1,649 research outputs found
On the X-rays of permutations
The X-ray of a permutation is defined as the sequence of antidiagonal sums in
the associated permutation matrix. X-rays of permutation are interesting in the
context of Discrete Tomography since many types of integral matrices can be
written as linear combinations of permutation matrices. This paper is an
invitation to the study of X-rays of permutations from a combinatorial point of
view. We present connections between these objects and nondecreasing
differences of permutations, zero-sum arrays, decomposable permutations, score
sequences of tournaments, queens' problems and rooks' problems.Comment: 7 page
Decoupling Multivariate Polynomials Using First-Order Information
We present a method to decompose a set of multivariate real polynomials into
linear combinations of univariate polynomials in linear forms of the input
variables. The method proceeds by collecting the first-order information of the
polynomials in a set of operating points, which is captured by the Jacobian
matrix evaluated at the operating points. The polyadic canonical decomposition
of the three-way tensor of Jacobian matrices directly returns the unknown
linear relations, as well as the necessary information to reconstruct the
univariate polynomials. The conditions under which this decoupling procedure
works are discussed, and the method is illustrated on several numerical
examples
On the reconstruction of binary and permutation matrices under (binary) tomographic constraints
AbstractThe paper studies the problem of reconstructing binary matrices constrained by binary tomographic information. We prove new NP-hardness results that sharpen previous complexity results in the realm of discrete tomography but also allow applications to related problems for permutation matrices. Hence our results can be interpreted in terms of other combinatorial problems including the queens’ problem
Uniform random generation of large acyclic digraphs
Directed acyclic graphs are the basic representation of the structure
underlying Bayesian networks, which represent multivariate probability
distributions. In many practical applications, such as the reverse engineering
of gene regulatory networks, not only the estimation of model parameters but
the reconstruction of the structure itself is of great interest. As well as for
the assessment of different structure learning algorithms in simulation
studies, a uniform sample from the space of directed acyclic graphs is required
to evaluate the prevalence of certain structural features. Here we analyse how
to sample acyclic digraphs uniformly at random through recursive enumeration,
an approach previously thought too computationally involved. Based on
complexity considerations, we discuss in particular how the enumeration
directly provides an exact method, which avoids the convergence issues of the
alternative Markov chain methods and is actually computationally much faster.
The limiting behaviour of the distribution of acyclic digraphs then allows us
to sample arbitrarily large graphs. Building on the ideas of recursive
enumeration based sampling we also introduce a novel hybrid Markov chain with
much faster convergence than current alternatives while still being easy to
adapt to various restrictions. Finally we discuss how to include such
restrictions in the combinatorial enumeration and the new hybrid Markov chain
method for efficient uniform sampling of the corresponding graphs.Comment: 15 pages, 2 figures. To appear in Statistics and Computin
Quantum field tomography
We introduce the concept of quantum field tomography, the efficient and
reliable reconstruction of unknown quantum fields based on data of correlation
functions. At the basis of the analysis is the concept of continuous matrix
product states, a complete set of variational states grasping states in quantum
field theory. We innovate a practical method, making use of and developing
tools in estimation theory used in the context of compressed sensing such as
Prony methods and matrix pencils, allowing us to faithfully reconstruct quantum
field states based on low-order correlation functions. In the absence of a
phase reference, we highlight how specific higher order correlation functions
can still be predicted. We exemplify the functioning of the approach by
reconstructing randomised continuous matrix product states from their
correlation data and study the robustness of the reconstruction for different
noise models. We also apply the method to data generated by simulations based
on continuous matrix product states and using the time-dependent variational
principle. The presented approach is expected to open up a new window into
experimentally studying continuous quantum systems, such as encountered in
experiments with ultra-cold atoms on top of atom chips. By virtue of the
analogy with the input-output formalism in quantum optics, it also allows for
studying open quantum systems.Comment: 31 pages, 5 figures, minor change
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