10,196 research outputs found
Matched subspace detection with hypothesis dependent noise power
We consider the problem of detecting a subspace signal in white Gaussian noise when the noise power may be different under the null hypothesisâwhere it is assumed to be knownâand the alternative hypothesis. This situation occurs when the presence of the signal of interest (SOI) triggers an increase in the noise power. Accordingly, it may be
relevant in the case of a mismatch between the actual SOI subspace and its presumed value, resulting in a modelling error. We derive the generalized likelihood ratio test
(GLRT) for the problem at hand and contrast it with the GLRT which assumes known and equal noise power under the two
hypotheses. A performance analysis is carried out and the distributions of the two test statistics are derived. From this analysis, we discuss the differences between the two detectors and provide explanations for the improved performance of the new detector. Numerical simulations attest to the validity of the analysis
Conformal flow on and weak field integrability in AdS
We consider the conformally invariant cubic wave equation on the Einstein
cylinder for small rotationally symmetric
initial data. This simple equation captures many key challenges of nonlinear
wave dynamics in confining geometries, while a conformal transformation relates
it to a self-interacting conformally coupled scalar in four-dimensional anti-de
Sitter spacetime (AdS) and connects it to various questions of AdS
stability. We construct an effective infinite-dimensional time-averaged
dynamical system accurately approximating the original equation in the weak
field regime. It turns out that this effective system, which we call the
conformal flow, exhibits some remarkable features, such as low-dimensional
invariant subspaces, a wealth of stationary states (for which energy does not
flow between the modes), as well as solutions with nontrivial exactly periodic
energy flows. Based on these observations and close parallels to the cubic
Szego equation, which was shown by Gerard and Grellier to be Lax-integrable, it
is tempting to conjecture that the conformal flow and the corresponding weak
field dynamics in AdS are integrable as well.Comment: 22 pages, v2: minor revisions, several references added, v3: typos
corrected, v4: typos corrected, one reference added, matches version accepted
by CM
Macrostate Data Clustering
We develop an effective nonhierarchical data clustering method using an
analogy to the dynamic coarse graining of a stochastic system. Analyzing the
eigensystem of an interitem transition matrix identifies fuzzy clusters
corresponding to the metastable macroscopic states (macrostates) of a diffusive
system. A "minimum uncertainty criterion" determines the linear transformation
from eigenvectors to cluster-defining window functions. Eigenspectrum gap and
cluster certainty conditions identify the proper number of clusters. The
physically motivated fuzzy representation and associated uncertainty analysis
distinguishes macrostate clustering from spectral partitioning methods.
Macrostate data clustering solves a variety of test cases that challenge other
methods.Comment: keywords: cluster analysis, clustering, pattern recognition, spectral
graph theory, dynamic eigenvectors, machine learning, macrostates,
classificatio
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