108 research outputs found
Nearest Points on Toric Varieties
We determine the Euclidean distance degree of a projective toric variety.
This extends the formula of Matsui and Takeuchi for the degree of the
-discriminant in terms of Euler obstructions. Our primary goal is the
development of reliable algorithmic tools for computing the points on a real
toric variety that are closest to a given data point.Comment: 20 page
Implicitization of curves and (hyper)surfaces using predicted support
We reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation.
For predicting the implicit support, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory.
Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial.
We implement our methods on Maple, and some on Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces.
We apply our approach to approximate implicitization,
and quantify the accuracy of the approximate output,
which turns out to be satisfactory on all tested examples; we also relate our measures to Hausdorff distance.
In building a square or rectangular matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complexes offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice.
We compare our prototype to existing software and find that it is rather competitive
An Implicitization Challenge for Binary Factor Analysis
We use tropical geometry to compute the multidegree and Newton polytope of
the hypersurface of a statistical model with two hidden and four observed
binary random variables, solving an open question stated by Drton, Sturmfels
and Sullivant in "Lectures on Algebraic Statistics" (Problem 7.7). The model is
obtained from the undirected graphical model of the complete bipartite graph
by marginalizing two of the six binary random variables. We present
algorithms for computing the Newton polytope of its defining equation by
parallel walks along the polytope and its normal fan. In this way we compute
vertices of the polytope. Finally, we also compute and certify its facets by
studying tangent cones of the polytope at the symmetry classes vertices. The
Newton polytope has 17214912 vertices in 44938 symmetry classes and 70646
facets in 246 symmetry classes.Comment: 25 pages, 5 figures, presented at Mega 09 (Barcelona, Spain
Tropical Monte Carlo quadrature for Feynman integrals
We introduce a new method to evaluate algebraic integrals over the simplex
numerically. This new approach employs techniques from tropical geometry and
exceeds the capabilities of existing numerical methods by an order of
magnitude. The method can be improved further by exploiting the geometric
structure of the underlying integrand. As an illustration of this, we give a
specialized integration algorithm for a class of integrands that exhibit the
form of a generalized permutahedron. This class includes integrands for
scattering amplitudes and parametric Feynman integrals with tame kinematics. A
proof-of-concept implementation is provided with which Feynman integrals up to
loop order 17 can be evaluated.Comment: 6 figures, see http://github.com/michibo/tropical-feynman-quadrature
for the referenced program code; v2: typos corrected, version accepted for
publication in Annales de l'Institut Henri Poincar\'e
Tropical Ehrhart theory and tropical volume
We introduce a novel intrinsic volume concept in tropical geometry. This is achieved by developing the foundations of a tropical analog of lattice point counting in polytopes. We exhibit the basic properties and compare it to existing measures. Our exposition is complemented by a brief study of arising complexity questions
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