23,167 research outputs found
Actions on permutations and unimodality of descent polynomials
We study a group action on permutations due to Foata and Strehl and use it to
prove that the descent generating polynomial of certain sets of permutations
has a nonnegative expansion in the basis ,
. This property implies symmetry and unimodality. We
prove that the action is invariant under stack-sorting which strengthens recent
unimodality results of B\'ona. We prove that the generalized permutation
patterns and are invariant under the action and use this to
prove unimodality properties for a -analog of the Eulerian numbers recently
studied by Corteel, Postnikov, Steingr\'{\i}msson and Williams.
We also extend the action to linear extensions of sign-graded posets to give
a new proof of the unimodality of the -Eulerian polynomials of
sign-graded posets and a combinatorial interpretations (in terms of
Stembridge's peak polynomials) of the corresponding coefficients when expanded
in the above basis.
Finally, we prove that the statistic defined as the number of vertices of
even height in the unordered decreasing tree of a permutation has the same
distribution as the number of descents on any set of permutations invariant
under the action. When restricted to the set of stack-sortable permutations we
recover a result of Kreweras.Comment: 19 pages, revised version to appear in Europ. J. Combi
Sampling of stochastic operators
We develop sampling methodology aimed at determining stochastic operators
that satisfy a support size restriction on the autocorrelation of the operators
stochastic spreading function. The data that we use to reconstruct the operator
(or, in some cases only the autocorrelation of the spreading function) is based
on the response of the unknown operator to a known, deterministic test signal
A Distance-Based Test of Association Between Paired Heterogeneous Genomic Data
Due to rapid technological advances, a wide range of different measurements
can be obtained from a given biological sample including single nucleotide
polymorphisms, copy number variation, gene expression levels, DNA methylation
and proteomic profiles. Each of these distinct measurements provides the means
to characterize a certain aspect of biological diversity, and a fundamental
problem of broad interest concerns the discovery of shared patterns of
variation across different data types. Such data types are heterogeneous in the
sense that they represent measurements taken at very different scales or
described by very different data structures. We propose a distance-based
statistical test, the generalized RV (GRV) test, to assess whether there is a
common and non-random pattern of variability between paired biological
measurements obtained from the same random sample. The measurements enter the
test through distance measures which can be chosen to capture particular
aspects of the data. An approximate null distribution is proposed to compute
p-values in closed-form and without the need to perform costly Monte Carlo
permutation procedures. Compared to the classical Mantel test for association
between distance matrices, the GRV test has been found to be more powerful in a
number of simulation settings. We also report on an application of the GRV test
to detect biological pathways in which genetic variability is associated to
variation in gene expression levels in ovarian cancer samples, and present
results obtained from two independent cohorts
Graph-based task libraries for robots: generalization and autocompletion
In this paper, we consider an autonomous robot that persists
over time performing tasks and the problem of providing one additional
task to the robot's task library. We present an approach to generalize
tasks, represented as parameterized graphs with sequences, conditionals,
and looping constructs of sensing and actuation primitives. Our approach
performs graph-structure task generalization, while maintaining task ex-
ecutability and parameter value distributions. We present an algorithm
that, given the initial steps of a new task, proposes an autocompletion
based on a recognized past similar task. Our generalization and auto-
completion contributions are eective on dierent real robots. We show
concrete examples of the robot primitives and task graphs, as well as
results, with Baxter. In experiments with multiple tasks, we show a sig-
nicant reduction in the number of new task steps to be provided
Generalized Separable Nonnegative Matrix Factorization
Nonnegative matrix factorization (NMF) is a linear dimensionality technique
for nonnegative data with applications such as image analysis, text mining,
audio source separation and hyperspectral unmixing. Given a data matrix and
a factorization rank , NMF looks for a nonnegative matrix with
columns and a nonnegative matrix with rows such that .
NMF is NP-hard to solve in general. However, it can be computed efficiently
under the separability assumption which requires that the basis vectors appear
as data points, that is, that there exists an index set such that
. In this paper, we generalize the separability
assumption: We only require that for each rank-one factor for
, either for some or for
some . We refer to the corresponding problem as generalized separable NMF
(GS-NMF). We discuss some properties of GS-NMF and propose a convex
optimization model which we solve using a fast gradient method. We also propose
a heuristic algorithm inspired by the successive projection algorithm. To
verify the effectiveness of our methods, we compare them with several
state-of-the-art separable NMF algorithms on synthetic, document and image data
sets.Comment: 31 pages, 12 figures, 4 tables. We have added discussions about the
identifiability of the model, we have modified the first synthetic
experiment, we have clarified some aspects of the contributio
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