Geometry of orthogonally invariant matrix varieties

Transcription

1 Geometry of orthogonally invariant matrix varieties Dmitriy Drusvyatskiy Mathematics, University of Washington Joint work with H.-L. Lee (UW), G. Ottaviano (Florence), and R.R. Thomas (UW) UC Davis Algebra & Discrete Mathematics AFOSR YIP

2 The Euclidean distance degree of orthogonally invariant matrix varieties, D-Lee-Ottaviani-Thomas, To appear in Israel J. Math., Counting real critical points of the distance to orthogonally invariant matrix sets, D-Lee-Thomas, SIAM J. Matrix Anal. Applic. 36(3): , /14

3 Absolute symmetry Signed permutation group: Π n ± = {signed permutations} Subset S R n is Π n ±-invariant if πs = S for all π Π n ±. 3/14

4 Absolute symmetry Signed permutation group: Π n ± = {signed permutations} Subset S R n is Π n ±-invariant if πs = S for all π Π n ±. Real orthogonal group: O n = {n n orthogonal matrices} Subset M R n n is O n -invariant if UM = MU = M for all U O n 3/14

5 Universal examples Singular values decomp.: any X R n n can be written as σ 1 (X) σ 2 (X) X = U... V T σn(x) with U, V O n and σ 1 (X) σ 2 (x)... σ n (X) 0. 4/14

6 Universal examples Singular values decomp.: any X R n n can be written as σ 1 (X) σ 2 (X) X = U... V T σn(x) with U, V O n and σ 1 (X) σ 2 (x)... σ n (X) 0. Singular values σ i (X) = eigenvalues of X T X 4/14

7 Universal examples Singular values decomp.: any X R n n can be written as σ 1 (X) σ 2 (X) X = U... V T σn(x) with U, V O n and σ 1 (X) σ 2 (x)... σ n (X) 0. Singular values σ i (X) = eigenvalues of X T X Examples of O n -invariant sets: Rr n n = {X : rank X r} B = {X : σ i (X) 1} i E 3 3 = {X R 3 3 : σ 1 (X) = σ 2 (X), σ 3 (X) = 0} 4/14

8 Universal examples Singular values decomp.: any X R n n can be written as σ 1 (X) σ 2 (X) X = U... V T σn(x) with U, V O n and σ 1 (X) σ 2 (x)... σ n (X) 0. Singular values σ i (X) = eigenvalues of X T X Examples of O n -invariant sets: Rr n n = {X : rank X r} B = {X : σ i (X) 1} i E 3 3 = {X R 3 3 : σ 1 (X) = σ 2 (X), σ 3 (X) = 0} Elementary fact: M is O n -invariant M = σ 1 (S) for Π n ±-invariant S 4/14

9 Universal examples Singular values decomp.: any X R n n can be written as σ 1 (X) σ 2 (X) X = U... V T σn(x) with U, V O n and σ 1 (X) σ 2 (x)... σ n (X) 0. Singular values σ i (X) = eigenvalues of X T X Examples of O n -invariant sets: Rr n n = {X : rank X r} B = {X : σ i (X) 1} i E 3 3 = {X R 3 3 : σ 1 (X) = σ 2 (X), σ 3 (X) = 0} Elementary fact: M is O n -invariant M = σ 1 (S) for Π n ±-invariant S Concisely S = {x R n : Diag (x) M} is diagonal restriction 4/14

10 Guiding philosophy Transfer Principle: Properties of σ 1 (S) and S are in one-to-one correspondence. 5/14

11 Guiding philosophy Transfer Principle: Properties of σ 1 (S) and S are in one-to-one correspondence. Examples: σ 1 (S) convex S convex (von Neumann 37, Davis 57, Lewis 96) 5/14

12 Guiding philosophy Transfer Principle: Properties of σ 1 (S) and S are in one-to-one correspondence. Examples: σ 1 (S) convex S convex (von Neumann 37, Davis 57, Lewis 96) Reason: V, X = max σ(v ), x x S max X σ 1 (S) 5/14

13 Guiding philosophy Transfer Principle: Properties of σ 1 (S) and S are in one-to-one correspondence. Examples: σ 1 (S) convex S convex (von Neumann 37, Davis 57, Lewis 96) Reason: max V, X = max σ(v ), x X σ 1 (S) x S Here V, X = tr (V T X) is the trace product. 5/14

14 Guiding philosophy Transfer Principle: Properties of σ 1 (S) and S are in one-to-one correspondence. Examples: σ 1 (S) convex S convex (von Neumann 37, Davis 57, Lewis 96) Reason: max V, X = max σ(v ), x X σ 1 (S) x S Here V, X = tr (V T X) is the trace product. σ 1 (S) is C p -smooth S is C p -smooth (Sylvester 85, Šilhavý 99, Daniilidis-D-Lewis 14) 5/14

15 Guiding philosophy Transfer Principle: Properties of σ 1 (S) and S are in one-to-one correspondence. Examples: σ 1 (S) convex S convex (von Neumann 37, Davis 57, Lewis 96) Reason: max V, X = max σ(v ), x X σ 1 (S) x S Here V, X = tr (V T X) is the trace product. σ 1 (S) is C p -smooth S is C p -smooth (Sylvester 85, Šilhavý 99, Daniilidis-D-Lewis 14) Reason: dist σ 1 (S)(Y ) = dist S (σ(y )) 5/14

16 Guiding philosophy Transfer Principle: Properties of σ 1 (S) and S are in one-to-one correspondence. Examples: σ 1 (S) convex S convex (von Neumann 37, Davis 57, Lewis 96) Reason: max V, X = max σ(v ), x X σ 1 (S) x S Here V, X = tr (V T X) is the trace product. σ 1 (S) is C p -smooth S is C p -smooth (Sylvester 85, Šilhavý 99, Daniilidis-D-Lewis 14) Reason: dist σ 1 (S)(Y ) = dist S (σ(y )) σ 1 (S) is algebraic S is algebraic (Daniilidis-Malick-Sendov 09) 5/14

17 Guiding philosophy Transfer Principle: Properties of σ 1 (S) and S are in one-to-one correspondence. Examples: σ 1 (S) convex S convex (von Neumann 37, Davis 57, Lewis 96) Reason: max V, X = max σ(v ), x X σ 1 (S) x S Here V, X = tr (V T X) is the trace product. σ 1 (S) is C p -smooth S is C p -smooth (Sylvester 85, Šilhavý 99, Daniilidis-D-Lewis 14) Reason: dist σ 1 (S)(Y ) = dist S (σ(y )) σ 1 (S) is algebraic S is algebraic (Daniilidis-Malick-Sendov 09) Reason: e i (σ 2 1 (X),..., σ2 n(x)) coefficients of det(λi X T X). 5/14

18 Metric comparison Metric comparison: (von Neumann 37, Fan 49) Always σ(x) σ(y ) 2 X Y F. 6/14

19 Metric comparison Metric comparison: (von Neumann 37, Fan 49) Always σ(x) σ(y ) 2 X Y F. Equality holds U, V O n with U T XV = Diag σ(x) and U T Y V = Diag σ(y ) (simultaneous ordered singular value decomp.) 6/14

20 Metric comparison Metric comparison: (von Neumann 37, Fan 49) Always σ(x) σ(y ) 2 X Y F. Equality holds U, V O n with U T XV = Diag σ(x) and U T Y V = Diag σ(y ) (simultaneous ordered singular value decomp.) Special case: (Hardy-Littlewood-Pólya 52) x y x y 6/14

21 Euclidean Distance Degree Consider variety V = {x R n : f 1 (x) = f 2 (x) =... = f k (x) = 0}. with f i polynomials. 7/14

22 Euclidean Distance Degree Consider variety V = {x R n : f 1 (x) = f 2 (x) =... = f k (x) = 0}. with f i polynomials. Defn (ED critical point): x EDcrit(y) x smooth on V, y x T x V C. EDdegree(V) = EDcrit(y) (constant for general y!). (Draisma-Horobeţ-Ottaviani-Sturmfels-Thomas 15) 7/14

23 Euclidean Distance Degree Hilbert & Cohn-Vossen: Anschauliche Geometrie (Geometry and the Imagination) Springer-Verlag, Berlin 1932 The simplest curves are the planar curves. Among them the simplest one is the line. The next simplest is the circle. After that comes the parabola, and finally, general conics. 8/14

24 Euclidean Distance Degree Hilbert & Cohn-Vossen: Anschauliche Geometrie (Geometry and the Imagination) Springer-Verlag, Berlin 1932 The simplest curves are the planar curves. Among them the simplest one is the line (EDdegree 1). The next simplest is the circle (EDdegree 2). After that comes the parabola (EDdegree 3), and finally, general conics (EDdegree 4). 8/14

25 ED degree of O n -invariant varieties Theorem: (D-Lee-Ottaviani-Thomas 15) EDdegree(σ 1 (S)) = EDdegree(S) 9/14

26 ED degree of O n -invariant varieties Theorem: (D-Lee-Ottaviani-Thomas 15) EDdegree(σ 1 (S)) = EDdegree(S) Examples: Orthogonal Group: O n = {X : X T X = I} EDdegree(O n ) = 2 n (Draisma-Baaijens 14) 9/14

27 ED degree of O n -invariant varieties Theorem: (D-Lee-Ottaviani-Thomas 15) Examples: Low rank matrices: EDdegree(σ 1 (S)) = EDdegree(S) R n n r = {X : rank X r} ( ) n EDdegree(Rr n n ) = r (Draisma-Horobeţ-Ottaviani-Sturmfels-Thomas 15) 9/14

28 ED degree of O n -invariant varieties Theorem: (D-Lee-Ottaviani-Thomas 15) EDdegree(σ 1 (S)) = EDdegree(S) Examples: Essential Variety: E 3 3 = {X : σ 1 (X) = σ 2 (X), σ 3 (X) = 0} EDdegree(E n ) = 6 (D-Lee-Ottaviani-Thomas 15) 9/14

29 ED degree of O n -invariant varieties Theorem: (D-Lee-Ottaviani-Thomas 15) EDdegree(σ 1 (S)) = EDdegree(S) Examples: Special Linear Group: SL n ± = {X : det(x) = ±1} EDdegree(SL n ±) = n2 n (Draisma-Baaijens 14) 9/14

30 ED degree of O n -invariant varieties Theorem: (D-Lee-Ottaviani-Thomas 15) EDdegree(σ 1 (S)) = EDdegree(S) Examples: Schatten hypersurface: F n,d = {X : i σ i(x) d = 1} ( ) n n 1 EDdegree(F n,d ) = d (d 1) i n δ(j, d 2). j + 1 i=1 j=1 (Lee 14) 9/14

31 Why is this surprising? Main challenges: 1) No explicit polynomial description of σ 1 (S). Eg. E 3 3 = {X : det(x) = 0, 2XX T X tr (XX T )X = 0} 2) 10/14

32 Why is this surprising? Main challenges: 1) No explicit polynomial description of σ 1 (S). Eg. E 3 3 = {X : det(x) = 0, 2XX T X tr (XX T )X = 0} 2) Smoothness is finicky: Eg. Cartan Umbrella V = {(x, y, z) : z(x 2 + y 2 ) x 3 = 0} 10/14

33 Strategy: step I (complexification) Setting M = σ 1 (S), we have M = U Diag (S) V T U,V O n 11/14

34 Strategy: step I (complexification) Setting M = σ 1 (S), we have M = U Diag (S) V T U,V O n What about M C? 11/14

35 Strategy: step I (complexification) Setting M = σ 1 (S), we have M = U Diag (S) V T U,V O n What about M C? Obstruction: (Choudhury-Horn 87) A matrix X C n n admits an algebraic SVD X = UDiag (s)v T with U, V O n C and s C n if and only if XX T is diagonalizable and rank X = rank XX T. 11/14

36 Strategy: step I (complexification) Setting M = σ 1 (S), we have M = U Diag (S) V T U,V O n What about M C? Obstruction: (Choudhury-Horn 87) A matrix X C n n admits an algebraic SVD X = UDiag (s)v T with U, V O n C and s C n if and only if XX T is diagonalizable and rank X = rank XX T. Thm: (D-Lee-Ottaviani-Thomas 15) Equalities hold: S C = {x : Diag (x) M C } and M C = U Diag (S C ) V T U,V O n C 11/14

37 Strategy: step II (GIT perspective) Suppose M C is invariant under G := OC n On C Invariant ring: as before. Inv = {f C[M C ] : f is G-invariant} 12/14

38 Strategy: step II (GIT perspective) Suppose M C is invariant under G := OC n On C Invariant ring: as before. Inv = {f C[M C ] : f is G-invariant} Then Inv = C[M C /G] where M C /G is the GIT quotient M C /G = {y C n : y i = e i (x 2 1,..., x 2 n) for some x S C } (Geometric Invariant Theory (GIT) developed by Mumford 65) 12/14

39 Strategy: step II (GIT perspective) Suppose M C is invariant under G := OC n On C Invariant ring: as before. Inv = {f C[M C ] : f is G-invariant} Then Inv = C[M C /G] where M C /G is the GIT quotient M C /G = {y C n : y i = e i (x 2 1,..., x 2 n) for some x S C } (Geometric Invariant Theory (GIT) developed by Mumford 65) Then the quotient map is π : M C M C /G X (e 1 (XX T ),..., e n (XX T )) 12/14

40 Strategy: step II (GIT perspective) Suppose M C is invariant under G := OC n On C Invariant ring: as before. Inv = {f C[M C ] : f is G-invariant} Then Inv = C[M C /G] where M C /G is the GIT quotient M C /G = {y C n : y i = e i (x 2 1,..., x 2 n) for some x S C } (Geometric Invariant Theory (GIT) developed by Mumford 65) Then the quotient map is π : M C M C /G X (e 1 (XX T ),..., e n (XX T )) The fiber π 1 (y) is a union of orbits, and π 1 (y) is a single orbit π 1 (y) contains a diagonal matrix. 12/14

41 Conclusion Classical ideas in algebra/geometry/matrix analysis come together to explain behavior of O n -invariant varieties. 13/14

42 Thank you. 14/14