29 research outputs found
Ranks and Symmetric Ranks of Cubic Surfaces
We study cubic surfaces as symmetric tensors of format .
We consider the non-symmetric tensor rank and the symmetric Waring rank of
cubic surfaces, and show that the two notions coincide over the complex
numbers. The corresponding algebraic problem concerns border ranks. We show
that the non-symmetric border rank coincides with the symmetric border rank for
cubic surfaces. As part of our analysis, we obtain minimal ideal generators for
the symmetric analogue to the secant variety from the salmon conjecture. We
also give a test for symmetric rank given by the non-vanishing of certain
discriminants. The results extend to order three tensors of all sizes, implying
the equality of rank and symmetric rank when the symmetric rank is at most
seven, and the equality of border rank and symmetric border rank when the
symmetric border rank is at most five. We also study real ranks via the real
substitution method.Comment: 16 page
Duality of Graphical Models and Tensor Networks
In this article we show the duality between tensor networks and undirected
graphical models with discrete variables. We study tensor networks on
hypergraphs, which we call tensor hypernetworks. We show that the tensor
hypernetwork on a hypergraph exactly corresponds to the graphical model given
by the dual hypergraph. We translate various notions under duality. For
example, marginalization in a graphical model is dual to contraction in the
tensor network. Algorithms also translate under duality. We show that belief
propagation corresponds to a known algorithm for tensor network contraction.
This article is a reminder that the research areas of graphical models and
tensor networks can benefit from interaction
Mixtures and products in two graphical models
We compare two statistical models of three binary random variables. One is a
mixture model and the other is a product of mixtures model called a restricted
Boltzmann machine. Although the two models we study look different from their
parametrizations, we show that they represent the same set of distributions on
the interior of the probability simplex, and are equal up to closure. We give a
semi-algebraic description of the model in terms of six binomial inequalities
and obtain closed form expressions for the maximum likelihood estimates. We
briefly discuss extensions to larger models.Comment: 18 pages, 7 figure
Singular Vectors of Orthogonally Decomposable Tensors
Orthogonal decomposition of tensors is a generalization of the singular value
decomposition of matrices. In this paper, we study the spectral theory of
orthogonally decomposable tensors. For such a tensor, we give a description of
its singular vector tuples as a variety in a product of projective spaces.Comment: 15 pages, 6 figure
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The Algebraic Statistics of an Oberwolfach Workshop
Algebraic Statistics builds on the idea that statistical models can be understood via polynomials. Many statistical models are parameterized by polynomials in the model parameters; others are described implicitly by polynomial equalities and inequalities. We explore the connection between algebra and statistics for some small statistical models
Lower bounds on the rank and symmetric rank of real tensors
We lower bound the rank of a tensor by a linear combination of the ranks of
three of its unfoldings, using Sylvester's rank inequality. In a similar way,
we lower bound the symmetric rank by a linear combination of the symmetric
ranks of three unfoldings. Lower bounds on the rank and symmetric rank of
tensors are important for finding counterexamples to Comon's conjecture. A real
counterexample to Comon's conjecture is a tensor whose real rank and real
symmetric rank differ. Previously, only one real counterexample was known. We
divide the construction into three steps. The first step involves linear spaces
of binary tensors. The second step considers a linear space of larger
decomposable tensors. The third step is to verify a conjecture that lower
bounds the symmetric rank, on a tensor of interest. We use the construction to
build an order six real tensor whose real rank and real symmetric rank differ.Comment: 26 pages, 3 figures, v2: updated to match published versio
Ranks and singularities of cubic surfaces
We explore the connection between the rank of a polynomial and the singularities of its vanishing locus. We first describe the singularity of generic polynomials of fixed rank. We then focus on cubic surfaces. Cubic surfaces with isolated singularities are known to fall into 22 singularity types. We compute the rank of a cubic surface of each singularity type. This enables us to find the possible singular loci of a cubic surface of fixed rank. Finally, we study connections to the Hessian discriminant. We show that a cubic surface with singularities that are not ordinary double points lies on the Hessian discriminant, and that the Hessian discriminant is the closure of the rank six cubic surfaces
Learning Paths from Signature Tensors
Matrix congruence extends naturally to the setting of tensors. We apply
methods from tensor decomposition, algebraic geometry and numerical
optimization to this group action. Given a tensor in the orbit of another
tensor, we compute a matrix which transforms one to the other. Our primary
application is an inverse problem from stochastic analysis: the recovery of
paths from their third order signature tensors. We establish identifiability
results, both exact and numerical, for piecewise linear paths, polynomial
paths, and generic dictionaries. Numerical optimization is applied for recovery
from inexact data. We also compute the shortest path with a given signature
tensor.Comment: 22 pages, 3 figure