao’08]
Input B = (B1,B2,…,Bm) Bj:RnRnj
(BL data) p = (p1,p2,…,pm) pj ≥0
∫xRn ∏j fj(Bj(x)) ≤ C ∏j |fj|1/pj fi
[BCCT’08] C<∞ iff p  PB (the Polytope of B)
PB: ∑j pj nj = n
∑j pj dim(BjV) ≥ dim(V) subspace V in Rm
pj ≥0
(Exponentially many inequalities)
Bennett, Carbery, Christ, Tao
Rank non-decreasing

16. for some completely positive operator L  (B,p)
BL-constant [Lieb’90]
Input B = (B1,B2,…,Bm) Bj:RnRnj
(BL data) p = (p1,p2,…,pm) pj ≥0
∫xRn ∏j fj(Bj(x)) ≤ C ∏j |fj|1/pj fi
[Lieb’90] BL(B,p) is optimized when fj are Gaussian
Non-
convex
sup ∏j det(Aj)pj
Aj>0 det(∑j pj BjtAjBj)
1/cap(L) =
BL(B,p)2 =
for some completely positive operator L  (B,p) A1 A2 A3 A4 A5 B1 B2 B3
Quiver
reduction

17. Algorithms

18. Geometric BL [Ball’89,Barthe’98]
Input B = (B1,B2,…,Bm) Bj:RnRnj
(BL data) p = (p1,p2,…,pm) pj ≥0
∫xRn ∏j fj(Bj(x)) ≤ C ∏j |fj|1/pj fi
[B’89] (B,p) is geometric if
(Projection) BjBjt = Inj j
(Isotropy) ∑j pj BjtBj = In
[B’89] (B,p) geometric  BL(B,p)=1
Doubly-
stochastic

19. Alternate Minimization [GGOW’16]
[B’89] (B,p) is geometric if
(1) BjBjt = Inj j [Projection property]
(2) ∑j pj BjtBj = In [Isotropy property]
On input (B,p): attempt to make it geometric
Converges quickly iff (B,p) is feasible
[GGOW’16]
Feasibility (testing C<∞, p  PB) in polynomial time
Weak separation oracle
Feasible (B,p) converges to geometric in polytime
Keeping track of changes approximates BL(B,p)
Structure: bounds & continuity of BL(B,p), LP bounds.
Repeat t=nc times:
- Satisfy Projection (Right basis change)
- Satisfy Isotropy (Left basis change)

20. Optimization

21. Linear programming & Polytopes
P = conv {0, e1, e2,… em}  Rm
= { pRm: ∑j pj ≤ 1
pj ≥ 0 j[m] }
Membership Problem: Test if pP?
Easy if P has few inequalities, in a large
variety of settings with many inequalities, and when… [GGOW’16] P is a BL-polytope!
B = (B1,B2,…,Bm) Bj:RnRnj
PB: { pRm: ∑j pj nj = n
∑j pj dim(BjV) ≥ dim(V)  V ≤ Rn
pj ≥ ??Applications?? e2 e1

22. Optimization: linear programs with exponentially many inequalities
BL polytopes capture Matroids
M = {v1, v2,…… vm} vjRn
VJ = {vj : jJ}
PM = conv {1J: VJ is a basis}  Rm
= { pRm: ∑jJ pj ≤ dim(VJ) J[m]
pj ≥ j[m] }
Bj:RnR Bjx=<vj,x> j[m]
[Fact] PB = PM
Exponentially many
Inequalities

23. Optimization: linear programs with exponentially many inequalities
BL polytopes capture Matroid Intersection
M = {v1, v2,…… vm} vjRn
N = {u1, u2,…… um} ujRn
PM,N = conv {1J: VJ,UJ are bases}  Rm
[Edmonds] = {pRm: ∑j pj ≤ dim(VJ) J[m]
∑j pj ≤ dim(UJ) J[m]
pj ≥ j[m] }
Bj:R2nR2 Bjx=<vj,xL>,<uj,xR> j[m]
[Vishnoi] PB = PM,N

24. Optimization: linear programs with exponentially many inequalities
General matching as BL polytopes??
G = (V,E) |V|=2n, |E|=m
PG = conv {1S: SE perfect matching}  Rm
[Edmonds] = {pRm: ∑ijE pij =n
∑iS jS pij ≥1 SV odd
pij ≥ ijE }
Is this a BL-polytope?
Other nontrivial examples? Optimization?

25. Optimization, Complexity & Invariant Theory
Summary
One problem : Singularity of Symbolic Matrices
One algorithm: Alternating minimization
Non-commutative Algebra: Word problem
Invariant Theory: Nullcone & orbit problems
Quantum Information Theory: Positive operators
Analysis: Brascamp-Lieb inequalities
Optimization Exponentially large linear programs
Computational complexity VP=VNP?
Tools, applications, structure, connections,…
Optimization, Complexity & Invariant Theory
IAS workshop, June 4-8, 2018