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

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

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

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,…

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

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

Examples

Cauchy-Schwarz, Holder 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

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

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)]

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

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/p3 p1+p2+p3=2 1≥pi≥0 q1q1q2q2q3q3 p1p1p2p2p3p3 qi=1-pi C =

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

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

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

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

Algorithms

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

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)

Optimization

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 ≥0 ??Applications?? e2 e1

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 ≥ 0 j[m] } Bj:RnR Bjx=<vj,x> j[m] [Fact] PB = PM Exponentially many Inequalities

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 ≥ 0 j[m] } Bj:R2nR2 Bjx=<vj,xL>,<uj,xR> j[m] [Vishnoi] PB = PM,N

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 ≥ 0 ijE } Is this a BL-polytope? Other nontrivial examples? Optimization?

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