Proving Analytic Inequalities Avi Wigderson IAS, Princeton

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. ∫∏j fj≤C ∏j |fj|pj Propaganda:special cases & extensions Cauchy-Schwarz,HolderPrecopa-Leindler Loomis-WhitneyNelson Hypercontractive Young’s convolution Brunn-Minkowski Lieb’s Non-commutativeBL Barthe ReverseBL Bennett-Bez NonlinearBL Quantitative Helly Analysis, Geometry, Probability, Information Theory,… Brascamp-Lieb Inequalities [BL’76,Lieb’90]

4. 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 Cin P (through Operator Scaling & Alternate Minimization) Optimization: solving (some) exponential size LPs Brascamp-Lieb Inequalities [BL’76,Lieb’90]

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

6. Examples

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

8. d=2, x=(x1,x2) f1,f2:RR+ [Trivial] ∫xR2f1(x1)f2(x2) = |f1|1|f2|1 p1=p2=1 x2 Loomis-Whitney I a2 A x1 a1 area(A)≤len(a1)len(a2) H(Z1Z2)≤ H(Z1)+H(Z2)

9. d=3, x=(x1,x2,x3) f1,f2,f3:R2 R+ [LW] ∫xR3f1(x2x3)f2(x1x3)f3(x1x2) ≤ |f1|2|f2|2|f3|2 pi=½ Loomis-Whitney II x2 A12 A23 S any other norms? x1 x3 A13 H(Z1Z2Z2)≤ ½[H(Z1Z2)+H(Z2Z3)+H(Z1Z3)] vol(S) ≤ [area(A12)area(A13)area(A23)]1/2

10. x2 d=2, x=(x1,x2) f1,f2,f3:RR+ [Young] ∫xR2f1(x1)f2(x2)f3(x1+2) ≤ (√3)/2 |f1|3/2|f2|3/2|f3|3/2 pi=2/3 Young I x1+x2 area(A)≤ (√3)/2[len(a1)len(a2)len(a3)]2/3 a2 A a3 x1 Any other norms? a1

11. d=2, x=(x1,x2) f1,f2,f3:R2 R+ [Young] ∫xR2f1(x1)f2(x2)f3(x1+2) ≤ C |f1|1/p1|f2|1/p2|f3|1/p3 p1+p2+p3=2 1≥pi≥0 q1q1q2q2q3q3 p1p1p2p2p3p3 qi=1-pi Young II C =

12. 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 Cin P Brascamp-Lieb Inequalities [BL’76,Lieb’90]

13. Structure

14. 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:∑jpjnj = n ∑jpjdim(BjV) ≥ dim(V) subspace VinRm pj≥0 (Exponentially many inequalities) Feasibility: C<∞ [Bennett-Carbery-Christ-Tao’08]

15. 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 B1 B2 B3 BL-constant [Lieb’90] A1 A2 sup ∏j det(Aj)pj Aj>0 det(∑j pj BjtAjBj) A3 1/cap(L) = BL(B,p)2= A4 for some completely positive operator L (B,p) A5 Quiver reduction

16. Algorithms

17. 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 Geometric BL [Ball’89,Barthe’98] Doubly stochastic

18. [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 iff (B,p) is feasible • [GGOW’16] • - Feasibility (testingC<∞, p  PB) in polynomial time • 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. Alternate Minimization [GGOW’16] Repeat t=nc times: - Satisfy Projection (Right basis change) - Satisfy Isotropy (Left basis change)

19. Optimization

20. e3 P = conv {0, e1, e2,… em}  Rm = { pRm: ∑j pj≤ 1 pj≥ 0 j[m] } Membership Problem: Test if pP? Easyif few inequalities…or [GGOW’16] BL-polytope! B= (B1,B2,…,Bm) Bj:RnRnj PB: { pRm: ∑jpjnj = n ∑jpjdim(BjV) ≥ dim(V)  V ≤Rm pj≥0 } ??Applications?? e2 Linear programming & Polytopes 0 e1

21. 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 pj≤ dim(VJ) J[m] pj≥ 0 j[m] } Bj:RnR Bjx=<vj,x> j[m] [Fact] PB = PM Optimization: linear programs with exponentially many inequalities Exponentially many Inequalities

22. 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≥ 0 j[m] } Bj:R2nR2Bjx=<vj,x>,<uj,x> j[m] [Vishnoi] PB = PM,N Optimization: linear programs with exponentially many inequalities

23. General matching as BL polytopes?? G = (V,E) |V|=2n, |E|=m PG = conv {1S:SE perfect matching}  Rm [Edmonds] = {pRm: ∑ijEpij=n ∑iS jSpij≥1 SV odd pij≥ 0 ijE } Is this a BL-polytope? Other nontrivial examples? Optimization? Optimization: linear programs with exponentially many inequalities

24. 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,… Math and Computation New book on my website