30,579 research outputs found
Stochastic Behavior Analysis of the Gaussian Kernel Least-Mean-Square Algorithm
The kernel least-mean-square (KLMS) algorithm is a popular algorithm in nonlinear adaptive filtering due to its
simplicity and robustness. In kernel adaptive filters, the statistics of the input to the linear filter depends on the parameters of the kernel employed. Moreover, practical implementations require a finite nonlinearity model order. A Gaussian KLMS has two design parameters, the step size and the Gaussian kernel bandwidth. Thus, its design requires analytical models for the algorithm behavior as a function of these two parameters. This paper studies the steady-state behavior and the transient behavior of the
Gaussian KLMS algorithm for Gaussian inputs and a finite order nonlinearity model. In particular, we derive recursive expressions for the mean-weight-error vector and the mean-square-error. The model predictions show excellent agreement with Monte Carlo simulations in transient and steady state. This allows the explicit analytical determination of stability limits, and gives opportunity
to choose the algorithm parameters a priori in order to achieve prescribed convergence speed and quality of the estimate. Design examples are presented which validate the theoretical analysis and illustrates its application
Adaptive Graph Signal Processing: Algorithms and Optimal Sampling Strategies
The goal of this paper is to propose novel strategies for adaptive learning
of signals defined over graphs, which are observed over a (randomly
time-varying) subset of vertices. We recast two classical adaptive algorithms
in the graph signal processing framework, namely, the least mean squares (LMS)
and the recursive least squares (RLS) adaptive estimation strategies. For both
methods, a detailed mean-square analysis illustrates the effect of random
sampling on the adaptive reconstruction capability and the steady-state
performance. Then, several probabilistic sampling strategies are proposed to
design the sampling probability at each node in the graph, with the aim of
optimizing the tradeoff between steady-state performance, graph sampling rate,
and convergence rate of the adaptive algorithms. Finally, a distributed RLS
strategy is derived and is shown to be convergent to its centralized
counterpart. Numerical simulations carried out over both synthetic and real
data illustrate the good performance of the proposed sampling and
reconstruction strategies for (possibly distributed) adaptive learning of
signals defined over graphs.Comment: Submitted to IEEE Transactions on Signal Processing, September 201
Fast Cross-Validation via Sequential Testing
With the increasing size of today's data sets, finding the right parameter
configuration in model selection via cross-validation can be an extremely
time-consuming task. In this paper we propose an improved cross-validation
procedure which uses nonparametric testing coupled with sequential analysis to
determine the best parameter set on linearly increasing subsets of the data. By
eliminating underperforming candidates quickly and keeping promising candidates
as long as possible, the method speeds up the computation while preserving the
capability of the full cross-validation. Theoretical considerations underline
the statistical power of our procedure. The experimental evaluation shows that
our method reduces the computation time by a factor of up to 120 compared to a
full cross-validation with a negligible impact on the accuracy
Distributed Adaptive Learning with Multiple Kernels in Diffusion Networks
We propose an adaptive scheme for distributed learning of nonlinear functions
by a network of nodes. The proposed algorithm consists of a local adaptation
stage utilizing multiple kernels with projections onto hyperslabs and a
diffusion stage to achieve consensus on the estimates over the whole network.
Multiple kernels are incorporated to enhance the approximation of functions
with several high and low frequency components common in practical scenarios.
We provide a thorough convergence analysis of the proposed scheme based on the
metric of the Cartesian product of multiple reproducing kernel Hilbert spaces.
To this end, we introduce a modified consensus matrix considering this specific
metric and prove its equivalence to the ordinary consensus matrix. Besides, the
use of hyperslabs enables a significant reduction of the computational demand
with only a minor loss in the performance. Numerical evaluations with synthetic
and real data are conducted showing the efficacy of the proposed algorithm
compared to the state of the art schemes.Comment: Double-column 15 pages, 10 figures, submitted to IEEE Trans. Signal
Processin
Statistical properties of determinantal point processes in high-dimensional Euclidean spaces
The goal of this paper is to quantitatively describe some statistical
properties of higher-dimensional determinantal point processes with a primary
focus on the nearest-neighbor distribution functions. Toward this end, we
express these functions as determinants of matrices and then
extrapolate to . This formulation allows for a quick and accurate
numerical evaluation of these quantities for point processes in Euclidean
spaces of dimension . We also implement an algorithm due to Hough \emph{et.
al.} \cite{hough2006dpa} for generating configurations of determinantal point
processes in arbitrary Euclidean spaces, and we utilize this algorithm in
conjunction with the aforementioned numerical results to characterize the
statistical properties of what we call the Fermi-sphere point process for to 4. This homogeneous, isotropic determinantal point process, discussed
also in a companion paper \cite{ToScZa08}, is the high-dimensional
generalization of the distribution of eigenvalues on the unit circle of a
random matrix from the circular unitary ensemble (CUE). In addition to the
nearest-neighbor probability distribution, we are able to calculate Voronoi
cells and nearest-neighbor extrema statistics for the Fermi-sphere point
process and discuss these as the dimension is varied. The results in this
paper accompany and complement analytical properties of higher-dimensional
determinantal point processes developed in \cite{ToScZa08}.Comment: 42 pages, 17 figure
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