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Proving Analytic Inequalities Avi Wigderson IAS, Princeton

Explore new book by Avi Wigderson focusing on proving analytic inequalities through algorithmic methods. Learn about the mathematical concepts, including polynomial identities and symbolic matrices, used to tackle algebraic problems, analysis, and optimization. The book delves into applications in analysis, geometry, probability, and information theory using techniques like alternate minimization and operator scaling. Discover key inequalities such as Brascamp-Lieb, Holder, and Young's convolution, and their implications in solving large-scale optimization problems. Gain insights into geometric properties, structural theory, and algorithmic consequences related to inequalities in mathematical frameworks. The text discusses the optimization of BL constants, feasibility testing, and the convergence of geometric configurations. Explore the intersection of analytic algorithms, linear programming, and polytope theory in addressing complex mathematical challenges.

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Proving Analytic Inequalities Avi Wigderson IAS, Princeton

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  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

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