Ankit Garg

Microsoft Research India

N O N C O M M U T A T I V E A N A L Y S I S , C O M P U T A T I O N A L C O M P L E X I T Y , A N D Q U A N T U M I N F O R M A T I O N

October 16, 2019

Invariant theory, noncommutative optimization and applications: part 1

Overview

! Optimization problems in invariant theory.! Connections to several areas of computer science, mathematics and

physics.

! Surprising avenues for convexity: geodesic convexity and moment polytopes.

Geometric complexity theory – asymptotic vanishing of Kronecker coefficients.

Quantum information theory– one-body quantum marginal problem.

Functional analysis – Brascamp-Liebinequalities.

Optimization– Geodesic convexity. Captures general linear programming.

Complexity theory and derandomization –Special cases of polynomial identity

testing.

Noncommutative algebra – Special cases of polynomial identity testing.

Outline

! Invariant theory! Optimization in invariant theory! Moment polytopes! Open problems

Invariant theory

Linear actions of groups

Group ! acts linearly on vector space " (= %&).(: ! → !+(") ,×, matrices group homomorphism.67: " → " invertible linear map ∀ 9 ∈ !.67;7< = 67;67< and 6=& = >,.

Example 1! = @A acts on " = %A by permuting coordinates.

6B CD, … , CA = CB(D), … , CB(A) .

Example 2! = !+A % acts on " = 6A(%) by conjugation.

6H I = JIJKD.

Objects of study

Group ! acts linearly on vector space ".• Invariant polynomials: Polynomial functions on "

invariant under action of !. # s.t. # $%& = #(&) for all * ∈!, & ∈ ".

• Orbits: Orbit of vector &, -. = $%& ∶ * ∈ ! . • Orbit-closures: Orbits may not be closed. Take their

closures. Orbit-closure of vector & , -. = cl $%& ∶ * ∈ ! .

Null cone

Group ! acts linearly on vector space ".Null cone: Vectors # s.t. 0 lies in the orbit-closure of #.

#: 0 ∈ '( .Sequence of group elements )*,… , )-, … s.t. lim-→2345 # = 0.

Problem: Given # ∈ ", decide if it is in the null cone.Captures many interesting questions.

[Hilbert 1893; Mumford 1965]: # in null cone iff 7 # = 0 for all homogeneous invariant polynomials 7.! One direction clear (polynomials are continuous).! Other direction uses Nullstellansatz and some algebraic

geometry.

Example 1

" = $% acts on & = '% by permuting coordinates. () *+,… , *% → *)(+), … , *)(%) .

Null cone = {0}.No closures.

Example 2

" = "$% & acts on ' = (%(&) by conjugation. (+ , = -,-./.

! Invariants: generated by tr ,2 .! Null cone: nilpotent matrices.

Example 3

" = $%& ' ×$%&(') acts on + = ,&(') by left-right multiplication.

,(-,/) 0 = 102.• Invariants: generated by Det(0).• Null cone: Singular matrices.

Example 4

"#$: group*of*,×, diagonal*matrices*with*determinant*1.< = "#$×"#$ acts on > = ?$(A) by left-right multiplication.

?(C,E) F = GFH.• Invariants: generated by FI,J(I) ⋅ FL,J(L) ⋯F$,J($).• Null cone: perfect matching.GN is in null cone iff O has no perfect matching.

O1 0 11 0 01 1 1

GN

Example 5: Linear programming

" = $%: (Abelian!) group5of57×7 diagonal5matrices.?:5(Laurent)5polynomials.

" acts on ? by scaling variables. C ∈ $%, C =diag(EF, … , E%).

IC J(KF, … , K%) = J(EFKF, … , E%K%).

J = ∑M∈NOMKM. supp J = P ∈ Ω: OM ≠ 0 .

Null cone ↔ Linear ProgrammingJ not in null cone ↔ 0 ∈ conv P ∶ P ∈ supp(J) .In nonVAbelian5groups,5the5null cone (membership) problem is a non-commutative analogue of linear programming.

Example 6

" = $%& ' ×$%&(') acts on + = ,&(')⊕. by simultaneous left-right multiplication.

,(/,1) (23, … , 2.) = (5236,… , 52.6).! Invariants [DW 00, DZ 01, SdB 01, ANS 10]: generated by Det ∑= >= ⊗ 2= .

! Null cone: Non-commutative singularity. Captures non-commutative rational identity testing.

[GGOW 16, DM 16, IQS 16]: Deterministic polynomial time algorithms.

Non-commutative rational expressions

! ≮ #$, #&, … , #( ≯: non-commutative rational functions and their representations.Example: ((#$#&+$ + #-+$)+$+#/)+$No easy canonical form.

#$ #& 1

+ +

×

# #$

INV

+

Non-commutative RIT

Given two non-commutative rational expressions as formulas/circuits, determine if they represent the same element.

What does it mean for two expressions represent the same element? – No easy canonical form.

Operational definition [Amitsur 66].

Equivalence

Given a rational expression ! "#, "%, … , "' =(("#"%*# + ",*#)*#+".)*#: Dom(!) := (2#, 2%, … , 2') s. t. !(2#, 2%, … , 2') de8ined

Example: ! "#, "% = ("#"% − "%"#)*#. Dom ! = 2#, 2% s. t. Det(2#2% − 2%2#) ≠ 0 .

[Definition]: Two rational expressions !# and !% are equivalent if !# 2#, … , 2' = !% 2#,… , 2' for all 2#, … , 2' ∈ Dom(!#) ∩

Dom(!%).

Non-commutative rational identity testing

Given two valid rational expressions as formulas/circuits, are they equivalent?Same as, given a valid rational expression, is it equivalent to 0?Reduces to example 6. Previous algorithms exponential time.Connection to invariant theory crucial for efficient algs.

Optimization in invariant theory: geometric invariant theory (GIT)

GIT: computational perspective

What is complexity of null cone membership?GIT puts it in !" ∩ $%!" (morally).! Hilbert-Mumford criterion: how to certify

membership in null cone.! Kempf-Ness theorem: how to certify non-

membership in null cone.

Kempf-Ness

Group ! acts linearly on vector space ".How to certify # not in null cone?Exhibit invariant polynomial $ s.t. $ # ≠ 0.Not feasible in general.Invariants hard to find, high degree, high complexity etc.Kempf-Ness provides an efficient way.

An optimization perspective

Finding minimal norm elements in orbit-closures!Group ! acts linearly on vector space ".

cap & = inf+∈- .+ & //.

Null cone: & s.t. cap & = 0.

12

&′

Moment map

Group ! acts linearly on vector space ".

#$ % = '( ) **.

Moment map +,()): gradient of #$ % at % = /0.How much norm of ) decreases by infinitesimal action around /0.

Example 1

" = $∗ acts on & = $.' ( ) = ( ).*+ ( = |(|-|)|-.Moment map: consider action of ( = exp(2), 2 ≈ 0.

67 ) = 899: exp 22 ) -

:<== 2 ) -.

Example 2

" = $%&×$%& acts on ( = )&.)(+,,+.) 0 = 12 0 13.42, 43 ∈ 6&: ∑8 42 9 = ∑: 43(;) = 0. Directional derivative: action of (exp @ 42 , exp(@ 43)), @ ≈ 0.! BC 0 = (D2, D3), ∑8 D2 9 = ∑: D3(;) = 0 s.t.

! ⟨D2, 42⟩ + ⟨D3, 43⟩ = HIIJ

K(exp @ 42 , exp @ 43 ) 0 L3

JMN= 2 OP, 42 + 2 ⟨QP, 43⟩= 2 OP − ST, 42 + 2 ⟨QP − ST, 43⟩

OP, QP vectors of row and column U33 norms of 0. BC 0 = 2(OP − ST, QP − ST).

Kempf-Ness

Group ! acts linearly on vector space ".[Kempf, Ness 79]: # not in null cone iff non-zero $ in orbit-closure of # s.t. %& $ = 0.$ certifies # not in null cone.One direction easy.! # not in null cone. Take $ vector of minimal norm in orbit-

closure of #. $ non-zero.! $ minimal norm in its orbit. ⇒ Norm does not decrease by

infinitesimal action around *+. ⇒ %& $ = 0.! Global minimum ⇒ local minimum.

Kempf-Ness

Other direction: local minimum ⇒ global. Some “convexity”.! Commutative group actions – Euclidean convexity

(change of variables) [exercise].! Non-commutative group actions: geodesic

convexity.

Moment polytopes

Commutative case

! = #$: (Abelian!) group4of46×6 diagonal4matrices.>:4(Laurent)4polynomials.! acts on > by scaling variables. B ∈ #$, B = diag(DE, … , D$).

HB I(JE, … , J$) = I(DEJE, … , D$J$).I = ∑L∈MNLJL. supp I = O ∈ Ω: NL ≠ 0 .

Moment map: ST I = ∑L∈M NL UO.

Moment polytope: V = all gradients= {ST I : I U = 1}= conv O ∶ O ∈ supp(I)

Noncommutative case

Collection of gradients not convex.! = !#$ acts on % = &$. Matrix vector mult.'( ) = 2 ))+.Theorem [Kostant 73, Atiyah 82, GS 82, NM 84, Kirwan 84, Brion 87]: Collection of spectra convex polytope!

spec 4 = 56, 58, … , 5$ , 56 ≥ 58 ≥ ⋯ ≥ 5$< = {spec('( ) ): ) 8 = 1} convex polytope. Called the moment polytope.True even if vary over orbit-closures.

Quantum marginals

Pure quantum state |"⟩$%,…,$( () quantum systems).Characterize marginals *$%, … , *$( (marginal states on systems)?Only the spectra matter (local rotations for free).! Collection of such spectra convex polytope!! Follows from theory of moment polytopes.! Underlying group action: Products of +,’s on

tensors.[BFGOWW 18]: Efficient algorithms via

alternating minimization.[BFGOWW 19]: Efficient algorithms for all group actions via geodesic gradient descent.Algorithms run in time poly(1/6). 6 error parameter.

More examples

! Newton: ! ∈ # $%, … , $( , ! = ∑+∈,-+$+ homogeneous polynomial../ = conv 4: 4 ∈ Ω ⊆ 8(

! Schur-Horn: 9 :×: symmetric matrix..< = diag A : A similar to 9 ⊆ 8(

! Horn: . = G<, GH, GI : 9 + A = K ⊆ 8L(

! Edmonds: M,M′ linear matroids on [:]..Q,QR = conv{1U: V basis for M,M′} ⊆ 8(

Open problems

Open problems

! Complexity of null cone, moment polytope membership. !" ∩ $%!"? Polynomial time?

! Ellipsoid/interior point methods for geodesicallyconvex problems. poly(log(1/.)) running time.

! More applications?

Lectures and videos

! IAS workshop videos:https://www.math.ias.edu/ocit2018! Avi’s CCC 2017 tutorial:http://computationalcomplexity.org/Archive/2017/tutorial.php

Thank You

References

! [Hilbert 1893; Mumford 1965]: Geometric invariant theory, book.

! [DW 00, DZ 01, SdB 01, ANS 10]: Harm Derksen and Jerzy Weyman. Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients.

Matyas Domokos and A. N. Zubkov. Semi-invariants of quivers as determinants.Aidan Schofield and Michel Van den Bergh. Semi-invariants of quivers for arbitrary dimension vectors.Bharat Adsul, Suresh Nayak, and K. V. Subrahmanyam. A geometric approach to the kronecker problem ii : rectangular shapes, invariants of n n matrices, and a generalization of the artin-procesi theorem.

References

! [GGOW 16, DM 16, IQS 16]:Ankit Garg, Leonid Gurvits, Rafael Oliveira, Avi Wigderson. Operator scaling: theory and applications.Harm Derksen, Visu Makam. Polynomial degree bounds for matrix semi-invariants.Gábor Ivanyos, Youming Qiao, K. V. Subrahmanyam. Constructive noncommutative rank computation is in deterministic polynomial time

References

! [Amitsur 66]: Shimshon Amitsur. Rational identities and applications to algebra and geometry.

! [Kempf, Ness 79]: Kempf, George; Ness, Linda (1979), "The length of vectors in representation spaces“.

! [Kostant 73, Atiyah 82, GS 82, NM 84, Kirwan 84, Brion 87]:B. Kostant. On convexity, the Weyl group and the Iwasawa decomposition.M. F. Atiyah. Convexity and commuting hamiltonians.Victor Guillemin and Shlomo Sternberg. Convexity properties of the moment mapping.Linda Ness and David Mumford. A stratification of the null cone via the moment map.Frances Kirwan. Convexity properties of the moment mapping, III.Michel Brion. Sur l’image de l’application moment.! [BFGOWW 18]: Peter Bürgisser, Cole Franks, Ankit Garg, Rafael Mendes de

Oliveira, Michael Walter, Avi Wigderson. Efficient algorithms for tensor scaling, quantum marginals and moment polytopes.

! [BFGOWW 19]: Upcoming paper by the same authors as above. Almost ready!