arXiv subjects

Kaie Kubjas

Publications and source records attributed to Kaie Kubjas.

30 records · Page 2Linked to original sources

arXiv2017

Geometry of symmetric group-based models

Phylogenetic models have polynomial parametrization maps. For symmetric group-based models, Matsen studied the polynomial inequalities that characterize the joint probabilities in the image of these parametrizations. We employ this description for maximum likelihood estimation via numerical algebraic geometry. In particular, we explore an example where the maximum likelihood estimate does not exist, which would be difficult to discover without using algebraic methods. We also study the embedding problem for symmetric group-based models, i.e. we identify which mutation matrices are matrix exponentials of rate matrices that are invariant under a group action.

q-bio.PE↗

arXiv2017

Determinantal Generalizations of Instrumental Variables

Linear structural equation models relate the components of a random vector using linear interdependencies and Gaussian noise. Each such model can be naturally associated with a mixed graph whose vertices correspond to the components of the random vector. The graph contains directed edges that represent the linear relationships between components, and bidirected edges that encode unobserved confounding. We study the problem of generic identifiability, that is, whether a generic choice of linear and confounding effects can be uniquely recovered from the joint covariance matrix of the observed random vector. An existing combinatorial criterion for establishing generic identifiability is the half-trek criterion (HTC), which uses the existence of trek systems in the mixed graph to iteratively discover generically invertible linear equation systems in polynomial time. By focusing on edges one at a time, we establish new sufficient and necessary conditions for generic identifiability of edge effects extending those of the HTC. In particular, we show how edge coefficients can be recovered as quotients of subdeterminants of the covariance matrix, which constitutes a determinantal generalization of formulas obtained when using instrumental variables for identification.

math.ST↗

arXiv2017

How to flatten a soccer ball

This is an experimental case study in real algebraic geometry, aimed at computing the image of a semialgebraic subset of 3-space under a polynomial map into the plane. For general instances, the boundary of the image is given by two highly singular curves. We determine these curves and show how they demarcate the "flattened soccer ball". We explore cylindrical algebraic decompositions, by working through concrete examples. Maps onto convex polygons and connections to convex optimization are also discussed.

math.AG↗

arXiv2017

Positive semidefinite rank and nested spectrahedra

The set of matrices of given positive semidefinite rank is semialgebraic. In this paper we study the geometry of this set, and in small cases we describe its boundary. For general values of positive semidefinite rank we provide a conjecture for the description of this boundary. Our proof techniques are geometric in nature and rely on nesting spectrahedra between polytopes.

math.AG↗

arXiv2016

The geometry of rank-one tensor completion

The geometry of the set of restrictions of rank-one tensors to some of their coordinates is studied. This gives insight into the problem of rank-one completion of partial tensors. Particular emphasis is put on the semialgebraic nature of the problem, which arises for real tensors with constraints on the parameters. The algebraic boundary of the completable region is described for tensors parametrized by probability distributions and where the number of observed entries equals the number of parameters. If the observations are on the diagonal of a tensor of format $d\times\dots\times d$, the complete semialgebraic description of the completable region is found.

math.AG↗

arXiv2016

Matrix Completion for the Independence Model

We investigate the problem of completing partial matrices to rank-one matrices in the standard simplex. The motivation for studying this problem comes from statistics: A lack of eligible completion can provide a falsification test for partial observations to come from the independence model. For each pattern of specified entries, we give equations and inequalities which are satisfied if and only if an eligible completion exists. We also describe the set of valid completions, and we optimize over this set.

math.ST↗

arXiv2015

Fixed points of the EM algorithm and nonnegative rank boundaries

Mixtures of $r$ independent distributions for two discrete random variables can be represented by matrices of nonnegative rank $r$. Likelihood inference for the model of such joint distributions leads to problems in real algebraic geometry that are addressed here for the first time. We characterize the set of fixed points of the Expectation-Maximization algorithm, and we study the boundary of the space of matrices with nonnegative rank at most $3$. Both of these sets correspond to algebraic varieties with many irreducible components.

math.ST↗

arXiv2014

Algebraic boundary of matrices of nonnegative rank at most three

The Zariski closure of the boundary of the set of matrices of nonnegative rank at most 3 is reducible. We give a minimal generating set for the ideal of each irreducible component. In fact, this generating set is a Grobner basis with respect to the graded reverse lexicographic order. This solves a conjecture by Robeva, Sturmfels and the last author.

math.AG↗

arXiv2013

Conformal blocks, Berenstein-Zelevinsky triangles and group-based models

Work of Buczynska, Wisniewski, Sturmfels and Xu, and the second author has linked the group-based phylogenetic statistical model associated with the group Z/2Z with the Wess-Zumino-Witten (WZW) model of conformal field theory associated to SL(2,C). In this article we explain how this connection can be generalized to establish a relationship between the phylogenetic statistical model for the cyclic group Z/mZ and the WZW model for the special linear group SL(m,C). We use this relationship to also show how a combinatorial device from representation theory, the Berenstein-Zelevinsky triangles, correspond to elements in the affine semigroup algebra of the Z/3Z phylogenetic statistical model.

math.AG↗

arXiv2013

Low degree minimal generators of phylogenetic semigroups

The phylogenetic semigroup on a graph generalizes the Jukes-Cantor binary model on a tree. Minimal generating sets of phylogenetic semigroups have been described for trivalent trees by Buczyńska and Wiśniewski, and for trivalent graphs with first Betti number 1 by Buczyńska. We characterize degree two minimal generators of the phylogenetic semigroup on any trivalent graph. Moreover, for any graph with first Betti number 1 and for any trivalent graph with first Betti number 2 we describe the minimal generating set of its phylogenetic semigroup.

math.CO↗

arXiv2012

On the graph labellings arising from phylogenetics

We study semigroups of labellings associated to a graph. These generalize the Jukes-Cantor model and phylogenetic toric varieties defined by Buczyńska. Our main theorem bounds the degree of the generators of the semigroup by g+1 when the graph has first Betti number g. Also, we provide a series of examples where the bound is sharp.

math.CO↗

arXiv2010

Hilbert polynomial of the Kimura 3-parameter model

Buczyńska and Wiśniewski showed that for the Jukes Cantor binary model of a 3-valent tree the Hilbert polynomial depends only on the number of leaves of the tree and not on its shape. We ask if this can be generalized to other group-based models. In this paper we consider the Kimura 3-parameter model and show that the generalization of the statement about the Hilbert polynomials to the Kimura 3-parameter model is not possible as the Hilbert polynomial depends on the shape of a 3-valent tree.

math.AC↗