49,168 research outputs found
Uniform determinantal representations
The problem of expressing a specific polynomial as the determinant of a
square matrix of affine-linear forms arises from algebraic geometry,
optimisation, complexity theory, and scientific computing. Motivated by recent
developments in this last area, we introduce the notion of a uniform
determinantal representation, not of a single polynomial but rather of all
polynomials in a given number of variables and of a given maximal degree. We
derive a lower bound on the size of the matrix, and present a construction
achieving that lower bound up to a constant factor as the number of variables
is fixed and the degree grows. This construction marks an improvement upon a
recent construction due to Plestenjak-Hochstenbach, and we investigate the
performance of new representations in their root-finding technique for
bivariate systems. Furthermore, we relate uniform determinantal representations
to vector spaces of singular matrices, and we conclude with a number of future
research directions.Comment: 23 pages, 3 figures, 4 table
Lower Bounds for Two-Sample Structural Change Detection in Ising and Gaussian Models
The change detection problem is to determine if the Markov network structures
of two Markov random fields differ from one another given two sets of samples
drawn from the respective underlying distributions. We study the trade-off
between the sample sizes and the reliability of change detection, measured as a
minimax risk, for the important cases of the Ising models and the Gaussian
Markov random fields restricted to the models which have network structures
with nodes and degree at most , and obtain information-theoretic lower
bounds for reliable change detection over these models. We show that for the
Ising model, samples are
required from each dataset to detect even the sparsest possible changes, and
that for the Gaussian, samples are
required from each dataset to detect change, where is the smallest
ratio of off-diagonal to diagonal terms in the precision matrices of the
distributions. These bounds are compared to the corresponding results in
structure learning, and closely match them under mild conditions on the model
parameters. Thus, our change detection bounds inherit partial tightness from
the structure learning schemes in previous literature, demonstrating that in
certain parameter regimes, the naive structure learning based approach to
change detection is minimax optimal up to constant factors.Comment: Presented at the 55th Annual Allerton Conference on Communication,
Control, and Computing, Oct. 201
A note on palindromicity
Two results on palindromicity of bi-infinite words in a finite alphabet are
presented. The first is a simple, but efficient criterion to exclude
palindromicity of minimal sequences and applies, in particular, to the
Rudin-Shapiro sequence. The second provides a constructive method to build
palindromic minimal sequences based upon regular, generic model sets with
centro-symmetric window. These give rise to diagonal tight-binding models in
one dimension with purely singular continuous spectrum.Comment: 12 page
On the Complexity of Nondeterministically Testable Hypergraph Parameters
The paper proves the equivalence of the notions of nondeterministic and
deterministic parameter testing for uniform dense hypergraphs of arbitrary
order. It generalizes the result previously known only for the case of simple
graphs. By a similar method we establish also the equivalence between
nondeterministic and deterministic hypergraph property testing, answering the
open problem in the area. We introduce a new notion of a cut norm for
hypergraphs of higher order, and employ regularity techniques combined with the
ultralimit method.Comment: 33 page
Community detection in temporal multilayer networks, with an application to correlation networks
Networks are a convenient way to represent complex systems of interacting
entities. Many networks contain "communities" of nodes that are more densely
connected to each other than to nodes in the rest of the network. In this
paper, we investigate the detection of communities in temporal networks
represented as multilayer networks. As a focal example, we study time-dependent
financial-asset correlation networks. We first argue that the use of the
"modularity" quality function---which is defined by comparing edge weights in
an observed network to expected edge weights in a "null network"---is
application-dependent. We differentiate between "null networks" and "null
models" in our discussion of modularity maximization, and we highlight that the
same null network can correspond to different null models. We then investigate
a multilayer modularity-maximization problem to identify communities in
temporal networks. Our multilayer analysis only depends on the form of the
maximization problem and not on the specific quality function that one chooses.
We introduce a diagnostic to measure \emph{persistence} of community structure
in a multilayer network partition. We prove several results that describe how
the multilayer maximization problem measures a trade-off between static
community structure within layers and larger values of persistence across
layers. We also discuss some computational issues that the popular "Louvain"
heuristic faces with temporal multilayer networks and suggest ways to mitigate
them.Comment: 42 pages, many figures, final accepted version before typesettin
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