Cutting Corners

Onn, Shmuel; Sturmfels, Bernd

doi:10.1006/aama.1999.0645

Cited by 40 publications

(59 citation statements)

References 16 publications

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“…This explicitly establishes the link between the combinatorial structure of the basis of the factor space and the structure of the ideal: whether an order ideal supports a border basis is solely determined by the combinatorial structure of the order ideal polytope. A related result for Gröbner bases of the vanishing ideal of generic points was established in [46], where it was shown that distinct reduced Gröbner bases of the vanishing ideal are in bijection with the vertices of the corner cut polyhedron.…”

Section: Our Contribution Our Contribution Is the Followingmentioning

confidence: 93%

A Polyhedral Characterization of Border Bases

Braun¹,

Pokutta²

2016

SIAM J. Discrete Math.

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A. Border bases arise as a canonical generalization of Gröbner bases, using order ideals instead of term orderings. We provide a polyhedral characterization of all order ideals (and hence all border bases) that are supported by a zero-dimensional ideal: order ideals that support a border basis correspond one-to-one to integral points of the order ideal polytope. In particular, we establish a crucial connection between the ideal and its combinatorial structure. Based on this characterization we also provide an adaptation of the border basis algorithm of Kehrein and Kreuzer [35] to allow for computing border bases for arbitrary order ideals, given implicitly via maximizing a preference on monomials (variable selection problem), independent of term orderings. The algorithm requires the same size of resources as the border basis algorithm except for some minor overhead. We also show that the underlying variable selection problem of finding an order ideal that supports a border basis is NP-hard and that any linear description of the associated convex hull of all order ideals requires a superpolynomial number of inequalities. IIn many different disciplines and real-world applications one is faced with solving systems of polynomial equations. Often this is simply due to a physical or dynamical system having a natural representation as a system of polynomial equations, but equally often it is due to the sheer expressive power of polynomial systems of equations that allow for easy reformulation. To give an example of the latter, an inequality ax ≤ b with a ∈ R n and b ∈ R can be expressed via a single polynomial equation: ax + u 2 = b. A slightly more involved example is that of the feasible region of a binary program {x | Ax ≤ b, x ∈ {0, 1} n }, which can be captured via rewriting each individual inequality as before, and adding quadratic polynomials x 2 i − x i = 0 for each coordinate i = 1, . . . , n of x. As a consequence, there is a huge need to computationally model, understand, manipulate, and extract the solution set of systems of polynomial equations.A key insight in (computational) commutative algebra is that one can choose a smart ordering on the monomials and compute a special set of generators of the ideal generated by the system of equations that makes many operations easy, and provides a structural insight into the system. One such special set of generators is a Gröbner basis. By now, Gröbner bases are fundamental and standard tools in commutative algebra to actually perform important operations on ideals such as intersection, membership test, elimination, projection, and many more. Border bases arise as a natural generalization of Gröbner bases that can be computed for zero-dimensional ideals, i.e., the associated factor space is a finite-dimensional vector space (see Section 2 for details). While this might seem to be a severe restriction, for many applications it is sufficient. Roughly speaking, whenever the solution set is finite, we are dealing with a zero-dimensional ideal. For example, systems of p...

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Section: Our Contribution Our Contribution Is the Followingmentioning

confidence: 93%

A Polyhedral Characterization of Border Bases

Braun¹,

Pokutta²

2016

SIAM J. Discrete Math.

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“…The thick line shown encloses all the terms satisfying |α| ≤ 3. This representation is a generalization of the corner cut diagram in algebraic methods proposed in [21] by compressing a d-dimensional representation onto two dimensions. A (d − 1) dimensional corner cut hyperplane separating the multi-indices with |α| ≤ 3 is now represented in two dimensions by the thick line.…”

Section: Uncertainty Quantificationmentioning

confidence: 99%

“…Figure 1 (b) shows the terms satisfying |α| ≤ 3. Using corner-cut staircase introduced by [21], these terms can also be represented as a hypertetrahedron satisfying |α| ≤ p in α ∈ N d . Indeed, Figure 1 (b) illustrate in two-dimensions the concept of a generic set which will be discussed in the following section.…”

Section: Uncertainty Quantificationmentioning

confidence: 99%

The Algebraic Method in Quadrature for Uncertainty Quantification

Ko¹,

Wynn²

2016

SIAM/ASA J. Uncertainty Quantification

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Abstract. A general method of quadrature for uncertainty quantification (UQ) is introduced based on the algebraic method in experimental design. This is a method based on the theory of zero dimensional algebraic varieties. It allows quadrature of polynomials or polynomial approximands for quite general sets of quadrature points, here called 'designs'. The method goes some way to explaining when quadrature weights are non-negative and gives exact quadrature for monomials in the quotient ring defined by the algebraic method. The relationship to the classical methods based on zeros of orthogonal polynomials is discussed and numerical comparisons made with methods such as Gaussian quadrature and Smolyak grids. Application to UQ is examined in the context of polynomial chaos expansion and probabilistic collocation method where solution statistics are estimated.

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“…. , x k ] modulo the ideal of polynomials vanishing on the design points; see [3,13] and references therein for more on this.) If B is identifiable then, given any vector of measurements y ∈ R m at the design points, the vector of coefficients c ∈ R m of the unique polynomial Φ B (x) = m j=1 c j k h=1 x β j,h h supported on B that is consistent with the measurements, is given by c := T −1 y.…”

Section: Experimental Designmentioning

confidence: 99%

“…In [3,13], in the context of the theory of Gröbner bases in commutative algebra, it was shown that the smallest set containing all m-point hierarchical models B (termed staircases therein) in N k is the following set, consisting of roughly O(m log m) points,…”

Section: Experimental Designmentioning

confidence: 99%

Nonlinear Matroid Optimization and Experimental Design

Berstein¹,

Lee²,

Maruri-Aguilar³

et al. 2008

SIAM J. Discrete Math.

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We study the problem of optimizing nonlinear objective functions over matroids presented by oracles or explicitly. Such functions can be interpreted as the balancing of multi-criteria optimization. We provide a combinatorial polynomial time algorithm for arbitrary oracle-presented matroids, that makes repeated use of matroid intersection, and an algebraic algorithm for vectorial matroids.Our work is partly motivated by applications to minimum-aberration model-fitting in experimental design in statistics, which we discuss and demonstrate in detail.

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Cutting Corners

Cited by 40 publications

References 16 publications

A Polyhedral Characterization of Border Bases

A Polyhedral Characterization of Border Bases

The Algebraic Method in Quadrature for Uncertainty Quantification

Nonlinear Matroid Optimization and Experimental Design

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