## A Sparse Flat Extension Theorem for Moment Matrices

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@MISC{Laurent_asparse,

author = {Monique Laurent and Bernard Mourrain},

title = {A Sparse Flat Extension Theorem for Moment Matrices},

year = {}

}

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### Abstract

Abstract. In this note we prove a generalization of the flat extension theorem of Curto and Fialkow [4] for truncated moment matrices. It applies to moment matrices indexed by an arbitrary set of monomials and its border, assuming that this set is connected to 1. When formulated in a basis-free setting, this gives an equivalent result for truncated Hankel operators.

### Citations

851 | Semidefinite programming
- Vandenberghe, Boyd
- 1996
(Show Context)
Citation Context ...nting measure suported by the set K (cf. [6, 11]). Using moment matrices, the program (3.1) can be formulated as an instance of semidefinite programming for which efficient algorithms exist (see e.g. =-=[20, 21]-=-). We have: p∗ t ≤ p∗ , with equality if H Mn,t Λ is a flat extension of H Mn,t−d Λ for an optimum solution Λ to (3.1) (d := maxj dj). In that case, the atoms of the representing measure (which exists... |

359 | Global optimization with polynomials and the problem of moments
- Lasserre
- 2001
(Show Context)
Citation Context ...oblem has recently attracted a lot of attention also within the optimization community, since it can be used to formulate semidefinite programming relaxations to polynomial optimization problems (see =-=[11]-=-). Moreover the flat extension theorem of Curto and Fialkow permits to detect optimality of the relaxations and to extract global optimizers to the original optimization problem (see [8]). Here is a b... |

204 |
The Classical Moment Problem
- Akhiezer
- 1965
(Show Context)
Citation Context ... K[x] on K[x] ∗ is denoted by where (p · Λ)(q) := Λ(pq) for q ∈ K[x]. 1.1. The moment problem (p, Λ) ∈ K[x] × K[x] ∗ ↦→ p · Λ ∈ K[x] ∗ In this section, we consider K = R. The moment problem (see e.g. =-=[1, 7]-=-) deals with the characterization of the sequences of moments of measures. Given a probability measure µ on R n , its moment of order a = x α ∈ Mn is the quantity ∫ x α µ(dx). The moment problem conce... |

65 | Sums of squares, moment matrices and optimization over polynomials
- Laurent
- 2009
(Show Context)
Citation Context ...n the sequence y is bounded [2] or, more generally, exponentially bounded [3]. The next result of Curto and Fialkow [4] shows that this is also the case when the matrix M(y) has finite rank (cf. also =-=[14, 15]-=- for a short proof). Theorem 1.1. [4] If M(y) is positive semidefinite and the rank of M(y) is finite, then y has a (unique) representing measure (which is finitely atomic with rank M(y) atoms). In th... |

58 | Solution of the truncated complex moment problem for flat data
- Curto, Fialkow
- 1996
(Show Context)
Citation Context ...nting measure under some additional assumptions. This is the case, for instance, when the sequence y is bounded [2] or, more generally, exponentially bounded [3]. The next result of Curto and Fialkow =-=[4]-=- shows that this is also the case when the matrix M(y) has finite rank (cf. also [14, 15] for a short proof). Theorem 1.1. [4] If M(y) is positive semidefinite and the rank of M(y) is finite, then y h... |

54 | The truncated complex K-moment problem
- Curto, Fialkow
(Show Context)
Citation Context ...uncated sequence (ya)a∈Mn,t has a representing measure for all t ∈ N. 1.2. The flat extension theorem of Curto and Fialkow Curto and Fialkow studied intensively the truncated moment problem (cf. e.g. =-=[4, 5, 6]-=- and further references therein). In particular, they observed that the notion of flat extension of matrices plays a central role in this problem. Given matrices MC and MB indexed, respectively, by C ... |

54 | A new criterion for normal form algorithms
- Mourrain
- 1999
(Show Context)
Citation Context ...which H˜ Λ is a flat extension of HC+ Λ , i.e. ˜ Λ coincides with Λ on Span(C + · C +) and rank H˜ Λ = rank HC+ Λ . 1.5. Border bases and commuting multiplication operators We recall here a result of =-=[16]-=- about border bases of polynomial ideals that we exploit to prove our flat extension theorem. Let B := {b1, . . . , bN} be a finite set of monomials. Assume that, for each border monomial xibj ∈ ∂B, w... |

51 | Detecting global optimality and extracting solutions in GloptiPoly”, LAAS-CNRS
- Henrion, Lasserre
- 2003
(Show Context)
Citation Context ...roblems (see [11]). Moreover the flat extension theorem of Curto and Fialkow permits to detect optimality of the relaxations and to extract global optimizers to the original optimization problem (see =-=[8]-=-). Here is a brief sketch; see e.g. [15] and references therein for details. Suppose we want to compute the infimum p ∗ of a polynomial p over a semialgebraic set K defined by the polynomial inequalit... |

36 |
The multidimensional moment problem
- Fuglede
(Show Context)
Citation Context ... K[x] on K[x] ∗ is denoted by where (p · Λ)(q) := Λ(pq) for q ∈ K[x]. 1.1. The moment problem (p, Λ) ∈ K[x] × K[x] ∗ ↦→ p · Λ ∈ K[x] ∗ In this section, we consider K = R. The moment problem (see e.g. =-=[1, 7]-=-) deals with the characterization of the sequences of moments of measures. Given a probability measure µ on R n , its moment of order a = x α ∈ Mn is the quantity ∫ x α µ(dx). The moment problem conce... |

28 | Flat extensions of positive moment matrices: Recursively generated relations, Memoirs
- Curto, Fialkow
- 1998
(Show Context)
Citation Context ...uncated sequence (ya)a∈Mn,t has a representing measure for all t ∈ N. 1.2. The flat extension theorem of Curto and Fialkow Curto and Fialkow studied intensively the truncated moment problem (cf. e.g. =-=[4, 5, 6]-=- and further references therein). In particular, they observed that the notion of flat extension of matrices plays a central role in this problem. Given matrices MC and MB indexed, respectively, by C ... |

22 | Revisiting two theorems of Curto and Fialkow on moment matrices
- Laurent
(Show Context)
Citation Context ...n the sequence y is bounded [2] or, more generally, exponentially bounded [3]. The next result of Curto and Fialkow [4] shows that this is also the case when the matrix M(y) has finite rank (cf. also =-=[14, 15]-=- for a short proof). Theorem 1.1. [4] If M(y) is positive semidefinite and the rank of M(y) is finite, then y has a (unique) representing measure (which is finitely atomic with rank M(y) atoms). In th... |

20 |
eds.), Handbook of semidefinite programming
- Wolkowicz, Saigal, et al.
- 2000
(Show Context)
Citation Context ...nting measure suported by the set K (cf. [6, 11]). Using moment matrices, the program (3.1) can be formulated as an instance of semidefinite programming for which efficient algorithms exist (see e.g. =-=[20, 21]-=-). We have: p∗ t ≤ p∗ , with equality if H Mn,t Λ is a flat extension of H Mn,t−d Λ for an optimum solution Λ to (3.1) (d := maxj dj). In that case, the atoms of the representing measure (which exists... |

15 |
Ressel: Positive definite functions on Abelian semigroups
- Berg, Christensen, et al.
- 1976
(Show Context)
Citation Context ...tivariate case (n ≥ 2). However, positivity is sufficient for the existence of a representing measure under some additional assumptions. This is the case, for instance, when the sequence y is bounded =-=[2]-=- or, more generally, exponentially bounded [3]. The next result of Curto and Fialkow [4] shows that this is also the case when the matrix M(y) has finite rank (cf. also [14, 15] for a short proof). Th... |

14 | Semidefinite characterization and computation of zero-dimensional real radical ideals
- Lasserre, Laurent, et al.
- 2007
(Show Context)
Citation Context ...nd they can be computed from Λ [8]. Moreover, they are all the global minimizers when H Mn,t Λ has the maximum possible rank among all optimum solutions to the semidefinite program (3.1). As shown in =-=[12]-=-, the truncated moment problem also yields an algorithmic approach to the problem of computing the real roots to polynomial equations g1 = 0, . . .,gm = 0 (assuming their number is finite). Indeed, th... |

13 |
An algebraist’s view on border bases
- Kehrein, Kreuzer, et al.
- 2005
(Show Context)
Citation Context ...we are given a polynomial of the form g (ij) := xibj − N∑ h=1 a (ij) h bh where a (ij) h ∈ K. The set F := {g (ij) | i = 1, . . .,n, j = 1, . . .,N with xibj ∈ ∂B} (1.3) is known as a border prebasis =-=[9]-=- or a rewriting family for B [16]. When the set B contains the constant monomial 1, one can easily verify that B is a generating set for the quotient space K[x]/(F), where (F) is the ideal generated b... |

10 |
Exponentially bounded positive definite functions,” Illinois journal of mathematics
- BERG, MASERICK
- 1984
(Show Context)
Citation Context ... sufficient for the existence of a representing measure under some additional assumptions. This is the case, for instance, when the sequence y is bounded [2] or, more generally, exponentially bounded =-=[3]-=-. The next result of Curto and Fialkow [4] shows that this is also the case when the matrix M(y) has finite rank (cf. also [14, 15] for a short proof). Theorem 1.1. [4] If M(y) is positive semidefinit... |

10 |
Solving the truncated moment problem solves the moment problem
- Stochel
(Show Context)
Citation Context ...e of a representing measure for a truncated sequence indexed by a subset of monomials. A solution to the truncated moment problem would in fact imply a solution to the moment problem. Indeed, Stochel =-=[19]-=- shows that a sequence y = (ya)a∈Mn has a representing measure if and only if the truncated sequence (ya)a∈Mn,t has a representing measure for all t ∈ N. 1.2. The flat extension theorem of Curto and F... |

3 | Moment matrices, border bases and real radical ideals
- Lasserre, Laurent, et al.
- 2008
(Show Context)
Citation Context ...ent matrices indexed by sparse sets of monomials instead of the full degree levels Mn,t. This is where our new sparse flat extension theorem may become very useful. It will be used, in particular, in =-=[10]-=-.A Sparse Flat Extension Theorem for Moment Matrices 11 The approach in [12] also permits to find the real radical of the ideal generated by the polynomials g1, . . . , gm. Indeed, if Λ ∈ (R[x]) ∗ is... |

3 | A prolongation-projection algorithm for computing the finite real variety of an ideal. Theoretical Computer Science
- Lasserre, Laurent, et al.
- 2009
(Show Context)
Citation Context ...et B ⊆ C index a maximum linearly independent set of columns of M which is constructed by the greedy algorithm using the ordering ≼. One can easily verify that B is closed under taking divisions (cf. =-=[13]-=-). □ The following example shows that, even if C is connected to 1, there may not always exist a base B connected to 1 for HC Λ (which justifies our generalisation of Theorem 1.6 to kernels of Hankel ... |

1 | A Gröbner basis proof of the flat extension theorem for moment matrices. arXiv:0801.4243v1
- Schweighofer
- 2008
(Show Context)
Citation Context ...ar system of equations in order to construct the flat extension Mt+1(˜y) of Mt(y) (and then iteratively the infinite flat extension M(˜y)). See also [15] for an exposition of this proof. Schweighofer =-=[18]-=- proposes an alternative proof which is less technical and relies on properties of Gröbner bases. We propose in this note another simple alternative proof, which applies more generally to truncated mo... |