34 research outputs found
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
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
Fixed points of the EM algorithm and nonnegative rank boundaries
Mixtures of independent distributions for two discrete random variables
can be represented by matrices of nonnegative rank . 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 . Both of
these sets correspond to algebraic varieties with many irreducible components.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1282 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Geometry of Log-Concave Density Estimation
Shape-constrained density estimation is an important topic in mathematical
statistics. We focus on densities on that are log-concave, and
we study geometric properties of the maximum likelihood estimator (MLE) for
weighted samples. Cule, Samworth, and Stewart showed that the logarithm of the
optimal log-concave density is piecewise linear and supported on a regular
subdivision of the samples. This defines a map from the space of weights to the
set of regular subdivisions of the samples, i.e. the face poset of their
secondary polytope. We prove that this map is surjective. In fact, every
regular subdivision arises in the MLE for some set of weights with positive
probability, but coarser subdivisions appear to be more likely to arise than
finer ones. To quantify these results, we introduce a continuous version of the
secondary polytope, whose dual we name the Samworth body. This article
establishes a new link between geometric combinatorics and nonparametric
statistics, and it suggests numerous open problems.Comment: 22 pages, 3 figure
Superresolution without Separation
This paper provides a theoretical analysis of diffraction-limited
superresolution, demonstrating that arbitrarily close point sources can be
resolved in ideal situations. Precisely, we assume that the incoming signal is
a linear combination of M shifted copies of a known waveform with unknown
shifts and amplitudes, and one only observes a finite collection of evaluations
of this signal. We characterize properties of the base waveform such that the
exact translations and amplitudes can be recovered from 2M + 1 observations.
This recovery is achieved by solving a a weighted version of basis pursuit over
a continuous dictionary. Our methods combine classical polynomial interpolation
techniques with contemporary tools from compressed sensing.Comment: 23 pages, 8 figure
Convolutions of Totally Positive Distributions with applications to Kernel Density Estimation
In this work we study the estimation of the density of a totally positive
random vector. Total positivity of the distribution of a random vector implies
a strong form of positive dependence between its coordinates and, in
particular, it implies positive association. Since estimating a totally
positive density is a non-parametric problem, we take on a (modified) kernel
density estimation approach. Our main result is that the sum of scaled standard
Gaussian bumps centered at a min-max closed set provably yields a totally
positive distribution. Hence, our strategy for producing a totally positive
estimator is to form the min-max closure of the set of samples, and output a
sum of Gaussian bumps centered at the points in this set. We can frame this sum
as a convolution between the uniform distribution on a min-max closed set and a
scaled standard Gaussian. We further conjecture that convolving any totally
positive density with a standard Gaussian remains totally positive