Eigenvectors of Tensors and Waring Decomposition

Transcription

1 Eigenvectors of Tensors and Waring Decomposition Luke Oeding University of California Berkeley Oeding, Ottaviani (UCB, Firenze) Giorgio Ottaviani Universita degli Studi di Firenze Waring Decomposition January 16, / 24

2 Polynomial Waring decomposition Let V = C n+1, f S d V homogeneous polynomial. Waring decomposition: f = r i=1 c iv d i, with c i C, and v i V. Goals: Algorithms that quickly decompose low rank forms. (naive algorithms always exist, but are infeasible) Uniform treatment (Eigenvectors and vector bundles). Non-Goal: One algorithm to decompose them all (NP-hard! -[Lim-Hillar 12]). Motivation: CDMA-like communication scheme: Send (the coefficients of) f = r i=1 c iv d Recover v i uniquely. Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24 i.

3 Main Results Theorem (O.-Ottaviani 13) Let f S d C n+1, with d = 2m + 1, n + 1 4, and general among forms of rank r. If r ( ) m+n n then the Koszul Flattening Algorithm produces the unique Waring decomposition. We implemented our algorithm in Macaulay2 and you can download it from the ancillary files accompanying the arxiv version of our paper. Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

4 Algebraic Geometry helps Engineering: generic rank (/C) Theorem (Campbell 1891, Terracini 1916, Alexander-Hirschowitz 1995) The general f ins d C n+1, d 3 has rank ) ( n+d d n + 1, the generic rank, except 2 n 4, d = 4 generic rank is ( ) n+2 2, (n, d) = (4, 3) generic rank is 8. Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

5 Algebraic Geometry helps Engineering: Uniqueness (/C) Theorem (... A-H, 95 ) The general f ins d C n+1 ( n+d, d 3 has the generic rank 2 n 4, d = 4 generic rank is ( ) n+2 2, (n, d) = (4, 3) generic rank is 8. d ) n+1 Theorem (Sylvester 1851, Chiantini-Ciliberto, Mella, Ballico ), except The general f S d C n+1 among the forms of subgeneric rank has a unique decomposition, except 2 n 4, d = 4, r = ( ) n+2 2 1, -ly many decomps. defective (n, d) = (4, 3), r = 7, -ly many decomps. defective rank 9 in S 6 C 3, 2 decomps. weakly defective rank 8 in S 4 C 4, 2 decomps. weakly defective Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

6 Algebraic Geometry helps Engineering: Non-Uniqueness (/C) Expected: If (n+d d ) n+1 is an integer, then uniqueness fails for the general form. Mella showed in 2006 that when d > n this is true. The only known failures are (and we give a uniform proof): S 2m+1 C 2 rank m + 1 Sylvester 1851, S 5 C 3 rank 7 Hilbert-Palatini-Richmond 1902, S 3 C 4 rank 5 Sylvester Pentahedral Theorem. Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

7 From equations to decompositions General approach: Find nice (determinantal) equations for secant varieties Get an algorithm for decomposition. A Flattenings / Catalecticants / truncated moment matrices B Koszul Flattenings and eigenvectors of tensors Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

8 Koszul Flattenings: Examples / Overview Equations of secant varieties from Koszul flattenings: Strassen: σ r (P 2 P 2 P 2 ) P ( C 3 C 3 C 3) Toeplitz: σ r (P 2 ν 2 (P 3 )) P ( C 3 S 2 C 4) Aronhold: σ r (ν 3 (P 2 )) P ( S 3 C 3) Cartwright-Erman-O. 11: σ r (P 2 ν 2 (P n )) P ( C 3 S 2 C n+1), r 5. Landsberg-Ottaviani 2012: Many more cases, much more general. Our decomposition algorithms via Koszul Flattenings Sylvester Pentahedral Thm.: S 3 C 4, r 5, HPR quintics: S 5 C 3, r 7, More generally: S 2m+1 C n+1, r ( ) n+m n. Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

9 Review: The catalecticant algorithm via an example Decompose f = 7x 3 30x 2 y + 42xy 2 19y 3 S 3 (C 2 ): Compute the flattening: S 2 (C 2 ) C 2, ( ) C f =, with kernel: The kernel K (in the space of polynomials on the dual) is spanned by 6 2 x + 7 x y y = (2 x + y )(3 x + 2 y ). Notice (2 x + y ) kills ( x + 2y) and ( x + 2y) d for all d. Also, (3 x + 2 y ) kills (2x 3y) and (2x 3y) d for all d. K annihilates precisely (up to scalar) {( x + 2y), (2x 3y)}. Therefore f = c 1 ( x + 2y) 3 + c 2 (2x 3y) 3. Solve: c 1 = c 2 = 1. Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24 C f

10 Catalecticant algorithm in general [Iarrobino-Kanev 1999] Input: f S d (V ) V = C n+1. 1 Construct C m f = C f, m = d 2 C m f : S m V S d m V x i1 x im m f xi1 xim 2 Compute ker C f, note Rank(f ) rank(c f ). 3 Compute Z = zeros(ker C f ) if #Z =, fail else Z = {[v 1 ],..., [v s ]} 4 Solve f = s i=1 c i v d i, c i C. Output: The unique Waring decomposition of f. Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

11 Catalecticant algorithm in general [Iarrobino-Kanev 1999] The catalecticant algorithm appears in work of Sylvester, Iarrobino-Kanev, Brachat-Comon-Mourrain-Tsigaridas, Bernardi-Idá-Gimigliano. Iarrobino and Kanev gave bounds for the success of the catalecticant algorithm. Here is a slight improvement: Theorem (O.-Ottaviani 2013) Let r i=1 v i d = f be general among forms of rank r in S d V. Set z i := [v i ], Z := {z 1,..., z r } and let m = d 2. 1 If d is even and r ( ) n+m n n 1, or if d is odd and ( ) n+m 1 n, then ker C f = I Z,m (subspace of deg. m polys vanishing on Z). the catalecticant algorithm succeeds with Z = Z = zeros(ker C f ). 2 If d is even n 3 and r = ( ) n+m n n, Z Z is possible. the catalecticant algorithm succeeds after finitely many checks. Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

12 Why the catalecticant algorithm works Given f S d V, we have the catalecticant: Rank conditions: : S m V S d m V x i1 x im C m f f has rank 1 rank C f = 1. subadditivity of matrix rank implies that (f has rank r rank C f r). m f xi1 xim The zero set of the kernel is polar to the linear forms in the decomposition: Notice that (αx+βy) (βx αy)d = 0 (α x + β y is apolar to βx αy). In the case of binary forms, a general elt. F of the kernel factors (FTA). i.e. F = l1 l r kills all linear forms in decomposition. There exist c i C such that f = r i=1 c ili d if and only if l1 l r f = 0. One inclusion is obvious, the other is by dimension count. Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

13 Eigenvectors of tensors An essential ingredient is the notion of an eigenvector of a tensor. The eigenvector equation for matrices: M C n n, v C n, Mv = λv, λ C M(v) v = 0 Definition Let M Hom(S m V, a V ). v V is an eigenvector of the tensor M if M(v m ) v = 0. When a = m = 1 this is the classical definition. When a = 1, [Lim 05] and [Qi 05] independently introduced this notion. Further generalizations: Ottaviani-Sturmfels, Sam (Kalman varieties), and Qi et.al. (Spectral theory of tensors). Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

14 The number of eigenvectors of different types of tensors Theorem (O.-Ottaviani 13) For a general M Hom(S m C n+1, a C n+1 ) the number e(m) of eigenvectors is e(m) = m, when n = 1 and a {0, 2}, e(m) =, when n > 1 and a {0, n + 1}, (classical) e(m) = mn+1 1 m 1, when a = 1 [CS 10], e(m) = 0, for 2 a n 2, e(m) = (m+1)n+1 +( 1) n m+2, for a = n 1. Our result includes a result of Cartwright-Sturmfels. Our proofs rely on the simple observation that the a Chern class computation for the appropriate vector bundle gives the number of eigenvectors. Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

15 The Koszul complex and Koszul matrices The Koszul complex arises via the minimal free resolution of the maximal ideal x 0,..., x n. Let V be the span of the x i. 0 n+1 V k n+1 n V 2 V k 2 1 V k 1 C 0 Some examples: for n = 2, k 1 = ( w x y ) x y 0 y, k 2 = w 0 y k 3 = x, 0 w x w x y 0 z 0 0 for n = 3, k 1 = ( w x y z ), k 2 = w 0 y 0 z 0 0 w x 0 0 z, w x y k 3 Note: these complexes are exact. Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

16 Sections of vector bundles to eigenvectors of tensors Construct a map (tensor a Koszul map with a catalecticant map) A f : Hom(S m V, a V ) Hom( n a V, S d m 1 V ) M Hom(S m V, a V ), v is an eigenvector of M iff M(v m ) v = 0. Lemma M Hom(S m V, a V ), 1 v is an eigenvector of M iff M ker A f. 2 Let f = r i=1 v d i. If each v i is an eigenvector of M, then M ker A f. Lemma Let Q be the quotient bundle on P n. 1 The fiber of a Q at x = [v] is isomorphic to Hom([v m ], a V / v a 1 V. 2 the section s M vanishes if and only if v is an eigenvector of M. Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

17 Koszul Algorithm examples: HPR Quinitics Let V = C 3 a general form f S 5 C 3 has rank 7. Catalecticants: C f : S 3 V S 2 V is a 6 10 matrix - with max rank 6, so too small to detect rank 7. Koszul Flattening: S 5 V S 2 V V S 2 V S 2 V 2 V V S 2 V. Get a map: A f : S 2 V 2 V V S 2 V Hom(S 2 V, V ) Hom(V, S 2 V ) x y 0 C fx C fy 0 A f = w 0 y C f = C fw 0 C fy, 0 w x 0 C fw C fx where C fz is the 6 6 catalecticant of f z. Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

18 Koszul Algorithm examples: HPR Quinitics Koszul Flattening: S 5 V S 2 V V S 2 V S 2 V 2 V V S 2 V. Get a map: A f : S 2 V 2 V V S 2 V Hom(S 2 V, V ) Hom(V, S 2 V ) x y 0 C fx C fy 0 A f = w 0 y C f = C fw 0 C fy, 0 w x 0 C fw C fx A f is skew-symmetrizable, so even has rank. If f has rank 7, A f has rank 14. The Pfaffians vanish on the locus of border rank 7 forms. The general M in Hom(S 2 V, V ) has 7 eigenvectors, [Cartwright-Sturmfels]. By our theorem, the 7 eigenvector of a general M ker A f are the linear forms in the decomposition of f (up to scalars). Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

19 Computing eigenvectors of tensors In the HPR example, had ( Cfx C fy 0 ) A f : S 2 V V C fw 0 C fy 0 C fw C fx V S 2 V, with A f, an matrix composed of 6 6 blocks. An element of the kernel can be blocked as (h 1, h 2, h 3 ), where h i are quadrics in S 2 V by viewing S 2 V V as ( S 2 V x ) ( S 2 V y ) ( S 2 V z ). The 2-minors of ( ) h1 h 2 h 3 define the locus of eigenvectors. x y z In the general case the construction is similar: concatenate the (blocked) elements of the kernel with a Koszul matrix and compute the zero set of the minors. Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

20 Koszul Algorithm examples: Sylvester Pentahedral Let V = C 4. The general f S 3 V has rank 5. The most-square catalecticant is 10 4, so not big enough to detect rank 5. Koszul flattening: f S 3 V V V V V 2 V V V A f : V 2 V V V Hom(C 4, 2 C 4 ) Hom(C 4, C 4 ), x y 0 z 0 0 A f = k 2 C f, where k 2 = w 0 y 0 z 0 0 w x 0 0 z w x y General element of Hom(C 4, 2 C 4 ) has 5 eigenvectors! The eigenvectors of a general element of the kernel provide the linear forms in the Waring decomposition. Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

21 Koszul Flattening Algorithm Algorithm Input f S d V, V = C n+1. 1 Construct A f : Hom(S m V, V ) Hom( n 1 V, S d m 1 V ). 2 Compute ker A f. Note Rank(f ) rank(a f )/n. 3 Set Z = common eigenvectors of a basis of ker A f. a) if #Z =, fail. b) else Z = {[v 1 ],..., [v s ]}. 4 Solve f = s i=1 c iv d i. Output: unique Waring decomposition of f. Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

22 Success of the Koszul Flattening Algorithm Here are some effective bounds for the success of our algorithm. Theorem (O.-Ottaviani 13) Let n = 2, d = 2m + 1, f = r i=1 v i d, and set z i = [v i ], Z = {z 1,..., z r }. The Koszul Flattening algorithm succeeds when 1 2r m 2 + 3m + 4, 2 2r m 2 + 4m + 2 (after finitely many tries). and if n 3, The Koszul Flattening algorithm succeeds when 1 n-even, r ( ) n+m n (eigenvectors of ker Af = Z = Z), 2 n-odd, r ( ) n+m n (e.-vects of ker Af e.vects of (Im(A f )) = Z), 3 n = 3, r 1 3 ( 1 2 (m + 4)(m + 3)(m + 1) m2 /2 m 8)... set a = 2 in the algorithm. Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

23 General Vector Bundle Method Consider a line bundle L giving the embedding X L P ( H 0 (X, L) ) = PW. Let E X be a vector bundle on X. We get natural maps: H 0 (E) H 0 (E L) H 0 (L), H 0 (E) H 0 (L) H 0 (E L), H 0 (E) A f H 0 (E L), where A f depends linearly on H 0 (L). Get the matrix presentation via Koszul matrices when E = a Q, where Q is (at twist of) the quotient bundle on P n. Proposition (Landsberg-Ottaviani 12) Let f = r i=1 v i, and set z i = [v i ] X PW, Z = {z 1,..., z r }. Then H 0 (I Z E) ker A f, with equality if H 0 (E L) H 0 (E L Z ), and H 0 (I Z E L) (ImA f ), with equality if H 0 (E) H 0 (E Z ). Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24

24 General Vector Bundle Method Consider a line bundle L giving the embedding X L P ( H 0 (X, L) ) = PW. Let E X be a vector bundle on X. Theorem (O.-Ottaviani 13) Let f = r i=1 v i d, and set z i = [v i ] X PW, Z = {z 1,..., z r }. Assume rank(a f ) = k Rank(E) and H 0 (I Z E) H 0 (I Z E L) H 0 (I 2 Z L) is surjective. If X is not weakly k-defective, then the common base locus of ker(a f ) and Im(A f ) is given by Z (so one can reconstruct Z from f ). We use this general result to prove the specific results for each of our algorithms. Oeding, Ottaviani (UCB, Firenze) Waring Decomposition January 16, / 24