Cubature formulae, flat extensions and convex relaxation. B. - PowerPoint PPT Presentation

Cubature formulae, flat extensions and convex relaxation. B. Mourrain INRIA M´ editerran´ ee, Sophia Antipolis Bernard.Mourrain@inria.fr (collaboration with M. Abril Bucero & C. Bajaj)

The problem For any continuous function f , compute (an approximation of) � I [ f ] = w ( x ) f ( x ) d x Ω where Ω ⊂ R n and w is a positive function on Ω. Cubature formula: compute ξ j ∈ R n and weights w j ∈ R such that r � σ : f �→ � σ | f � = w j f ( ξ j ) j =1 satisfies: � σ | f � = I [ f ] , ∀ f ∈ V , where V is a finite dimensional vector space of polynomials. B. Mourrain Cubature formulae, flat extensions and convex relaxation. 2 / 24

Interest: ◮ Fast/accurate evaluation of integrals. ◮ Important ingredient in numerical methods. ◮ Applications in other domains : graph, algorithms (Lanczos), . . . . A long history: C.F. Gauss (1815), . . . J. Radon (1948), ◮ W. Gr¨ obner (1948), . . . ◮ A. H. Stroud (1971), I.P. Mysovskikh (1981), R. Cools (1993 ... 2003), . . . Many case studies on simplex, hyperspheres, hypercubes, for degree 1, 2,3,4,5, . . . B. Mourrain Cubature formulae, flat extensions and convex relaxation. 3 / 24

Solving the cubature of the disk (cf. [Cools’00]) B. Mourrain Cubature formulae, flat extensions and convex relaxation. 4 / 24

Example (1D) V = R [ x ] ≤ 2 r − 1 polynomials of degree ≤ 2 r − 1, spanned by 1 , x , . . . , x 2 r − 1 . Problem: Given σ 0 = I [1] , σ 1 = I [ x ] , . . . , σ 2 r − 1 = I [ x 2 r − 1 ], find ξ i ∈ R , ω i ∈ R s.t. r � ω i ξ k σ k = i . (1) i =1 Solution: If (1) is satisfied, then p ( x ) = p 0 + p 1 x + · · · + p r x r = � r i =1 ( x − ξ i ) is such that H σ � �� �       � r σ 0 σ 1 . . . σ r p 0 i =1 ω i p ( ξ i ) � r σ 1 σ r +1 p 1 i =1 ω i p ( ξ i ) ξ i              =  = 0 . . . .  . .   .   .  . . . .     � r i =1 ω i p ( ξ i ) ξ r − 1 σ r − 1 . . . σ 2 r − 1 σ 2 r − 1 p r i ☞ Compute an element p ( x ) in the kernel of H σ , its roots ξ 1 , . . . , ξ d and deduce the coefficients ω 1 , . . . , ω i s.t. σ k = � d i =1 ω i ξ k i . B. Mourrain Cubature formulae, flat extensions and convex relaxation. 5 / 24

In practice, for � f , g � = � � j ≤ k σ k f j g k − j , k ◮ Compute the orthogonal polynomials p i ( x ) such that � x j , p i � = 0 for j < i and � x i − p i , p i � = 0, which satisfies the recurrence relation p i +1 ( x ) = ( x − α i ) p i ( x ) + γ i p i − 1 ( x ) where α i = � x p i , p i � � p i , p i � , γ i = � x p i , p i − 1 � � p i , p i � � p i − 1 , p i − 1 � = � p i − 1 , p i − 1 � . ◮ Take the last polynomial p ( x ) = p r ( x ) for the quadrature rule. B. Mourrain Cubature formulae, flat extensions and convex relaxation. 6 / 24

What we are going to do ☞ Replace the cubature problem by a low-rank structured matrix-completion problem in a convex set . ☞ Relax the low-rank condition by a L 1 proxy and solve (a hierarchy of) convex optimization problems to obtain the minimal L 1 solutions. ☞ Deduce the cubature formula from the completed matrix . B. Mourrain Cubature formulae, flat extensions and convex relaxation. 7 / 24

From cubature formulae to structured matrix completion From cubature formulae to structured matrix completion 1 Reduction to a convex optimization problem 2 From moment matrices to cubature formulae 3 Why it is working 4 Illustration 5 B. Mourrain Cubature formulae, flat extensions and convex relaxation. 8 / 24

From cubature formulae to structured matrix completion ◮ Sequences in K N n : σ = ( σ α ) α ∈ N n ◮ Formal power series in K [[ z ]] = K [[ z 1 , . . . , z n ]]: z α � σ ( z ) = σ α α ! α ∈ N n ◮ Linear forms in the dual R ∗ where R = K [ x ] = K [ x 1 , . . . , x n ]: � � p α x α �→ � σ | p � = σ : p = σ α p α α α ◮ Isomorphism: K [ x ] ∗ ∼ K [[ z 1 , . . . , z n ]] . ◮ Structure of K [ x ] -module: ∀ a ∈ K [ x ] , ∀ σ ∈ K [ x ] ∗ , a ⋆ σ : b �→ � σ | a b � Example: x 1 ⋆ z α 1 1 z α 2 n = α 1 z α 1 − 1 z α 2 n = ∂ z 1 ( z α 1 1 z α 2 2 · · · z α n 2 · · · z α n 2 · · · z α n n ) . 1 B. Mourrain Cubature formulae, flat extensions and convex relaxation. 9 / 24

From cubature formulae to structured matrix completion Dictionary ◮ p �→ ∂ α 1 1 · · · ∂ α n n ( p )(0) represented by z α . ◮ p �→ p ( ξ ) represented by e ξ ( z ) = � α ∈ N n ξ α z α α ! = e z · ξ . � Ω p d µ represented by σ ( z ) = � � Ω e x · z dx . ◮ p �→ α ∈ N n i =1 ω i ξ i α represented by σ ( z ) = � r ◮ σ s.t. σ α = � r i =1 ω i e ξ i ( z ) where e ξ i ( z ) = e z · ξ i = e z 1 ξ 1 , i + ··· + z n ξ n , i . The cubature problem for V = R ≤ d over R : find ◮ frequencies ξ 1 , . . . , ξ r ∈ R n , ◮ weights ω 1 , . . . , ω r ∈ R , such that r � � e x · z dx ≡ ω i e ξ i ( z ) + O (( z ) d + 1 ) Ω i = 1 (Borel-Laplace transform). B. Mourrain Cubature formulae, flat extensions and convex relaxation. 10 / 24

From cubature formulae to structured matrix completion Vanishing ideal, Hankel operators and moments For σ ∈ K [ x ] ∗ = K [[ z ]], ◮ Hankel operator: K [ x ] ∗ H σ : K [ x ] → p �→ p ⋆ σ where p ⋆ σ : q �→ � σ | p q � . ◮ Vanishing ideal: 0 → I σ → K [ x ] → A ∗ σ → 0 with I σ := ker H σ , A σ := K [ x ] / I σ . ◮ Moments of σ ∈ � x A � ∗ : � σ | x α � ∈ K for α ∈ A ⊂ N n . ◮ Truncated moment matrix: If E 1 = � x A � , E 2 = � x B � , the matrix of H E 1 , E 2 E ∗ : E 1 → σ p ⋆ σ is the moment matrix of [ � σ | x α + β � ] α ∈ A ,β ∈ B . 2 p �→ B. Mourrain Cubature formulae, flat extensions and convex relaxation. 11 / 24

Reduction to a convex optimization problem From cubature formulae to structured matrix completion 1 Reduction to a convex optimization problem 2 From moment matrices to cubature formulae 3 Why it is working 4 Illustration 5 B. Mourrain Cubature formulae, flat extensions and convex relaxation. 12 / 24

Reduction to a convex optimization problem Semi-Definite Programming Relaxation If σ = � r i =1 w j e ξ j with ξ j ∈ R n , w j > 0, then H B , B � 0 and of rank ≤ r . σ For given moments i = ( i ( v )) v ∈ V , consider the convex set: H k ( i ) { H σ | σ ∈ R ∗ = 2 k , ∀ v ∈ V � σ | v � = i ( v ) , H σ � 0 } Cubature formulae with a minimal number of points as the solution of H ∈H k ( i ) rank ( H ) . min ☞ Relaxation of this NP-hard problem: H ∈H k ( i ) trace ( P t HP ) min (2) for a well-chosen matrix P . ☞ Optimization of a linear functional on a convex set (the cone of SDP matrices intersected with a linear space) by SDP solvers. B. Mourrain Cubature formulae, flat extensions and convex relaxation. 13 / 24

Reduction to a convex optimization problem Objective function: trace ( P t HP )= nuclear norm of P t HP . = trace ( HPP t ) = � H , Q � where Q = PP t . Convex optimization problem: argmin � H , Q � s.t. – H = ( h α,β ) α,β ∈ B � 0, – H satisfies the Hankel constraints h α,β = h α ′ ,β ′ =: h α + β if α + β = α ′ + β ′ . – h α = I [ x α ] for α ∈ A . Efficient solvers by interior point methods, with polynomial complexity (for a given precision ǫ ). Efficient tools: CSDP, SDPA, . . . . B. Mourrain Cubature formulae, flat extensions and convex relaxation. 14 / 24

From moment matrices to cubature formulae From cubature formulae to structured matrix completion 1 Reduction to a convex optimization problem 2 From moment matrices to cubature formulae 3 Why it is working 4 Illustration 5 B. Mourrain Cubature formulae, flat extensions and convex relaxation. 15 / 24

From moment matrices to cubature formulae Flat extension Let B ⊂ C , B ′ ⊂ C ′ , ∂ B = C \ B , ∂ B ′ = C ′ \ B ′ . Truncated moment matrix: � � H C , C ′ = � σ | x α + β � α ∈ C ,β ∈ C ′ Flat extension: � H B , B ′ � H B ,∂ B ′ H C , C ′ = , H ∂ B , B ′ H ∂ B ,∂ B ′ ◮ rank H C , C = rank H B , B ◮ there exist P ∈ C B × ∂ B ′ , P ′ ∈ C B ′ × ∂ B s.t. M = H t P , M ′ = H P ′ , N = P t H P ′ . (3) with H = H B , B ′ , M ′ = H B ,∂ B ′ , M = H ∂ B , B ′ , N = H ∂ B ,∂ B ′ B. Mourrain Cubature formulae, flat extensions and convex relaxation. 16 / 24

From moment matrices to cubature formulae When there is a flat extension for C = C ′ = B + ( B + = B ∪ x 1 B ∪ . . . ∪ x n B ; B connected to 1) ◮ The tables of multiplication in A σ = R [ x ] / I σ are M j := H B , x j B ( H B , B ) − 1 . ◮ Their common eigenvectors v i are, up to a scalar, the Lagrange interpolation polynomials u ξ i . ◮ The points of the cubature are ξ i = ( ξ i , 1 , . . . , ξ i , n ), where ξ i , j is an eigenvalue of M j . ◮ The decomposition is σ = � r 1 v i ( ξ i ) � σ | v i � e ξ i . i =1 B. Mourrain Cubature formulae, flat extensions and convex relaxation. 17 / 24

Why it is working From cubature formulae to structured matrix completion 1 Reduction to a convex optimization problem 2 From moment matrices to cubature formulae 3 Why it is working 4 Illustration 5 B. Mourrain Cubature formulae, flat extensions and convex relaxation. 18 / 24