On algebraic and geometric properties of spectral convex bodies - PowerPoint PPT Presentation

On algebraic and geometric properties of spectral convex bodies Raman Sanyal Goethe-Universit¨ at Frankfurt joint work with James Saunderson (Monash University) arXiv:2001.04361 1/ 17

Between you and I How many convex bodies do you really know? 2/ 17

Between you and I How many convex bodies do you really know? [A convex body is a convex and compact set.] 2/ 17

Between you and I How many convex bodies do you really know? [A convex body is a convex and compact set.] How many of them are not polytopes? [A polytope is the convex hull of finitely many points. Equivalently, the bounded set of solutions to finitely many linear inequalities.] 2/ 17

Between you and I How many convex bodies do you really know? [A convex body is a convex and compact set.] How many of them are not polytopes? [A polytope is the convex hull of finitely many points. Equivalently, the bounded set of solutions to finitely many linear inequalities.] Let me introduce you to Spectral convex bodies 2/ 17

Between you and I How many convex bodies do you really know? [A convex body is a convex and compact set.] How many of them are not polytopes? [A polytope is the convex hull of finitely many points. Equivalently, the bounded set of solutions to finitely many linear inequalities.] Let me introduce you to Spectral convex bodies Focus today: ◮ Operations and metric properties Minkowski sums, polarity, volume and Steiner polynomials ◮ Geometric and algebraic boundary faces, algebraic degree, hyperbolicity ◮ Representations spectrahedra and spectrahedral shadows 2/ 17

Spectral convex sets S d group of permutations acting on R d σ · ( x 1 , . . . , x d ) := ( x σ (1) , . . . , x σ ( d ) ) . 3/ 17

Spectral convex sets S d group of permutations acting on R d σ · ( x 1 , . . . , x d ) := ( x σ (1) , . . . , x σ ( d ) ) . A convex set K ⊆ R d is symmetric if σ K = K for all σ ∈ S d . 3/ 17

Spectral convex sets S d group of permutations acting on R d σ · ( x 1 , . . . , x d ) := ( x σ (1) , . . . , x σ ( d ) ) . A convex set K ⊆ R d is symmetric if σ K = K for all σ ∈ S d . Symmetric matrices S 2 R d = { A ∈ R d × d : A t = A } . 3/ 17

Spectral convex sets S d group of permutations acting on R d σ · ( x 1 , . . . , x d ) := ( x σ (1) , . . . , x σ ( d ) ) . A convex set K ⊆ R d is symmetric if σ K = K for all σ ∈ S d . Symmetric matrices S 2 R d = { A ∈ R d × d : A t = A } . Spectrum λ ( A ) = ( λ 1 , λ 2 , . . . , λ d ) ∈ R d are the d real eigenvalues of A ∈ S 2 R d . 3/ 17

Spectral convex sets S d group of permutations acting on R d σ · ( x 1 , . . . , x d ) := ( x σ (1) , . . . , x σ ( d ) ) . A convex set K ⊆ R d is symmetric if σ K = K for all σ ∈ S d . Symmetric matrices S 2 R d = { A ∈ R d × d : A t = A } . Spectrum λ ( A ) = ( λ 1 , λ 2 , . . . , λ d ) ∈ R d are the d real eigenvalues of A ∈ S 2 R d . A spectral convex set is a set of the form Λ( K ) := { A ∈ S 2 R d : λ ( A ) ∈ K } , where K is a symmetric convex set. 3/ 17

Spectral convex sets S d group of permutations acting on R d σ · ( x 1 , . . . , x d ) := ( x σ (1) , . . . , x σ ( d ) ) . A convex set K ⊆ R d is symmetric if σ K = K for all σ ∈ S d . Symmetric matrices S 2 R d = { A ∈ R d × d : A t = A } . Spectrum λ ( A ) = ( λ 1 , λ 2 , . . . , λ d ) ∈ R d are the d real eigenvalues of A ∈ S 2 R d . A spectral convex set is a set of the form Λ( K ) := { A ∈ S 2 R d : λ ( A ) ∈ K } , where K is a symmetric convex set. Proposition If K is a symmetric convex set/body, then Λ( K ) is a convex set/body. 3/ 17

Examples: Λ( K ) := { A ∈ S 2 R d : λ ( A ) ∈ K } ◮ Operator norm: K = { x : � x � ∞ ≤ 1 } = [ − 1 , 1] d Λ( K ) = { A ∈ S 2 R d : λ max ( A ) ≤ 1 } 4/ 17

Examples: Λ( K ) := { A ∈ S 2 R d : λ ( A ) ∈ K } ◮ Operator norm: K = { x : � x � ∞ ≤ 1 } = [ − 1 , 1] d Λ( K ) = { A ∈ S 2 R d : λ max ( A ) ≤ 1 } ◮ Nuclear norm: K = { x : � x � 1 ≤ 1 } Λ( K ) = { A ∈ S 2 R d : | λ 1 ( A ) | + · · · + | λ d ( A ) | ≤ 1 } 4/ 17

Examples: Λ( K ) := { A ∈ S 2 R d : λ ( A ) ∈ K } ◮ Operator norm: K = { x : � x � ∞ ≤ 1 } = [ − 1 , 1] d Λ( K ) = { A ∈ S 2 R d : λ max ( A ) ≤ 1 } ◮ Nuclear norm: K = { x : � x � 1 ≤ 1 } Λ( K ) = { A ∈ S 2 R d : | λ 1 ( A ) | + · · · + | λ d ( A ) | ≤ 1 } ◮ Frobenius norm: K = B ( R d ) = { x : � x � 2 ≤ 1 } Λ( K ) = { A ∈ S 2 R d : λ 1 ( A ) 2 + · · · + λ d ( A ) 2 ≤ 1 } = B ( S 2 R d ) 4/ 17

Examples: Λ( K ) := { A ∈ S 2 R d : λ ( A ) ∈ K } ◮ Operator norm: K = { x : � x � ∞ ≤ 1 } = [ − 1 , 1] d Λ( K ) = { A ∈ S 2 R d : λ max ( A ) ≤ 1 } ◮ Nuclear norm: K = { x : � x � 1 ≤ 1 } Λ( K ) = { A ∈ S 2 R d : | λ 1 ( A ) | + · · · + | λ d ( A ) | ≤ 1 } ◮ Frobenius norm: K = B ( R d ) = { x : � x � 2 ≤ 1 } Λ( K ) = { A ∈ S 2 R d : λ 1 ( A ) 2 + · · · + λ d ( A ) 2 ≤ 1 } = B ( S 2 R d ) ◮ PSD cone: K = R d ≥ 0 Λ( K ) = { A ∈ S 2 R d : A positive semidefinite } 4/ 17

Examples: Λ( K ) := { A ∈ S 2 R d : λ ( A ) ∈ K } ◮ Operator norm: K = { x : � x � ∞ ≤ 1 } = [ − 1 , 1] d Λ( K ) = { A ∈ S 2 R d : λ max ( A ) ≤ 1 } ◮ Nuclear norm: K = { x : � x � 1 ≤ 1 } Λ( K ) = { A ∈ S 2 R d : | λ 1 ( A ) | + · · · + | λ d ( A ) | ≤ 1 } ◮ Frobenius norm: K = B ( R d ) = { x : � x � 2 ≤ 1 } Λ( K ) = { A ∈ S 2 R d : λ 1 ( A ) 2 + · · · + λ d ( A ) 2 ≤ 1 } = B ( S 2 R d ) ◮ PSD cone: K = R d ≥ 0 Λ( K ) = { A ∈ S 2 R d : A positive semidefinite } ◮ Schur-Horn orbitopes [S-Sottile-Sturmfels’11]: K = conv ( S d · p ) Λ( K ) = { A ∈ S 2 R d : λ ( A ) majorized by p } 4/ 17

Spectral convex sets Proposition If K is a symmetric convex set/body, then Λ( K ) is a convex set/body. 5/ 17

Spectral convex sets Proposition If K is a symmetric convex set/body, then Λ( K ) is a convex set/body. D : S 2 R d → R d diagonal projection D ( A ) = ( A 11 , A 22 , . . . , A dd ) δ : R d → S 2 R d diagonal embedding Lemma Let K be a symmetric convex set. Then D (Λ( K )) = K = D (Λ( K ) ∩ δ ( R d )) . [Needs Schur’s insight: D ( A ) is majorized by λ ( A ).] 5/ 17

Spectral convex sets Proposition If K is a symmetric convex set/body, then Λ( K ) is a convex set/body. D : S 2 R d → R d diagonal projection D ( A ) = ( A 11 , A 22 , . . . , A dd ) δ : R d → S 2 R d diagonal embedding Lemma Let K be a symmetric convex set. Then D (Λ( K )) = K = D (Λ( K ) ∩ δ ( R d )) . [Needs Schur’s insight: D ( A ) is majorized by λ ( A ).] Proof of Proposition. ◮ Show A ∈ conv (Λ( K )), then A ∈ Λ( K ). 5/ 17

Spectral convex sets Proposition If K is a symmetric convex set/body, then Λ( K ) is a convex set/body. D : S 2 R d → R d diagonal projection D ( A ) = ( A 11 , A 22 , . . . , A dd ) δ : R d → S 2 R d diagonal embedding Lemma Let K be a symmetric convex set. Then D (Λ( K )) = K = D (Λ( K ) ∩ δ ( R d )) . [Needs Schur’s insight: D ( A ) is majorized by λ ( A ).] Proof of Proposition. ◮ Show A ∈ conv (Λ( K )), then A ∈ Λ( K ). ◮ O ( d )-invariance: g Λ( K ) g t = Λ( K ) for all g ∈ O ( d ). 5/ 17

Spectral convex sets Proposition If K is a symmetric convex set/body, then Λ( K ) is a convex set/body. D : S 2 R d → R d diagonal projection D ( A ) = ( A 11 , A 22 , . . . , A dd ) δ : R d → S 2 R d diagonal embedding Lemma Let K be a symmetric convex set. Then D (Λ( K )) = K = D (Λ( K ) ∩ δ ( R d )) . [Needs Schur’s insight: D ( A ) is majorized by λ ( A ).] Proof of Proposition. ◮ Show A ∈ conv (Λ( K )), then A ∈ Λ( K ). ◮ O ( d )-invariance: g Λ( K ) g t = Λ( K ) for all g ∈ O ( d ). ◮ Wlog A = δ ( p ) diagonal. 5/ 17

Spectral convex sets Proposition If K is a symmetric convex set/body, then Λ( K ) is a convex set/body. D : S 2 R d → R d diagonal projection D ( A ) = ( A 11 , A 22 , . . . , A dd ) δ : R d → S 2 R d diagonal embedding Lemma Let K be a symmetric convex set. Then D (Λ( K )) = K = D (Λ( K ) ∩ δ ( R d )) . [Needs Schur’s insight: D ( A ) is majorized by λ ( A ).] Proof of Proposition. ◮ Show A ∈ conv (Λ( K )), then A ∈ Λ( K ). ◮ O ( d )-invariance: g Λ( K ) g t = Λ( K ) for all g ∈ O ( d ). ◮ Wlog A = δ ( p ) diagonal. ◮ A = � i µ i A i for A i ∈ Λ( K ) implies p = D ( A ) = � i µ i D ( A i ) ∈ K . 5/ 17

Spectral convex sets Proposition If K is a symmetric convex set/body, then Λ( K ) is a convex set/body. D : S 2 R d → R d diagonal projection D ( A ) = ( A 11 , A 22 , . . . , A dd ) δ : R d → S 2 R d diagonal embedding Lemma Let K be a symmetric convex set. Then D (Λ( K )) = K = D (Λ( K ) ∩ δ ( R d )) . [Needs Schur’s insight: D ( A ) is majorized by λ ( A ).] Proof of Proposition. ◮ Show A ∈ conv (Λ( K )), then A ∈ Λ( K ). ◮ O ( d )-invariance: g Λ( K ) g t = Λ( K ) for all g ∈ O ( d ). ◮ Wlog A = δ ( p ) diagonal. ◮ A = � i µ i A i for A i ∈ Λ( K ) implies p = D ( A ) = � i µ i D ( A i ) ∈ K . ◮ Again by Lemma: A = δ ( p ) ∈ Λ( K ). 5/ 17

Rich class of convex sets Let K , L ⊂ R d be symmetric closed convex sets. ◮ Intersections Λ( K ) ∩ Λ( L ) = Λ( K ∩ L ) . 6/ 17

Rich class of convex sets Let K , L ⊂ R d be symmetric closed convex sets. ◮ Intersections Λ( K ) ∩ Λ( L ) = Λ( K ∩ L ) . ◮ Minkowski sums: K + L := { p + q : p ∈ K , q ∈ L } Λ( K ) + Λ( L ) = Λ( K + L ) . 6/ 17