26,127 research outputs found
A Probabilistic Interpretation of Sampling Theory of Graph Signals
We give a probabilistic interpretation of sampling theory of graph signals.
To do this, we first define a generative model for the data using a pairwise
Gaussian random field (GRF) which depends on the graph. We show that, under
certain conditions, reconstructing a graph signal from a subset of its samples
by least squares is equivalent to performing MAP inference on an approximation
of this GRF which has a low rank covariance matrix. We then show that a
sampling set of given size with the largest associated cut-off frequency, which
is optimal from a sampling theoretic point of view, minimizes the worst case
predictive covariance of the MAP estimate on the GRF. This interpretation also
gives an intuitive explanation for the superior performance of the sampling
theoretic approach to active semi-supervised classification.Comment: 5 pages, 2 figures, To appear in International Conference on
Acoustics, Speech, and Signal Processing (ICASSP) 201
Canonical correlation analysis of high-dimensional data with very small sample support
This paper is concerned with the analysis of correlation between two
high-dimensional data sets when there are only few correlated signal components
but the number of samples is very small, possibly much smaller than the
dimensions of the data. In such a scenario, a principal component analysis
(PCA) rank-reduction preprocessing step is commonly performed before applying
canonical correlation analysis (CCA). We present simple, yet very effective
approaches to the joint model-order selection of the number of dimensions that
should be retained through the PCA step and the number of correlated signals.
These approaches are based on reduced-rank versions of the Bartlett-Lawley
hypothesis test and the minimum description length information-theoretic
criterion. Simulation results show that the techniques perform well for very
small sample sizes even in colored noise
Estimation of the Number of Sources in Unbalanced Arrays via Information Theoretic Criteria
Estimating the number of sources impinging on an array of sensors is a well
known and well investigated problem. A common approach for solving this problem
is to use an information theoretic criterion, such as Minimum Description
Length (MDL) or the Akaike Information Criterion (AIC). The MDL estimator is
known to be a consistent estimator, robust against deviations from the Gaussian
assumption, and non-robust against deviations from the point source and/or
temporally or spatially white additive noise assumptions. Over the years
several alternative estimation algorithms have been proposed and tested.
Usually, these algorithms are shown, using computer simulations, to have
improved performance over the MDL estimator, and to be robust against
deviations from the assumed spatial model. Nevertheless, these robust
algorithms have high computational complexity, requiring several
multi-dimensional searches.
In this paper, motivated by real life problems, a systematic approach toward
the problem of robust estimation of the number of sources using information
theoretic criteria is taken. An MDL type estimator that is robust against
deviation from assumption of equal noise level across the array is studied. The
consistency of this estimator, even when deviations from the equal noise level
assumption occur, is proven. A novel low-complexity implementation method
avoiding the need for multi-dimensional searches is presented as well, making
this estimator a favorable choice for practical applications.Comment: To appear in the IEEE Transactions on Signal Processin
Optimal Sequential Investigation Rules in Competition Law
Although both in US antitrust and European competition law there is a clear evolution to a much broader application of "rule of reason" (instead of per-se rules), there is also an increasing awareness of the problems of a case-by-case approach. The "error costs approach" (minimizing the sum of welfare costs of decision errors and administrative costs) allows not only to decide between these two extremes, but also to design optimally differentiated rules (with an optimal depth of investigation) as intermediate solutions between simple per-se rules and a fullscale rule of reason. In this paper we present a decision-theoretic model that can be used as an instrument for deriving optimal rules for a sequential investigation process in competition law. Such a sequential investigation can be interpreted as a step-by-step sorting process into ever smaller subclasses of cases that help to discriminate better between pro- and anticompetitive cases. We analyze both the problem of optimal stopping of the investigation and optimal sequencing of the assessment criteria in an investigation. To illustrate, we show how a more differentiated rule on resale price maintenance could be derived after the rejection of its per-se prohibition by the US Supreme Court in the "Leegin" case 2007.Law Enforcement, Decision-Making, Competition Law, Antitrust Law
Minimax rank estimation for subspace tracking
Rank estimation is a classical model order selection problem that arises in a
variety of important statistical signal and array processing systems, yet is
addressed relatively infrequently in the extant literature. Here we present
sample covariance asymptotics stemming from random matrix theory, and bring
them to bear on the problem of optimal rank estimation in the context of the
standard array observation model with additive white Gaussian noise. The most
significant of these results demonstrates the existence of a phase transition
threshold, below which eigenvalues and associated eigenvectors of the sample
covariance fail to provide any information on population eigenvalues. We then
develop a decision-theoretic rank estimation framework that leads to a simple
ordered selection rule based on thresholding; in contrast to competing
approaches, however, it admits asymptotic minimax optimality and is free of
tuning parameters. We analyze the asymptotic performance of our rank selection
procedure and conclude with a brief simulation study demonstrating its
practical efficacy in the context of subspace tracking.Comment: 10 pages, 4 figures; final versio
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