A story... of one convex relaxation... and of the related - PDF document

A story... of one convex relaxation... and of the related revelation... Leonid Gurvits Los Alamos National Laboratory , Nuevo Mexico, USA. e-mail: gurvits@lanl.gov 1

The Mixed volume and the Mixed Discriminant, 1998, A. Barvinok’s paper in “Lectures on Mathematical Programming: ISMP-97” K = ( K 1 , ..., K n ) is a n -tuple of convex compact sub- sets (i.e. convex bodies ) in the Euclidean space R n ; V K ( λ 1 , ..., λ n ) =: V ol ( λ 1 K 1 + · · · + λ n K n ) , λ i ≥ 0 . Herman Minkowski proved that V K is a homogeneous polynomial with non-negative coefficients. The mixed volume: ∂ n V ( K 1 , ..., K n ) =: V K (0 , ..., 0) . ∂λ 1 ...∂λ n i.e. the mixed volume V ( K 1 , ..., K n ) is the coefficient 1 ≤ i ≤ n λ i in the Minkowski polynomial of the monomial � V K . 2

Let A = ( A 1 , ..., A n ) be an n -tuple of n × n complex matrices; the corresponding determinantal polynomial is defined as Det A ( λ 1 , ..., λ n ) = det( 1 ≤ i ≤ n λ i A i ) . � The mixed discriminant is defined as ∂ n D ( A 1 , ..., A n ) = Det A (0 , ..., 0) . ∂λ 1 ...∂λ n i.e. the mixed discriminant D ( A 1 , ..., A n ) is the coef- ficient of the monomial 1 ≤ i ≤ n λ i in the determinantal � polynomial Det A . 3

Examples. 1. K i = { ( t 1 , ..., t n ) : 0 ≤ t j ≤ A ( i, j ) , 1 ≤ j ≤ n } , the mixed volume of coordinate boxes K i : V ( K 1 ...K n ) = Per ( A ) = 1 ≤ i ≤ n A ( i, σ ( i )) . � � σ ∈ S n If each coordinate box K i is a rectangle(parallelogram) then computing the mixed volume V ( B 1 , ..., B n ) is “easy”. 2. K i = { ae i + bY i : 0 ≤ a, b ≤ 1 } is a parallelogram , A =: [ Y 1 , ..., Y n ]. Then the mixed volume V ( K 1 , ..., K n ) = MV ( A ) =: S ⊂{ 1 ,...,n } | det( A S,S ) | . � If Q i = e i e T i + Y i Y T then the mixed discriminant i S ⊂{ 1 ,...,n } (det( A S,S )) 2 . D ( Q 1 , ..., Q n ) = MD ( A ) =: � 3. Both MD ( A ) and MV ( A ) are # P − Complete even if the matrix A is unimodular . 4

From the mixed volume of Ellipsoids to the Mixed Discriminant The convex bodies K i are well-presented: Given a weak membership oracle for K i and a rational n × n matrix B i , a rational vector Y i ∈ R n such that √ Y i + B i ( Ball n (1)) ⊂ K i ⊂ Y i + n n + 1 B i ( Ball n (1)) (1) Let E B be the ellipsoid B ( Ball n (1)) in R n . Then √ n + 1) n V ( E B 1 , ..., E B n ) . V ( E B 1 , ..., E B n ) ≤ V ( K 1 , ..., K n ) ≤ ( n [Barvinok, 1997]: Define v n =: V ol n ( Ball n (1)). Then the following inequalities hold: 3 − n +1 1 1 2 v n D 2 ( A 1 ( A 1 ) T , ..., A n ( A n ) T ) ≤ V ( E A 1 ... E A n ) ≤ v n D 2 ( .. ) (2) 5

Suppose that we have an effectively computable esti- mate F such that γ ( n ) ≤ D ( A 1 ( A 1 ) T , ..., A n ( A n ) T )) ≤ 1 . F Then 2 ≤ V ( K 1 , ..., K n ) γ ( n )3 − n +1 � ≤ n 1 . 5 n √ Fv ( n ) Which gives the approximation factor γ ( n )) − 1 ≥ n O ( n ) . n +1 n 1 . 5 n 3 2 ( � Barvinok [1997] gave the poly-time randomized algo- rithm with γ ( n ) = c n , c < 1 . 6

A deterministic algorithm for the Mixed Discriminant, Geometric Programming, Quantum Entanglement: 1998-2005 Let p ∈ Hom + ( n, n ) be a homogeneous polynomial with nonnegative coefficients. Define the following quan- tity, called Capacity : p ( x 1 , . . . , x n ) Cap ( p ) =: inf . 1 ≤ i ≤ n x i x i > 0 � ∂ n Clearly ∂x 1 ··· ∂x n p (0 , 0 , ..., 0) ≤ Cap ( p ). Now, y 1 + ... + y n =0 log( p ( e y 1 , ..., e y n )) log( Cap ( p )) = inf and the functional log( p ( e y 1 , ..., e y n )) is convex. There- fore log( Cap ( p )) might be, with some extra care and luck, effectively additively approximated using convex programming tools and an oracle, deterministic or ran- dom, evaluating the polynomial p . 7

But we need a lower bound: ∂ n p (0 , 0 , ..., 0) ≥ γ ( n ) Cap ( p ) , γ ( n ) > 0 . ∂x 1 · · · ∂x n In the case of the mixed discriminant the corresonding polynomial p ( x 1 , ..., x n ) = det( 1 ≤ i ≤ n x i Q i ), where the matrices � Q i � 0 are PSD. Easy to evaluate deterministically! 8

Boils down to the following result: : Let n -tuple A = ( A 1 , . . . , A n ) of her- Theorem 0.1 mitian n × n PSD matrices be doubly-stochastic: tr ( A i ) = 1 , 1 ≤ i ≤ n ; 1 ≤ i ≤ n A i = I. � Then the mixed discriminant ∂ n Det A (0 , . . . , 0) ≥ n ! D ( A ) =: (3) n n ∂x 1 , . . . , ∂x n The equality in (3) is attained iff A i = 1 n I, 1 ≤ i ≤ n . Solution of R. Bapat’s conjecture (1989), stated for real symmetric PSD matrices, generalization of Van der Waerden conjecture for the permanent; proved by L.G. (1999), final publication (2006)) The reason for the result: optimality condition for min y 1 + ... + y n =0 log( Det Q ( e y 1 , ..., e y n )) states that the tu- 1 ≤ i ≤ n e y i Q i ) − 1 ple ( P ( e y 1 Q 1 ) P, ..., P ( e y n Q n ) P ) , P = ( 2 � 9

is doubly-stochastic. This observation and Theorem(0.1) imply that n n ≤ D ( Q 1 , ..., Q n ) n ! Cap ( Det Q ) ≤ 1 . Can put γ ( n ) = n ! n n ≈ e − n . My proof is a very non-trivial adaptation of Ego- rychev’s proof of Van Der Waerden conjecture for the permanent, which I learned from Knut’s 1981 Monthly exposition. Did not actually need doubly-stochasticity, it served as a tool; non-convex optimization with semi-definite con- straints. The proof is very matrix-oriented, crucially uses the group action: D ( XA 1 X ∗ , ..., XA n X ∗ ) = det( XX ∗ ) D ( A 1 , ..., A n ) . 10

Got a deterministic poly-time(not strongly polyno- mial) algorithm to approximate the mixed discrim- inant with the factor e n and the mixed volume with the factor n O ( n ) . Can we get a factor c n deterministi- cally for the mixed volume ? NO! In the oracle setting, even for the single volume the �� n 2 ( Barany-Furedi � � n factor is greater than ٠log n bound ). Can we get factor c n using a randomized poly-time algorithm? Can we get a better factor for the mixed discrim- inant if the ranks Rank ( Q i ) are small? Is there a simpler proof? 11

A revelation, 2003-2004-2005-... : A homogeneous polynomial p ∈ Hom C ( m, n ) Definition 0.2 is H-Stable if | p ( z 1 , ..., z m ) | > 0; Re ( z i ) > 0 , 1 ≤ i ≤ m. : Consider a bivariate homogeneous poly- Example 0.3 nomial p ( z 1 , z 2 ) = ( z 2 ) n P ( z 1 z 2 ), where P is some univari- ate polynomial. Then p is H-Stable iff the roots of P are non-positive real numbers. This assertion is just a rephrasing of the next set equality: C − { z 1 : Re ( z 1 ) , Re ( z 2 ) > 0 } = { x ∈ R : x ≤ 0 } . z 2 This simple bivariate observation gives the connection between H-Stability and Hyperbolicity : 12

: A homogeneous polynomial p ∈ Hom C ( m, n ) Fact 0.4 is H-Stable iff it is e -hyperbolic, e = (1 , ..., 1), i.e. the roots of p ( x 1 − t, ..., x m − t ) = 0 are real for all real vectors X ∈ R m , and its hyperbolic cone contains the positive orthant R m ++ , i.e. the roots of p ( X − te ) = 0 are positive real numbers for all positive real vectors X ∈ R m ++ . p p ( X ) ∈ Hom + ( m, n ) for all X ∈ R m Moreover ++ and | p ( z 1 , ..., z m ) | ≥ | p ( Re ( z 1 ) , ..., Re ( z m )) | : Re ( z i ) ≥ 0, 1 ≤ i ≤ m . 13

Note that a determinantal polynomial Det Q is H- Stable for non-trivial PSD tuples: Q i � 0 , 1 ≤ i ≤ n Q i ≻ 0. � A homogeneous polynomial q ∈ Hom + ( n, n ) is called doubly-stochastic if ∂ q (1 , 1 , . . . , 1) = 1 , 1 ≤ i ≤ n. ∂x i Alternative definition: q ( x 1 , ..., x n ) ≥ 1 ≤ i ≤ n x i , x i > 0; q ( e ) = 1 . (4) � A determinantal polynomial Det Q is H-Stable and doubly-stochastic for doubly-stochastic tuples ( Q 1 , ..., Q n ) is doubly-stochastic !?!?... A possible generalization of Van Der Waerden and Bapat’s conjectures, but how to prove it? All previous proofs heavily relied on the matrix struc- ture. 14

The Capacity , which appeared as an algorithmic tool, happened to be the “saviour”! 15

Theorem 0.5: Let p ∈ Hom + ( n, n ) be H-Stable polynomial and i − 1  i − 1   G ( i ) = , i > 1; G (1) = 1 .    i Then the following inequality holds ∂ n ∂x 1 ...∂x n p (0 , . . . , 0) 1 ≥ ≥ 2 ≤ i ≤ n G ( min( i, deg p ( i ))) . � Cap ( p ) (5) wdv ( i ) wdv ( i − 1) , where vdw ( i ) = i ! Actually, G ( i ) = i i and this function G is strictly decreasing on [0 , ∞ ). 2 ≤ i ≤ n G ( min( i, deg p ( i ))) ≥ G (2) · · · G ( n ) = n ! Thus � n n : Assume WLOG that Cap ( p ) > 0 . Corollary 0.6 Then ∂ n ∂x 1 ...∂x n p (0 , . . . , 0) ≥ n ! (6) n n Cap ( p ) Equality in (6) is attained iff p ( x 1 , ..., x n ) = ( a 1 x 1 + ... + a n x n ) n ; a i > 0 , 1 ≤ i ≤ n. 16

Proof: Step 1. Lemma 0.7: Consider an univariate polynomial 0 ≤ i ≤ k a i t i ; a i ≥ 0 , a k > 0 . If the roots of R ( t ) = � R are real( nec. non-positive) then k − 1 R ( t )  k − 1   R ′ (0) ≥ G ( k ) inf t , G ( k ) = (7)    k t> 0 Proof: The case R (0) = 0 is trivial: G ( k ) ≥ 1 and R ( t ) R ′ (0) = inf t> 0 t . Otherwise, R ( t ) = R (0) 1 ≤ i ≤ k (1 + b i t ) where b i > � 0 , 1 ≤ i ≤ k . Assume WLOG that R (0) = 1. We get, using AM/GM inequality, that R ( t ) ≤ Pow ( t ) =: (1 + R ′ (0) t ) k k Pow ( t ) = R ′ (0)( G ( k )) − 1 . Easy to compute that inf t> 0 t Which leads to Pow ( t ) R ( t ) R ′ (0)( G ( k )) − 1 = inf ≥ inf t> 0 t . t t> 0 17