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Some Real and Unreal Enumerative Geometry for Flag Manifolds
Build geometric shape numerical homotopy continuation algorithm
2014/12/29
We present a general method for constructing real solutions to some problems in enumerative geometry which gives lower bounds on the maximum number of real solutions. We apply this method to show that...
An Excursion from Enumerative Geometry to Solving Systems of Polynomial Equations with Macaulay 2
To solve the polynomial equations mathematics geometry cross theory
2014/12/29
Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics ha...
Enumerative Geometry is concerned with the number of solutions to a structured system of polynomial equations, when the structure comes from geometry. Enumerative real algebraic geometry studies real ...
Geometry of the tetrahedron space
Mark smooth space
2014/12/25
Let X be the space of all labeled tetrahedra in . In [E. Babson, P.E. Gunnells, R. Scott, A smooth space of tetrahedra, Adv. Math. 165(2) (2002) 285–312] we constructed a smooth symmetric compactifica...
Discrete holomorphic geometry I. Darboux transformations and spectral curves
Discrete regular concept map discrete Riemann discrete differential geometry the discrete surface
2014/12/24
Finding appropriate notions of discrete holomorphic maps and, more generally, conformal immersions of discrete Riemann surfaces into 3-space is an important problem of discrete differential geometry a...
Likelihood Geometry
Likelihood Geometry
2013/6/17
We study the critical points of monomial functions over an algebraic subset of the probability simplex. The number of critical points on the Zariski closure is a topological invariant of that embedded...
Computational information geometry: theory and practice
Information geometry Multinomial distribution Affne geometry Exponential family Extended exponential family
2012/11/22
This paper lays the foundations for a unified framework for numerically and computationally applying methods drawn from a range of currently distinct geometrical approaches to statistical modelling. I...
Geometry of faithfulness assumption in causal inference
causal inference PC-algorithm (strong) faithfulness conditional independence directed acyclic graph structural equation model real algebraic hypersurface Crofton's formula algebraic statistics.
2012/9/18
Many algorithms for inferring causality rely heavily on the faithfulness assumption.The main justi cation for imposing this assumption is that the set of unfaithful distribu-tions has Lebesgue measure...
We determine an explicit Gr¨obner basis, consisting of linear forms and determi-nantal quadrics, for the prime ideal of Raftery’s mixture transition distribution model for Markov chains. When the stat...
Riemannian statistics geometry: A counterpart approach of inference geometry
Geometrical Methods in Statistics
2011/7/19
Riemannian statistics geometry is proposed in this work as a counterpart approach of inference geometry.
Conformal geometry of statistical manifold with application to sequential estimation
Affi ne connections Curved exponential family Hyperboloid dis-tribution Information geometry Projective transformation Riemannian metric Space of constant curvature Totally umbilic von Mises-Fisher distribution
2011/3/22
We present a geometrical method for analyzing sequential estimating procedures. It is based on the design principle of the second-order efficient sequential estimation provided in Okamoto, Amari and T...
Conformal geometry of statistical manifold with application to sequential estimation
Affi ne connections Curved exponential family Hyperboloid dis-tribution Information geometry Projective transformation Riemannian metric Space of constant curvature Totally umbilic von Mises-Fisher distribution
2011/3/23
We present a geometrical method for analyzing sequential estimating procedures. It is based on the design principle of the second-order efficient sequential estimation provided in Okamoto, Amari and T...
Geometry of maximum likelihood estimation in Gaussian graphical models
Statistics Theory (math.ST) Algebraic Geometry (math.AG) Optimization and Control (math.OC)
2010/12/17
We study maximum likelihood estimation in Gaussian graphical models from a geometric point of view. An algebraic elimination criterion allows us to find exact lower bounds on the number of observation...
The Convex Geometry of Linear Inverse Problems
Optimization and Control (math.OC) Statistics Theory (math.ST)
2010/12/17
In applications throughout science and engineering one is often faced with the challenge of solving an ill-posed inverse problem, where the number of available measurements is smaller than the dimensi...
Consistent order estimation and the local geometry of mixtures
consistent order estimation penalized likelihood likelihood inequalities uniform law of iterated logarithm empirical process theory bracketing entropy
2010/3/10
Consider an i.i.d. sequence of random variables whose distribution f
lies in one of a nested family of models (Mq)q2N,Mq Mq+1. The smallest
index q such that Mq contains f is called the model orde...