39 research outputs found
Tensor Rank, Invariants, Inequalities, and Applications
Though algebraic geometry over is often used to describe the
closure of the tensors of a given size and complex rank, this variety includes
tensors of both smaller and larger rank. Here we focus on the tensors of rank over , which has as a dense subset the orbit
of a single tensor under a natural group action. We construct polynomial
invariants under this group action whose non-vanishing distinguishes this orbit
from points only in its closure. Together with an explicit subset of the
defining polynomials of the variety, this gives a semialgebraic description of
the tensors of rank and multilinear rank . The polynomials we
construct coincide with Cayley's hyperdeterminant in the case , and thus
generalize it. Though our construction is direct and explicit, we also recast
our functions in the language of representation theory for additional insights.
We give three applications in different directions: First, we develop basic
topological understanding of how the real tensors of complex rank and
multilinear rank form a collection of path-connected subsets, one of
which contains tensors of real rank . Second, we use the invariants to
develop a semialgebraic description of the set of probability distributions
that can arise from a simple stochastic model with a hidden variable, a model
that is important in phylogenetics and other fields. Third, we construct simple
examples of tensors of rank which lie in the closure of those of rank
.Comment: 31 pages, 1 figur
Latent tree models
Latent tree models are graphical models defined on trees, in which only a
subset of variables is observed. They were first discussed by Judea Pearl as
tree-decomposable distributions to generalise star-decomposable distributions
such as the latent class model. Latent tree models, or their submodels, are
widely used in: phylogenetic analysis, network tomography, computer vision,
causal modeling, and data clustering. They also contain other well-known
classes of models like hidden Markov models, Brownian motion tree model, the
Ising model on a tree, and many popular models used in phylogenetics. This
article offers a concise introduction to the theory of latent tree models. We
emphasise the role of tree metrics in the structural description of this model
class, in designing learning algorithms, and in understanding fundamental
limits of what and when can be learned
Multilinear algebra for phylogenetic reconstruction
Phylogenetic reconstruction tries to recover the ancestral relationships among a group of contemporary species and represent them in a phylogenetic tree. To do it, it is useful to model evolution adopting a parametric statistic model. Using these models one is able to deduce polynomial relationships between the observed probabilities, known as phylogenetic invariants. Mathematicians have recently begun to be interested in the study of these polynomials and have developed techniques from algebraic geometry that have already been used in the study of phylogenetics. Nowadays there exist some phylogenetic reconstruction methods based in these phylogenetic invariants. In this project we study some theoretical results on stochasticity conditions of the parameters of the model and we analyze whether they give some new information to these reconstruction methods. We implement the conditions and analyze the results comparing them with the results provided by the reconstruction method Erik+2. Finally we propose a new reconstruction method based in the same ideas, with different implementation, and with very good results on simulated data
SAQ: semi-algebraic quartet reconstruction
© 2021 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.We present the phylogenetic quartet reconstruction method SAQ (Semi-Algebraic Quartet reconstruction). SAQ is consistent with the most general Markov model of nucleotide substitution and, in particular, it allows for rate heterogeneity across lineages. Based on the algebraic and semi-algebraic description of distributions that arise from the general Markov model on a quartet, the method outputs normalized weights for the three trivalent quartets (which can be used as input of quartet-based methods). We show that SAQ is a highly competitive method that outperforms most of the well known reconstruction methods on data simulated under the general Markov model on 4-taxon trees. Moreover, it also achieves a high performance on data that violates the underlying assumptions.The authors were partially supported by Spanish government Secretar´ıa de Estado de Investigaci´on, Desarrollo e Innovaci´on [MTM2015-69135-P (MINECO/FEDER)] and [PID2019-
103849GB-I00 (MINECO)]; Generalitat de Catalunya [2014 SGR-634]. M. Garrote-L´opez
was also funded by Spanish government, Ministerio de Econom´ıa y Competitividad research project Maria de Maeztu [MDM-2014-0445].Peer ReviewedPostprint (author's final draft
SAQ: semi-algebraic quartet reconstruction method
We present the phylogenetic quartet reconstruction method SAQ (Semi-algebraic
quartet reconstruction). SAQ is consistent with the most general Markov model
of nucleotide substitution and, in particular, it allows for rate heterogeneity
across lineages. Based on the algebraic and semi-algebraic description of
distributions that arise from the general Markov model on a quartet, the method
outputs normalized weights for the three trivalent quartets (which can be used
as input of quartet-base methods). We show that SAQ is a highly competitive
method that outperforms most of the well known reconstruction methods on data
simulated under the general Markov model on 4-taxon trees. Moreover, it also
achieves a high performance on data that violates the underlying assumptions
Algebraic Aspects of Conditional Independence and Graphical Models
This chapter of the forthcoming Handbook of Graphical Models contains an
overview of basic theorems and techniques from algebraic geometry and how they
can be applied to the study of conditional independence and graphical models.
It also introduces binomial ideals and some ideas from real algebraic geometry.
When random variables are discrete or Gaussian, tools from computational
algebraic geometry can be used to understand implications between conditional
independence statements. This is accomplished by computing primary
decompositions of conditional independence ideals. As examples the chapter
presents in detail the graphical model of a four cycle and the intersection
axiom, a certain implication of conditional independence statements. Another
important problem in the area is to determine all constraints on a graphical
model, for example, equations determined by trek separation. The full set of
equality constraints can be determined by computing the model's vanishing
ideal. The chapter illustrates these techniques and ideas with examples from
the literature and provides references for further reading.Comment: 20 pages, 1 figur