Computing real points on determinantal varieties and spectrahedra - PowerPoint PPT Presentation

Computing real points on determinantal varieties and spectrahedra Didier Henrion 1 , 2 , 3 Simone Naldi 1 , 2 Mohab Safey El Din 4 1 LAAS - CNRS (Toulouse) 2 Universit´ e de Toulouse 3 Czech Technical University (Prague) 4 ´ Equipe POLSYS - Lip6 (Paris) Proj´ et GeoLMI: www.laas.fr/geolmi J N C F 2 0 1 4 Luminy — 4/11 0/13

Linear matrices and Spectrahedra A linear matrix is a polynomial Examples matrix of degree 1:  x 11 . . . x 1 m  A( x ) = A 0 + x 1 A 1 + . . . + x n A n , � x 1 1 − x 3 � . . ... . .   . . x 2 1   with x = ( x 1 , . . . , x n ) . x m 1 . . . x mm Suppose that A i ∈ Q m × m . For A i = A T i , a linear matrix Lyapounov Stability is an LMI inequality is the positivity condition Solve in P A 0 + x 1 A 1 + . . . + x n A n � 0 P ≻ 0 d f dt = M · f → M T P + PM ≺ 0 where � 0 is positive semidefinite. The set S = { x ∈ R n : A ( x ) � 0 } is � 1 x 1 0 x 1 � 1 x 1 x 2 x 3 called a spectrahedron . Properties: � � x 1 1 x 2 0 x 1 1 x 1 x 2 0 x 2 1 x 3 x 2 x 1 1 x 1 ◮ convex basic semi-algebraic x 1 0 x 3 1 x 3 x 2 x 1 1 ◮ exposed faces ◮ over Fr ( S ) → det( A ) = 0 1/13

Linear matrices and Spectrahedra A linear matrix is a polynomial Examples matrix of degree 1:  x 11 . . . x 1 m  A( x ) = A 0 + x 1 A 1 + . . . + x n A n , � x 1 1 − x 3 � . . ... . .   . . x 2 1   with x = ( x 1 , . . . , x n ) . x m 1 . . . x mm Suppose that A i ∈ Q m × m . For A i = A T i , a linear matrix Lyapounov Stability is an LMI inequality is the positivity condition Solve in P A 0 + x 1 A 1 + . . . + x n A n � 0 P ≻ 0 d f dt = M · f → M T P + PM ≺ 0 where � 0 is positive semidefinite. The set S = { x ∈ R n : A ( x ) � 0 } is called a spectrahedron . Properties: ◮ convex basic semi-algebraic ◮ exposed faces ◮ over Fr ( S ) → det( A ) = 0 1/13

Linear matrices and Spectrahedra Define the complex determinantal variety � x ∈ C n � � D r = � rank A( x ) ≤ r for r ≤ m − 1 . � Theorem Henrion-N.-Safey � � Let A ( x ) be symmetric. Let r = min rank A( x ) | x ∈ S and C a connected component of D r ∩ R n such that C ∩ S � = ∅ . Then C ⊂ S . Remark: compute points in D r ∩ R n → points in S . Semidefinite Programming: min c 1 x 1 + . . . + c n x n � x ∈ R n | A( x ) � 0 � s . t . x ∈ S = The “probability” to be solution is positive for small-rank points. 2/13

Problem statement Given X 2 ◮ m , n , r ∈ N , 0 ≤ r ≤ m − 1 ◮ A 0 , A 1 . . . A n ∈ Q m × m , let ◮ A( x ) = A 0 + x 1 A 1 + . . . + x n A n � x ∈ C n � � ◮ D r = � rank A( x ) ≤ r . � X 1 → D r is an algebraic set! Then Compute one point in each connected component of D r ∩ R n . * one point in each connected component = a good sample set * r = m − 1 : hypersurface det A = 0 * r = m − 1 , n = 1 : Real Eigenvalue Problem * n ≥ 2 : positive dimensional problem * first step for solving det A > 0 and det A ≥ 0 . 3/13

State of the art Existence/computation of real roots ◮ F ( x 1 . . . x n ) = 0 , deg F = m : complexity m O ( n ) , hard in practice [ Basu, Pollack, Roy, Grigoriev, Vorobjov, Heintz, Solerno ]; ◮ Using polar varieties [ Bank, Giusti, Heintz, Mbakop, Pardo, Safey, Schost ]: ◮ O ( m 3n ) : regular case ◮ O ( m 4n ) : singular case. ◮ Gr¨ obner Bases = to compute solutions to poly. equations [ FGb,RAGlib ] ◮ Quadratic case. Complexity: poly. in n , expon. in the codimension Determinantal structure ◮ Extensively studied in Algebraic Geometry ◮ Finite (0-dimensional) case: Gr¨ obner Bases methods ❀ complexity bounds [ Faug` ere, Safey, Spaenlehauer ] 4/13

Positive dimensional singular varieties How to avoid singularities? Input : P 1 = . . . = P a = 0 A system Q 1 = . . . = Q b = 0 − → V ( P 1 , . . . , P a ) possibly singular V ( Q 1 , . . . , Q b ) good properties How to reduce the dimension? A system R 1 = . . . = R c = 0 − → dim V ( Q 1 , . . . , Q b ) > 0 V ( R 1 , . . . , R c ) is finite and such that C ⊂ ( V ( P 1 , . . . , P a ) ∩ R n ) − → C ∩ ( V ( R 1 , . . . , R c ) ∩ R n ) � = ∅ 5/13

Removing singularities Room − Kempf desingularization : we build the bi-linear system Q . . .  y 1 , 1 y 1 , m − r  . . A( x ) · Y = A( x ) · . .  = 0 .   . .  y m , 1 . . . y m , m − r 1 . . . 0   . . ... U · Y = . .   . .   0 . . . 1 where U has full rank. It defines a set V r = V ( Q ) ⊂ C n + m ( m − r ) . ◮ rank A( x ) ≤ r ⇐ ⇒ dim(ker A( x )) ≥ m − r . � Y � = ker A( x ) . ◮ Π x ( V r ) ⊂ D r ◮ Real points in V r : ( x , y ) ∈ V r ∩ R n + m ( m − r ) → x ∈ D r ∩ R n Theorem: generic smoothness and equidimensionality of V r = V ( Q ) . → for generic linear matrices A , the set V r has no singular points. 6/13

Compute critical points Consider the projection map Π a ( x , y ) = a 1 x 1 + . . . + a n x n . Critical points → solutions to the multi-linear Lagrange system R : � jac X Q ( x , y ) � jac Y Q ( x , y ) z ′ Q ( x , y ) = 0 = 0 . a 1 , . . . , a n 0 · · · 0 where a = [ a 1 , . . . , a n ] T ∈ R n . ◮ R = Critical points of Π a ( x , y ) = a 1 x 1 + . . . + a n x n on V r ◮ z = Lagrange multipliers ◮ ( x , y ) critical for Π a ⇐ ⇒ ∃ z : ( x , y , z ) is a solution ◮ # polynomials = # variables Theorem: generically w.r.t. { A i , a } the solution set is finite. → for generic projections, finite number of critical points. 7/13

Projections on generic lines Degenerate situations Projecting { xy − 1 = 0 } on y = 0 y y one obtains an open set and no critical points for this map. x x change of variables two critical points ( x 2 , y ) ( x 2 , y ) C C π 1 π 1 π 1 π − 1 π − 1 1 1 x 1 x 1 Theorem: generic closedness of projections − → after a change of x − variables, only these two cases can hold. 8/13

Output and complexity The algorithms produces a zero-dimensional ideal � f � = � f 1 . . . f N � . Define: δ := max t =1 ... N deg � f 1 . . . f t � Data representation Rational Parametrization ( p , p 0 , p 1 , . . . , p n ) : x 1 = p 1 ( t ) p 0 ( t ) , . . . , x n = p n ( t ) p 0 ( t ) , p ( t ) = 0 . Complexity model: [ Giusti, Lecerf, Salvy , 2001, Geometric Resolution] the arithmetic complexity is essentially quadratic on δ. ezout bounds − → δ exponential in m , n !!! B´ → Multi-linear B´ ezout bounds Multi-linear structure − 9/13

Output and complexity Complexity analysis Henrion-N.-Safey The number of arithmetic operations over Q needed to compute one point per connected component of D r with parameters ( m , n , r ) is in: ◮ generic linear matrices � 6 � � � n + m ( m − r ) O Poly ( m , n , r ) · n ◮ symmetric linear matrices � 6 � � � n + ( m + r +1)( m − r ) O Poly ( m , n , r ) · 2 n ◮ Hankel/Toeplitz linear matrices � 6 � � � n + 2 m − r − 1 O Poly ( m , n , r ) · n 10/13

An algorithm for spectrahedra Input A 0 , A 1 , . . . , A n symmetric matrices. Output ◮ [ ] if S = ∅ ◮ a RP ( p , p 0 , p 1 , . . . , p n ) ∈ Q [ t ] , the min-rank r Procedure For r from 1 to m − 1 do ◮ apply Algorithm to (A( x ) , r ) ; ◮ for all x ∈ V ( p ( t ) , x i − p i ( t ) / p 0 ( t )) test whether x ∈ S ; ◮ if yes, return ( p , p 0 , p 1 , . . . , p n , r ) . 11/13

Timings Table: m − r = 1 10 Algorithm m n RAGlib 8 RealDeterminant RAGlib 2 4 0.22 s 2.25 s 6 log (time) 4 2 10 0.63 s 25.6 s 2 2 20 1.99 s 4065 s 0 3 3 0.49 s 2.8 s −2 −4 3 20 10.5 s ≃ 7 h 0 2 4 6 8 10 12 14 16 18 20 n : number of variables 4 2 0.35 s 0.35 s Figure: ( k , r ) = (3 , 2) 4 4 110 s 835 s 4 16 4736 s ∞ 10 4 20 7420 s ∞ 8 RealDeterminant 6 RAGlib Table: m − r = 2 log (time) 4 m n time (s) 2 0 3 2 0.23 s −2 0 2 4 6 8 10 12 14 16 18 20 3 8 10.3 s n : number of variables 3 12 175 s Figure: ( k , r ) = (4 , 3) 4 4 503 s 4 5 716 s First implementation under Maple . 5 2 3 s We use FGb for Grobner bases 5 3 7 s computations 12/13

Number of complex solutions A 0 , A 1 , . . . , A n are random or random-symmetric m n r generic symm. m n r generic symm. 2 2 1 4 4 3 9 2 39 26 2 3 1 6 5 3 15 2 39 26 2 4 1 6 5 3 20 2 39 26 2 8 1 6 5 4 3 3 52 42 2 20 1 6 5 4 4 3 120 80 3 3 2 21 17 4 6 3 264 152 3 4 2 33 23 4 7 3 284 162 3 5 2 39 26 4 10 3 284 162 3 6 2 39 26 4 20 3 284 162 min { n , m 2 − r 2 } r ( m − r ) � m ( m − r ) N − 1 �� r ( m − r ) �� � � � generic ≤ N − ( m − r ) 2 − ℓ N − ℓ ℓ N =( m − r ) 2 ℓ =0 min { n , c + r ( m − r ) } r ( m − r ) c n − 1 �� r ( m − r ) � �� � � � symmetric ≤ n − c + r ( m − r ) − ℓ n − ℓ ℓ N = c − r ( m − r ) ℓ =0 with c = ( m − r )( m + r + 1) 2 13/13

Thank you 13/13