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Spectrahedra Cynthia Vinzant, North Carolina State University Simons Institute, Berkeley (Fall 2017) Cynthia Vinzant Spectrahedra Spectrahedra . . . What? Why? Let PSD d denote the convex cone of positive semidefinite matrices in R d d sym .

  1. Spectrahedra Cynthia Vinzant, North Carolina State University Simons Institute, Berkeley (Fall 2017) Cynthia Vinzant Spectrahedra

  2. Spectrahedra . . . What? Why? Let PSD d denote the convex cone of positive semidefinite matrices in R d × d sym . A spectrahedron is the intersection PSD d with an affine linear space L . Cynthia Vinzant Spectrahedra

  3. Spectrahedra . . . What? Why? Let PSD d denote the convex cone of positive semidefinite matrices in R d × d sym . A spectrahedron is the intersection PSD d with an affine linear space L . Writing L = A 0 + span R { A 1 , . . . , A n } identifies L ∩ PSD d with x ∈ R n : A ( x ) � 0 A ( x ) = A 0 + � n � � S = where i =1 x i A i . These are feasible sets of semidefinite programs (extension of linear programming with applications in combinatorial optimization, control, polynomial optimization, quantum information, . . . ). Cynthia Vinzant Spectrahedra

  4. Some examples polytope cylinder elliptope   1 − x 0 0 y  ℓ 1 ( x ) 0    1 x y 1 + x 0 0 y ...   1 x z       0 0 1 − z 0     1 y z 0 ℓ 12 ( x ) 0 0 0 1 + z Some differences with polyhedra: ◮ S can have infinitely-many faces ◮ dim(face) + dim(normal cone) not always equal to n . Cynthia Vinzant Spectrahedra

  5. Positive semidefinite matrices A real symmetric matrix A is positive semidefinite if the following equivalent conditions hold: ◮ all eigenvalues of A are ≥ 0 ◮ all diagonal minors of A are ≥ 0 ◮ v T Av ≥ 0 for all v ∈ R d ◮ there exists B ∈ R d × k with k A = BB T = ( � r i , r j � ) ij = � c i c T i i =1 where r 1 , . . . , r d , c 1 , . . . , c k are the rows and cols of B Cynthia Vinzant Spectrahedra

  6. The convex cone of PSD matrices The cone of PSD matrices PSD d = conv( { xx T : x ∈ R d } ). PSD d is self-dual under the inner product � A , B � = trace( A · B ) : � A , B � ≥ 0 for all B ∈ PSD d ⇔ � A , bb T � ≥ 0 for all b ∈ R d ⇔ b T Ab ≥ 0 for all b ∈ R d ⇔ A ∈ PSD d Cynthia Vinzant Spectrahedra

  7. The convex cone of PSD matrices The cone of PSD matrices PSD d = conv( { xx T : x ∈ R d } ). PSD d is self-dual under the inner product � A , B � = trace( A · B ) : � A , B � ≥ 0 for all B ∈ PSD d ⇔ � A , bb T � ≥ 0 for all b ∈ R d ⇔ b T Ab ≥ 0 for all b ∈ R d ⇔ A ∈ PSD d � r +1 � Faces of PSD d have dim for r = 0 , 1 , . . . , d and look like 2 F L = { A ∈ PSD d : L ⊆ ker( A ) } . Ex: for L = span { e r +1 , . . . , e d } , �� A � � 0 ∼ F L = : A ∈ PSD r = PSD r 0 0 Cynthia Vinzant Spectrahedra

  8. Structure of spectrahedra A spectrahedron S = { x ∈ R n : A ( x ) � 0 } is a convex, basic-closed semi-algebraic set.  1 − x 0 0  y 1 + x 0 0 y   ↔ A ( x , y , z ) =   0 0 1 − z 0   0 0 0 1 + z Its faces are intersections of faces of PSD d with { A ( x ) : x ∈ R n } → characterized by kernels of A ( x ). Cynthia Vinzant Spectrahedra

  9. Spectrahedral Shadows: an interlude Caution: The projection of spectrahedron may not be a spectrahedron! not basic closed proj ( S ) = S = ⇒ not a spectrahedron Cynthia Vinzant Spectrahedra

  10. Spectrahedral Shadows: an interlude Caution: The projection of spectrahedron may not be a spectrahedron! not basic closed proj ( S ) = S = ⇒ not a spectrahedron Caution: The convex dual of spectrahedron may not be a spectrahedron! 0.5 S ◦ = still not a spectrahedron S = - - - - Cynthia Vinzant Spectrahedra

  11. Spectrahedral shadows: an interlude A spectrahedral shadow is the image of a spectrahedron under linear projection. These are convex semialgebraic sets. Unlike spectrahedra, the class of spectrahedral shadows is closed under projection, duality, convex hull of unions, . . . For more, come to “An Afternoon of Real Algebraic Geometry,” MSRI, Friday Sept. 15, 2-6pm. Cynthia Vinzant Spectrahedra

  12. Spectrahedral shadows: an interlude A spectrahedral shadow is the image of a spectrahedron under linear projection. These are convex semialgebraic sets. Unlike spectrahedra, the class of spectrahedral shadows is closed under projection, duality, convex hull of unions, . . . Helton-Nie Conjecture (2009): Every convex semialgebraic set is a spectrahedral shadow. For more, come to “An Afternoon of Real Algebraic Geometry,” MSRI, Friday Sept. 15, 2-6pm. Cynthia Vinzant Spectrahedra

  13. Spectrahedral shadows: an interlude A spectrahedral shadow is the image of a spectrahedron under linear projection. These are convex semialgebraic sets. Unlike spectrahedra, the class of spectrahedral shadows is closed under projection, duality, convex hull of unions, . . . Helton-Nie Conjecture (2009): Every convex semialgebraic set is a spectrahedral shadow. Counterexample by Scheiderer in 2016. Open: What is the smallest dimension of a counterexample? For more, come to “An Afternoon of Real Algebraic Geometry,” MSRI, Friday Sept. 15, 2-6pm. Cynthia Vinzant Spectrahedra

  14. Example: Elliptopes The d × d elliptope is E d = { A ∈ PSD d : A ii = 1 for all i } = { d × d correlation matrices } in stats E d has 2 d − 1 matrices of rank-one: { xx T : x ∈ {− 1 , 1 } d } , corresponding to cuts in the complete graph K d . Cynthia Vinzant Spectrahedra

  15. Example: Elliptopes The d × d elliptope is E d = { A ∈ PSD d : A ii = 1 for all i } = { d × d correlation matrices } in stats E d has 2 d − 1 matrices of rank-one: { xx T : x ∈ {− 1 , 1 } d } , corresponding to cuts in the complete graph K d . (1 − x i x j ) � � MAXCUT = max w ij = max w ij 2 S ⊂ [ d ] x ∈{− 1 , 1 } d i ∈ S , j ∈ S c i , j (1 − A ij ) (1 − A ij ) � � = max ≤ max . w ij w ij 2 2 A ∈E d A ∈E d , rk( A )=1 i , j i , j Goemans-Williamson use this to give ≈ . 87 optimal cuts of graphs. Cynthia Vinzant Spectrahedra

  16. Example: Univariate Moments S = conv { ( t , t 2 , . . . , t 2 d ) : t ∈ R } is a spectrahedron in R 2 d x ∈ R 2 d : M ( x ) � 0 � � S = where M ( x ) = ( x i + j − 2 ) 1 ≤ i , j ≤ d +1 Cynthia Vinzant Spectrahedra

  17. Example: Univariate Moments S = conv { ( t , t 2 , . . . , t 2 d ) : t ∈ R } is a spectrahedron in R 2 d x ∈ R 2 d : M ( x ) � 0 � � S = where M ( x ) = ( x i + j − 2 ) 1 ≤ i , j ≤ d +1 � 1 � � � x 1 Ex. (d=1): conv { ( t , t 2 ) : t ∈ R } = ( x 1 , x 2 ) : � 0 x 1 x 2 Cynthia Vinzant Spectrahedra

  18. Example: Univariate Moments S = conv { ( t , t 2 , . . . , t 2 d ) : t ∈ R } is a spectrahedron in R 2 d x ∈ R 2 d : M ( x ) � 0 � � S = where M ( x ) = ( x i + j − 2 ) 1 ≤ i , j ≤ d +1 � 1 � � � x 1 Ex. (d=1): conv { ( t , t 2 ) : t ∈ R } = ( x 1 , x 2 ) : � 0 x 1 x 2 Minimization of univariate polynomial of degree ≤ 2 d → Minimization of linear function over S Cynthia Vinzant Spectrahedra

  19. Example: Univariate Moments S = conv { ( t , t 2 , . . . , t 2 d ) : t ∈ R } is a spectrahedron in R 2 d x ∈ R 2 d : M ( x ) � 0 � � S = where M ( x ) = ( x i + j − 2 ) 1 ≤ i , j ≤ d +1 � 1 � � � x 1 Ex. (d=1): conv { ( t , t 2 ) : t ∈ R } = ( x 1 , x 2 ) : � 0 x 1 x 2 Minimization of univariate polynomial of degree ≤ 2 d → Minimization of linear function over S Ex: conv { ( t , t 2 , t 3 ) : t ∈ [ − 1 , 1] } � 1 ± x 1 � � � x 1 ± x 2 x ∈ R 3 : = � 0 x 1 ± x 2 x 2 ± x 3 Cynthia Vinzant Spectrahedra

  20. Extreme Points: Pataki range S = { x ∈ R n : A ( x ) � 0 } , dim( S ) = n , A i ∈ R d × d sym . If x is an extreme point of S and r is the rank of A ( x ) then � r + 1 � � d + 1 � + n ≤ 2 2 � d − r +1 � Furthermore if A 0 , . . . , A n are generic, then n ≥ . 2 The interval of r ∈ Z + satisfying both ≤ ’s is the Pataki range. Cynthia Vinzant Spectrahedra

  21. Extreme Points: Pataki range S = { x ∈ R n : A ( x ) � 0 } , dim( S ) = n , A i ∈ R d × d sym . If x is an extreme point of S and r is the rank of A ( x ) then � r + 1 � � d + 1 � + n ≤ 2 2 � d − r +1 � Furthermore if A 0 , . . . , A n are generic, then n ≥ . 2 The interval of r ∈ Z + satisfying both ≤ ’s is the Pataki range. Open: For each d , n , is there a spectrahedron with an extreme point of each rank in the Pataki range? Cynthia Vinzant Spectrahedra

  22. Facial structure Example: d = 3 , n = 3 Pataki range: r = 1 , 2 Cynthia Vinzant Spectrahedra

  23. Facial structure Example: d = 3 , n = 3 Pataki range: r = 1 , 2 Counting rank-1 matrices: { X : rank( X ) ≤ 1 } is variety of codim 3 and degree 4 in R 3 × 3 sym . ⇒ 0 , 1 , 2 , 3 , 4 or ∞ rank-1 matrices in S (generically 0,2, or 4) Cynthia Vinzant Spectrahedra

  24. Facial structure Example: d = 3 , n = 3 Pataki range: r = 1 , 2 Counting rank-1 matrices: { X : rank( X ) ≤ 1 } is variety of codim 3 and degree 4 in R 3 × 3 sym . ⇒ 0 , 1 , 2 , 3 , 4 or ∞ rank-1 matrices in S (generically 0,2, or 4) There must be ≥ 1 rank-1 matrix. Why? Topology! If ∂ S has no rank-1 matrices, then the map S 2 ∼ = ∂ S → P 2 ( R ) given by x �→ ker( A ( x )) is an embedding. ⇒⇐ (For more see Friedland, Robbin, Sylvester,1984) Cynthia Vinzant Spectrahedra

  25. Another connection with topology Suppose A 0 = I and let f ( x ) = det( A ( x )). ⇒ f is hyperbolic, i.e. for every x ∈ R n , f ( tx ) ∈ R [ t ] is real-rooted. Cynthia Vinzant Spectrahedra

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