Tensor decomposition and moment matrices

Transcription

1 Tensor decomposition and moment matrices J. Brachat (Ph.D.) & B. Mourrain GALAAD, INRIA Méditerranée, Sophia Antipolis SIAM AAG, Raleigh, NC, Oct. -9, 211 Collaboration with A. Bernardi, P. Comon and E. Tsigaridas.

2 Blind identification Observing x t with find H and s t?? x t = H s t If the sources are statistically independent, using high order statistics E(x i x j x k ) of the signal x, identifying H reduces to decompose the symmetric tensor as a sum of s tensors of rank 1. T i,j,k,... = E(x i x j x k ) B. Mourrain Tensor decomposition and moment matrices 2 / 29

3 cf. [T. Schultz, H.P. Seidel 8], [A. Ghosh, R. Deriche,... 9] B. Mourrain Tensor decomposition and moment matrices / 29 Recovering branching structures From some measurements, identify the main directions at the crossing points. Approximate the set of measurements on the sphere by a (symmetric) tensor T = (T i,j,k,l ). Decompose/approximate it as a minimal sum of tensors of rank 1: T = r i=1 λ i v i v i v i v i. From the decomposition of tensors, deduce the geometric structure:

4 Generalized Waring problem (177) Given a homogeneous polynomial T of degree d in the variables x = (x, x 1,..., x n ): T (x) = T α x α, α =d find a minimal decomposition of T of the form T (x) = r γ i (ζ i, x + ζ i,1 x ζ i,n x n ) d i=1 for ζ i = (ζ i,, ζ i,1,..., ζ i,n ) C n+1, γ i C. B. Mourrain Tensor decomposition and moment matrices 4 / 29

5 Sylvester approach (188) For all f = α =d f ( d α α) x α, g = α =d g ( d α α) x α S d, f, g = ( ) d f α g α. α α =d For homogeneous polynomials g(x,..., x n ) of degree d and k(x) = k x + + k n x n, g(x), k(x) d = g(k,..., k n ) = g(k). If T = k 1 (x) d + + k r (x) d and g(k i ) = for i = 1,..., r, then h with deg(h) = d deg(g) g h, T =. Find the polynomials apolar to T and their roots. B. Mourrain Tensor decomposition and moment matrices 5 / 29

6 Sylvester method for binary forms Theorem The binary form T (x, x 1 ) = d i= c d ) i( i x d i sum of r distinct powers of linear forms T = x i 1 r λ k (α k x + β k x 1 ) d k=1 iff there exists a polynomial q such that c c 1... c r c 1 c r+1.. c d r... c d 1 c d can be decomposed as a q q 1. q r = and of the form r q(x, x 1 ) := µ (β k x α k x 1 ). k=1 B. Mourrain Tensor decomposition and moment matrices / 29

7 Geometric point of view Definition (Veronese variety) ν d : P(E) P(S d (E)) v v(x) d Its image is the Veronese Variety, denoted Ξ(S d (E)). Definition (Secant of the Veronese variety) σ,d r = {[T ] P(S d ) v 1,..., v r K n+1 s.t. T = r i=1 v i(x) d } σ d r = σ,d r r(t ) = smallest r s.t. T σ,d r, called the rank of T. r σ (T ) = smallest r s.t. T σ d r, called the border rank of T. Example: T ɛ := 1 ( d ε (x + εx 1 ) d x d ), T := x d 1 x 1. r(t ɛ ) = r σ (T ) = 2 but r(t ) = d. B. Mourrain Tensor decomposition and moment matrices 7 / 29

8 Algebraic point of view S := K[x,..., x n ]; S d := {f S; deg(f ) = d}; Definition (Apolar ideal) (T ) = {g S h S d deg(g), g h, T = } S d+1. Problem: find an ideal I S such that I (T ); I is saturated zero dimensional; I defines a minimal number r of simple points. necessary and sufficient conditions. B. Mourrain Tensor decomposition and moment matrices 8 / 29

9 Schematic rank Definition (Punctual Hilbert Scheme) Hilb r (P n ) is the set of saturated ideals of r points (counted with mult.). Property: Hilb r (P n ) is a subvariety of the Grassmannian Gr r (S r ) defined by quadratic equations in the Plücker coordinates [ABM 1]. Definition (Cactus variety) K d r = {[T ] P(S d ) I Hilb r (P n ) s.t. I (T )} (def. with closure in [Buczynska-Buczynski 11]). r sch (T ) = smallest r s.t. T K d r, called the schematic rank of T. (see [Iarrobino-Kanev 99]) B. Mourrain Tensor decomposition and moment matrices 9 / 29

10 Duality S = K[x,..., x n ] = K[x], R = K[x 1,..., x n ] = K[x]. Dual space: E = Hom K (E, K). R K[[d]] = K[[d 1,..., d n ]] = K[[ 1,..., n ]]: Λ = α N n Λ(x α ) d α = α N n Λ(x α ) 1 α! 1 α. where (d α ) α N n is the dual basis in R. Example: for ζ = (ζ 1,..., ζ k ) K n 1 K n k, with 1 ζ = α N n 1 N n k (S d ) K[[d]] d = K[d,..., d n ] d : 1 ζ : R K p p(ζ) ζ α d α = k ζ α i i α N n 1 N n k i=1 d α i i. B. Mourrain Tensor decomposition and moment matrices 1 / 29

11 Duality cont d For T = T (x) = α N n+1 α =d T (d) = α N n+1 ; α =d T (d) = α N n+1 ; α =d ( d α T α x α S d, let ) 1Tα d α (S d ) such that T S d, T T = T (T ). ( d ) 1Tα α d α (R d ) such that T R d, T T = T (T ). B. Mourrain Tensor decomposition and moment matrices 11 / 29

12 Hankel operators H Λ : R R p p Λ where p Λ : q Λ(p q). Properties: I Λ := ker H Λ is an ideal of R. rankh Λ = r iff A Λ = R/I Λ is an algebra of dimension r over K. If rankh Λ = r, A Λ is a zero-dimensional Gorenstein algebra: 1 A Λ = A Λ Λ (free module of rank 1). 2 (a, b) Λ(ab) is non-degenerate in A Λ. Hom AΛ (A Λ, A Λ) = D A Λ where D = r i=1 b i ω i for (b i ) 1 i r a basis of A Λ and (ω i ) 1 i r its dual basis for Λ. B. Mourrain Tensor decomposition and moment matrices 12 / 29

13 If rankh Λ = r, then Λ : p 1 ζi θ i ( 1,..., n )(p) r i=1 for some ζ i C n and some differential polynomials θ i with r = r α d i=1 dim( d (θ α i ), α N N ), VC(I Λ ) = {ζ 1,..., ζ r }. If rank H Λ = r, (b i ) 1 i r a basis of A Λ and (ω i ) 1 i r its dual basis for Λ then IΛ = ker H Λ where = r i=1 b iω i. B. Mourrain Tensor decomposition and moment matrices 1 / 29

14 Generalized additive decomposition Definition A tensor T T has a generalized additive decomposition of size r iff there exists points ζ i K N and differential polynomials θ i s.t. after a change of coordinates, T 1 ζi θ i ( ) on R d r i=1 and r d α i=1 dim( d (θ α i ), α N n ) r. Definition G d r = {[T ] P(S d ) with a generalized additive decomposition of size r}. r G (T ) = smallest r such that T G d r, called the generalized additive rank of T. B. Mourrain Tensor decomposition and moment matrices 14 / 29

15 Truncated moment matrices Given (λ α ) α A, let Λ x A := Hom K ( x A, K) be Λ : p = p α x α p α λ α α A α A For B, B A with B B A, we define H B,B Λ : x B x B p = β B p β x β p Λ with p Λ : q B Λ(p q) K. The matrix of H B,B Λ := [λ β +β] β B,β B. Given Λ x A, find Λ R such that Λ extends Λ and H Λ of minimal rank. B. Mourrain Tensor decomposition and moment matrices 15 / 29

16 r H (T ) = smallest r such that T H d r, called the extension rank of T. B. Mourrain Tensor decomposition and moment matrices 1 / 29 Flat extension Definition T S d has a flat extension of rank r iff there exists T S m+m u S 1 {} with m = max{r, d 2 }, m = max{r 1, d 2 } s.t. and and is of rank r. u m+m d T = T H m,m T : S m (S m ) p p T After a change of variables (u = x ), we have T α = T α for α d (truncated moments). Definition (Flat extension variety) H d r = {[T ] P(S d ) s.t. T has a flat extension of rank r}.

17 Catalecticant operators H i,d i T : S i (S d i ) := Hom(S d i, K) p p T where T (S d ) s.t. g (S d ), T (g) = T, g ; p S i, p T : S d i K Remark: ker H i,d i T = (T ) i. Definition (Catalecticant variety) Γ i,d i r := {[T ] P(S d ) rank H i,d i T q p T (q) = T (p q) = T, p q r}. r Cat (T ) := maximal rank of H i,d i T for i =,..., d, called the catalecticant rank of T. B. Mourrain Tensor decomposition and moment matrices 17 / 29

18 The rank hierarchy Theorem (Brachat-M.-B. 11) 1 Kr d = Hr d = Gr d. 2 It is a closed variety of P(S d ). Theorem (Buczynska-Buczynski 11, Brachat 11) For d 2r and for any r i d r, we have K,d r = K d r = Γ i,d i r. (See also [Buczynski-Ginensky-Landsberg 1] and talk). For i d, Consequently, σ,d r σ d r K d r = H d r = G d r Γ i,d i r P(S d ) r(f ) r d σ (f ) r d sch (f ) = r d G (f ) = r d H(f ) r d Cat (f ) B. Mourrain Tensor decomposition and moment matrices 18 / 29

19 Computing a flat extension Theorem (LM 9, BBCM 1) If B, B are connected to 1 (m B implies m = 1 or m = x i m with m B) and Λ known on B + B + and H B,B is invertible then Λ extends uniquely Λ to R iff M B,B i M B,B j = M B,B j M B,B i (1 i, j n), where B + = i x i B B, M B,B i Equivalently Theorem (LM 9, BBCM 1) := H B,x i B Λ (H B,B ) Λ 1. Let B, B be connected to 1 of size r and Λ known on B + B +. If rankh B,B Λ = rankh B +,B + Λ = r, then there exists a unique Λ R = Hom K (R,K) which extends Λ. B. Mourrain Tensor decomposition and moment matrices 19 / 29

20 Recovering the decomposition If the truncated moment problem has a solution Λ, then: r = rankh Λ = dim R/(ker H Λ ) where r = B = B ; Let B, B be maximal sets of monomials R s.t. H B,B Λ invertible, then M B,B i := H B,x i B Λ (H B,B Λ ) 1. is the matrix of multiplication by x i in A Λ. If the decomposition is of size r, the eigenvectors of the operators (M t x i,j ) i,j are simple and equal (up to scalar) to {1 ζ1,..., 1 ζr }. ker H Λ = (ker H B +,B + Λ ) where B + = B x 1,1 B x nk,kb; For each x α B = B + \ B, there exists a unique f α = x α Σ β z α,β x β ker H B +,B + Λ. The (f α ) α B form a border basis of ker H Λ with respect to B. B. Mourrain Tensor decomposition and moment matrices 2 / 29

21 The results extend to general multihomogeneous tensors: x i = (x,i,..., x ni,i), x i = (x 1,i,..., x ni,i), i = 1,..., k; E i = x,i,..., x ni,i, i = 1,..., k; T := S δ1 (E 1 ) S δk (E k ); R = K[x 1,..., x k ]; T R δ := R δ1,...,δ k = {f R; deg xi (f ) δ i, i = 1,..., k}; f = f α x α1 1 xα k k. α=(α 1,...,α k ); α i δ i Apolar inner product: f g = α i δ i f α g α ( δ1 α 1 ) ( δk α k ). B. Mourrain Tensor decomposition and moment matrices 21 / 29

22 Algorithm For r = 1,..., 1 Choose B, B of size r, connected to 1; 2 Find an extension H B +,B + Λ s.t. the operators M i = H x i B,B commute. If this is not possible, start again with r := r + 1. Λ (H B,B Λ ) 1 4 Compute the n r eigenvalues s.t. M t i v j = ζ i,j v j, i = 1,..., n, j = 1,..., r. 5 Solve the linear system in (γ j ) j=1,...,k : Λ = r j=1 γ j1 ζj where ζ j C n are the vectors of eigenvalues found in (4). B. Mourrain Tensor decomposition and moment matrices 22 / 29

23 Example in Ξ (S 4 (K )) The tensor: T = 79 x x x 2 x x 2 1 x x x 1 x x x 1. The Hankel matrix: 1 x 1 x 2 x1 2 x 1 x 2 x2 2 x1 x1 2 x 2 x 1 x2 2 x2 x1 4 x1 x 2 x1 2 x2 2 x 1 x2 x x h 5 h 41 h 2 h 2 h 14 x h 41 h 2 h 2 h 14 h 5 x h 5 h 41 h 2 h 2 h h 51 h 42 h h 24 x 1 x h 41 h 2 h 2 h 14 h 51 h 42 h h 24 h 15 x h 2 h 2 h 14 h 5 h 42 h h 24 h 15 h x h 5 h 41 h 2 h h 51 h 42 h h 7 h 1 h 52 h 4 h 4 x1 2 x 2 49 h 41 h 2 h 2 h 51 h 42 h h 24 h 1 h 52 h 4 h 4 h 25 x 1 x h 2 h 2 h 14 h 42 h h 24 h 15 h 52 h 4 h 4 h 25 h 1 x2 h 2 h 14 h 5 h h 24 h 15 h h 4 h 4 h 25 h 1 h 7 1 x 4 h 5 h 41 h h 51 h 42 h 7 h 1 h 52 h 4 h 8 h 71 h 2 h 5 h 44 x1 x 2 h 41 h 2 h 51 h 42 h h 1 h 52 h 4 h 4 h 71 h 2 h 5 h 44 h 5 x1 2 x h 2 h 2 h 42 h h 24 h 52 h 4 h 4 h 25 h 2 h 5 h 44 h 5 h 2 x 1 x2 h 2 h 14 h h 24 h 15 h 4 h 4 h 25 h 1 h 5 h 44 h 5 h 2 h 17 x2 4 h 14 h 5 h 24 h 15 h h 4 h 25 h 1 h 7 h 44 h 5 h 2 h 17 h 8 B. Mourrain Tensor decomposition and moment matrices 2 / 29

24 Extract a ( ) principal minor of full rank: H B Λ = The columns (and the rows) of the matrix correspond to the monomials {1, x 1, x 2, x 2 1, x 1x 2, x 2 2 }. B. Mourrain Tensor decomposition and moment matrices 24 / 29

25 The shifted matrix H B x 1 Λ is H x1 Λ = h 5 h 41 h 2 49 h 41 h 2 h h 2 h 2 h 14 The columns of the matrix correspond to the monomials {x 1, x1 2, x 1x 2, x1, x 1 2x 2, x 1 x2 2} = {1, x 1, x 2, x1 2, x 1x 2, x2 2} x 1. Similarly, H B x = Λ 49 h 41 h 2 h h 2 h 2 h 14 h 2 h 14 h 5 B. Mourrain Tensor decomposition and moment matrices 25 / 29

26 We form (all) the possible matrix equations: M xi M xj M xj M xi = H B x 1 Λ (HB Λ ) 1 H B x 2 Λ (HB Λ ) 1 H B x 2 Λ (HB Λ ) 1 H B x 1 Λ (HB Λ ) 1 =. Many of the resuling equations are trivial. We have unknonws: h 5, h 41, h 2, h 2, h 14, h 5 and 15 non-trivial equations. A solution of the system is: h 5 = 1, h 41 = 2, h 2 =, h 2 = 1.5, h 14 = 4.9, h 5 =.5. B. Mourrain Tensor decomposition and moment matrices 2 / 29

27 We subsitute these values to H x1 Λ and solve the generalized eigenvalue problem (H x1 Λ ζ H Λ ) v =.The normalized eigenvectors are i.2.51 i i i i ,, i. +.7 i i i i i i i i.1.2 i,, i..7 i i i i As the coordinates of the eigenvectors correspond to the evaluations of {1, x 1, x 2, x 2 1, x 1x 2, x 2 2 }, we can recover the values of (x 1, x 2 ). B. Mourrain Tensor decomposition and moment matrices 27 / 29,.

28 After solving the over-constrained linear system obtained by expansion and looking coefficient-wise, we deduce the decomposition: ( i) (x ( i)x 1 ( i)x 2 ) 4 +( i) (x ( i)x 1 (.2.5 i)x 2 ) (x + (1.142)x 1 +.8x 2 ) (x + (.95)x 1.71x 2 ) 4 ( i) (x (.88.1 i)x 1 + (. +.7 i)x 2 ) 4 ( i) (x ( i)x 1 + (..7 i)x 2 ) 4 B. Mourrain Tensor decomposition and moment matrices 28 / 29

29 Concluding remarks/questions What about nearest r-decomposition(s) and SVD-like properties? What are the equations of G d r = H d r = K d r and σ d r? Extension of geometric properties to any type of tensors? Thanks for your attention B. Mourrain Tensor decomposition and moment matrices 29 / 29