Analytic algorithms for the moment polytope Cole Franks Rutgers - PowerPoint PPT Presentation

Analytic algorithms for the moment polytope Cole Franks Rutgers University

Based on joint work with Peter B¨ urgisser Ankit Garg Rafael Oliveira Michael Walter Avi Wigderson Mainly from “Towards a theory of non-commutative optimization: geodesic 1st and 2nd order methods for moment maps and polytopes” FOCS 2019 1

Outline 1. Moment polytopes by example 2. Algorithms for the general problem 2

Moment polytopes

Motivating question Horn’s problem: Are λ 1 , λ 2 , λ 3 ∈ R n the spectra of three n × n matrices H 1 , H 2 , H 3 such that H 1 + H 2 = H 3 ? If so, can one find the matrices efficiently? 3

Motivating question Horn’s problem: Are λ 1 , λ 2 , λ 3 ∈ R n the spectra of three n × n matrices H 1 , H 2 , H 3 such that H 1 + H 2 = H 3 ? If so, can one find the matrices efficiently? 3

Horn set Let V = P (Mat( n ) 2 ), define µ : V → Herm( n ) 3 by µ : [ A 1 , A 2 ] �→ ( A 1 A † A 2 A † A † 1 A 1 + A † 1 , 2 , 2 A 2 ) . � A 1 � 2 + � A 2 � 2 Note eigs( AA † ) = eigs( A † A ), so eigs( A 1 A † eigs( A 2 A † eigs( A † 1 A 1 + A † 1 ) , 2 ) , 2 A 2 ) is a “yes” instance to Horn’s problem (in fact, all such instances take this form). 4

Horn set Let V = P (Mat( n ) 2 ), define µ : V → Herm( n ) 3 by µ : [ A 1 , A 2 ] �→ ( A 1 A † A 2 A † A † 1 A 1 + A † 1 , 2 , 2 A 2 ) . � A 1 � 2 + � A 2 � 2 Note eigs( AA † ) = eigs( A † A ), so eigs( A 1 A † eigs( A 2 A † eigs( A † 1 A 1 + A † 1 ) , 2 ) , 2 A 2 ) is a “yes” instance to Horn’s problem (in fact, all such instances take this form). 4

Moment polytopes • G = GL( n ) • π : G → C m a representation of G where U ( n ) acts unitarily • V ⊂ P ( C m ) a projective variety fixed by G , Moment map is the map µ : V → n × n Hermitians =: Herm( n ) given by µ : v �→ ∇ H ∈ Herm( n ) log � e H · v � i µ is a moment map for U ( n ) in the physical sense! In particular: Theorem (Kirwan) Image of µ take eigs. R n V Herm( n ) is a convex polytope in R n known as moment polytope, denoted ∆( V ) 5

Moment polytopes • G = GL( n ) • π : G → C m a representation of G where U ( n ) acts unitarily • V ⊂ P ( C m ) a projective variety fixed by G , Moment map is the map µ : V → n × n Hermitians =: Herm( n ) given by µ : v �→ ∇ H ∈ Herm( n ) log � e H · v � i µ is a moment map for U ( n ) in the physical sense! In particular: Theorem (Kirwan) Image of µ take eigs. R n V Herm( n ) is a convex polytope in R n known as moment polytope, denoted ∆( V ) 5

Moment polytopes • G = GL( n ) • π : G → C m a representation of G where U ( n ) acts unitarily • V ⊂ P ( C m ) a projective variety fixed by G , Moment map is the map µ : V → n × n Hermitians =: Herm( n ) given by µ : v �→ ∇ H ∈ Herm( n ) log � e H · v � i µ is a moment map for U ( n ) in the physical sense! In particular: Theorem (Kirwan) Image of µ take eigs. R n V Herm( n ) is a convex polytope in R n known as moment polytope, denoted ∆( V ) 5

Moment polytopes • G = GL( n ) • π : G → C m a representation of G where U ( n ) acts unitarily • V ⊂ P ( C m ) a projective variety fixed by G , Moment map is the map µ : V → n × n Hermitians =: Herm( n ) given by µ : v �→ ∇ H ∈ Herm( n ) log � e H · v � i µ is a moment map for U ( n ) in the physical sense! In particular: Theorem (Kirwan) Image of µ take eigs. R n V Herm( n ) is a convex polytope in R n known as moment polytope, denoted ∆( V ) 5

Horn polytope • V = P (Mat( n ) 2 ) • G = GL( n ) 3 • π given by ( g 1 , g 2 , g 3 ) · ( A 1 , A 2 ) = ( g 1 A 1 g † 3 , g 2 A 2 g † 3 ) . • µ : V → Herm( n ) 3 given by µ : [ A 1 , A 2 ] �→ ( A 1 A † A 2 A † A † 1 A 1 + A † 1 , 2 , 2 A 2 ) . � A 1 � 2 + � A 2 � 2 Thus, image of µ take eigs. Herm( n ) 3 ( R n ) 3 V is the* solution set of the Horn problem! 6

Horn polytope • V = P (Mat( n ) 2 ) • G = GL( n ) 3 • π given by ( g 1 , g 2 , g 3 ) · ( A 1 , A 2 ) = ( g 1 A 1 g † 3 , g 2 A 2 g † 3 ) . • µ : V → Herm( n ) 3 given by µ : [ A 1 , A 2 ] �→ ( A 1 A † A 2 A † A † 1 A 1 + A † 1 , 2 , 2 A 2 ) . � A 1 � 2 + � A 2 � 2 Thus, image of µ take eigs. Herm( n ) 3 ( R n ) 3 V is the* solution set of the Horn problem! 6

Horn polytope • V = P (Mat( n ) 2 ) • G = GL( n ) 3 • π given by ( g 1 , g 2 , g 3 ) · ( A 1 , A 2 ) = ( g 1 A 1 g † 3 , g 2 A 2 g † 3 ) . • µ : V → Herm( n ) 3 given by µ : [ A 1 , A 2 ] �→ ( A 1 A † A 2 A † A † 1 A 1 + A † 1 , 2 , 2 A 2 ) . � A 1 � 2 + � A 2 � 2 Thus, image of µ take eigs. Herm( n ) 3 ( R n ) 3 V is the* solution set of the Horn problem! 6

Horn polytope • V = P (Mat( n ) 2 ) • G = GL( n ) 3 • π given by ( g 1 , g 2 , g 3 ) · ( A 1 , A 2 ) = ( g 1 A 1 g † 3 , g 2 A 2 g † 3 ) . • µ : V → Herm( n ) 3 given by µ : [ A 1 , A 2 ] �→ ( A 1 A † A 2 A † A † 1 A 1 + A † 1 , 2 , 2 A 2 ) . � A 1 � 2 + � A 2 � 2 Thus, image of µ take eigs. Herm( n ) 3 ( R n ) 3 V is the* solution set of the Horn problem! 6

Horn polytope • V = P (Mat( n ) 2 ) • G = GL( n ) 3 • π given by ( g 1 , g 2 , g 3 ) · ( A 1 , A 2 ) = ( g 1 A 1 g † 3 , g 2 A 2 g † 3 ) . • µ : V → Herm( n ) 3 given by µ : [ A 1 , A 2 ] �→ ( A 1 A † A 2 A † A † 1 A 1 + A † 1 , 2 , 2 A 2 ) . � A 1 � 2 + � A 2 � 2 Thus, image of µ take eigs. Herm( n ) 3 ( R n ) 3 V is the* solution set of the Horn problem! 6

Link to algebra Why are moment polytopes interesting? Encode asymptotic representation theory of coordinate ring of V ! Theorem (Mumford, Ness ’84, Brion ’87) Let V G ,λ denote irrep of G of type λ . Then 1 � k { λ : V G ,λ ⊂ C [ V ] k } = ∆( V ) ∩ Q n ! k Additional math (Schur-Weyl duality, Saturation [KT00]) = ⇒ Horn polytope ∩ ( Z n ) 3 = { ( λ 1 , λ 2 , λ 3 ) : V GL( n ) , λ 3 ∈ V GL( n ) , λ 1 ⊗ V GL( n ) , λ 2 } 7

Link to algebra Why are moment polytopes interesting? Encode asymptotic representation theory of coordinate ring of V ! Theorem (Mumford, Ness ’84, Brion ’87) Let V G ,λ denote irrep of G of type λ . Then 1 � k { λ : V G ,λ ⊂ C [ V ] k } = ∆( V ) ∩ Q n ! k Additional math (Schur-Weyl duality, Saturation [KT00]) = ⇒ Horn polytope ∩ ( Z n ) 3 = { ( λ 1 , λ 2 , λ 3 ) : V GL( n ) , λ 3 ∈ V GL( n ) , λ 1 ⊗ V GL( n ) , λ 2 } 7

Link to algebra Why are moment polytopes interesting? Encode asymptotic representation theory of coordinate ring of V ! Theorem (Mumford, Ness ’84, Brion ’87) Let V G ,λ denote irrep of G of type λ . Then 1 � k { λ : V G ,λ ⊂ C [ V ] k } = ∆( V ) ∩ Q n ! k Additional math (Schur-Weyl duality, Saturation [KT00]) = ⇒ Horn polytope ∩ ( Z n ) 3 = { ( λ 1 , λ 2 , λ 3 ) : V GL( n ) , λ 3 ∈ V GL( n ) , λ 1 ⊗ V GL( n ) , λ 2 } 7

Algorithmic tasks Input ( V , π, λ ) • Projective variety V as arithmetic circuit parametrizing it • Representation π as its list of irreducible subrepresentations as elements of Z n • Target λ ∈ Q n 1. membership: determine whether λ in ∆( V ). 2. ε -search: given λ ∈ R n , either find an element v ∈ λ such that • � µ ( v ) − diag( λ ) � < ε , OR • correctly declare λ �∈ ∆( V ). i.e. find an approximate preimage under µ ! 1 / exp (poly)-search suffices for membership! 8

Algorithmic tasks Input ( V , π, λ ) • Projective variety V as arithmetic circuit parametrizing it • Representation π as its list of irreducible subrepresentations as elements of Z n • Target λ ∈ Q n 1. membership: determine whether λ in ∆( V ). 2. ε -search: given λ ∈ R n , either find an element v ∈ λ such that • � µ ( v ) − diag( λ ) � < ε , OR • correctly declare λ �∈ ∆( V ). i.e. find an approximate preimage under µ ! 1 / exp (poly)-search suffices for membership! 8

Algorithmic tasks Input ( V , π, λ ) • Projective variety V as arithmetic circuit parametrizing it • Representation π as its list of irreducible subrepresentations as elements of Z n • Target λ ∈ Q n 1. membership: determine whether λ in ∆( V ). 2. ε -search: given λ ∈ R n , either find an element v ∈ λ such that • � µ ( v ) − diag( λ ) � < ε , OR • correctly declare λ �∈ ∆( V ). i.e. find an approximate preimage under µ ! 1 / exp (poly)-search suffices for membership! 8

Algorithmic tasks Input ( V , π, λ ) • Projective variety V as arithmetic circuit parametrizing it • Representation π as its list of irreducible subrepresentations as elements of Z n • Target λ ∈ Q n 1. membership: determine whether λ in ∆( V ). 2. ε -search: given λ ∈ R n , either find an element v ∈ λ such that • � µ ( v ) − diag( λ ) � < ε , OR • correctly declare λ �∈ ∆( V ). i.e. find an approximate preimage under µ ! 1 / exp (poly)-search suffices for membership! 8

Algorithmic tasks Input ( V , π, λ ) • Projective variety V as arithmetic circuit parametrizing it • Representation π as its list of irreducible subrepresentations as elements of Z n • Target λ ∈ Q n 1. membership: determine whether λ in ∆( V ). 2. ε -search: given λ ∈ R n , either find an element v ∈ λ such that • � µ ( v ) − diag( λ ) � < ε , OR • correctly declare λ �∈ ∆( V ). i.e. find an approximate preimage under µ ! 1 / exp (poly)-search suffices for membership! 8