828 research outputs found
Clustering via kernel decomposition
Spectral clustering methods were proposed recently which rely on the eigenvalue decomposition of an affinity matrix. In this letter, the affinity matrix is created from the elements of a nonparametric density estimator and then decomposed to obtain posterior probabilities of class membership. Hyperparameters are selected using standard cross-validation methods
Finding Structural Information of RF Power Amplifiers using an Orthogonal Non-Parametric Kernel Smoothing Estimator
A non-parametric technique for modeling the behavior of power amplifiers is
presented. The proposed technique relies on the principles of density
estimation using the kernel method and is suited for use in power amplifier
modeling. The proposed methodology transforms the input domain into an
orthogonal memory domain. In this domain, non-parametric static functions are
discovered using the kernel estimator. These orthogonal, non-parametric
functions can be fitted with any desired mathematical structure, thus
facilitating its implementation. Furthermore, due to the orthogonality, the
non-parametric functions can be analyzed and discarded individually, which
simplifies pruning basis functions and provides a tradeoff between complexity
and performance. The results show that the methodology can be employed to model
power amplifiers, therein yielding error performance similar to
state-of-the-art parametric models. Furthermore, a parameter-efficient model
structure with 6 coefficients was derived for a Doherty power amplifier,
therein significantly reducing the deployment's computational complexity.
Finally, the methodology can also be well exploited in digital linearization
techniques.Comment: Matlab sample code (15 MB):
https://dl.dropboxusercontent.com/u/106958743/SampleMatlabKernel.zi
Backstepping PDE Design: A Convex Optimization Approach
Abstract\u2014Backstepping design for boundary linear PDE is
formulated as a convex optimization problem. Some classes of
parabolic PDEs and a first-order hyperbolic PDE are studied,
with particular attention to non-strict feedback structures. Based
on the compactness of the Volterra and Fredholm-type operators
involved, their Kernels are approximated via polynomial
functions. The resulting Kernel-PDEs are optimized using Sumof-
Squares (SOS) decomposition and solved via semidefinite
programming, with sufficient precision to guarantee the stability
of the system in the L2-norm. This formulation allows optimizing
extra degrees of freedom where the Kernel-PDEs are included
as constraints. Uniqueness and invertibility of the Fredholm-type
transformation are proved for polynomial Kernels in the space
of continuous functions. The effectiveness and limitations of the
approach proposed are illustrated by numerical solutions of some
Kernel-PDEs
Stochastic Physics-Informed Neural Ordinary Differential Equations
Stochastic differential equations (SDEs) are used to describe a wide variety
of complex stochastic dynamical systems. Learning the hidden physics within
SDEs is crucial for unraveling fundamental understanding of these systems'
stochastic and nonlinear behavior. We propose a flexible and scalable framework
for training artificial neural networks to learn constitutive equations that
represent hidden physics within SDEs. The proposed stochastic physics-informed
neural ordinary differential equation framework (SPINODE) propagates
stochasticity through the known structure of the SDE (i.e., the known physics)
to yield a set of deterministic ODEs that describe the time evolution of
statistical moments of the stochastic states. SPINODE then uses ODE solvers to
predict moment trajectories. SPINODE learns neural network representations of
the hidden physics by matching the predicted moments to those estimated from
data. Recent advances in automatic differentiation and mini-batch gradient
descent with adjoint sensitivity are leveraged to establish the unknown
parameters of the neural networks. We demonstrate SPINODE on three benchmark
in-silico case studies and analyze the framework's numerical robustness and
stability. SPINODE provides a promising new direction for systematically
unraveling the hidden physics of multivariate stochastic dynamical systems with
multiplicative noise
Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives
Part 2 of this monograph builds on the introduction to tensor networks and
their operations presented in Part 1. It focuses on tensor network models for
super-compressed higher-order representation of data/parameters and related
cost functions, while providing an outline of their applications in machine
learning and data analytics. A particular emphasis is on the tensor train (TT)
and Hierarchical Tucker (HT) decompositions, and their physically meaningful
interpretations which reflect the scalability of the tensor network approach.
Through a graphical approach, we also elucidate how, by virtue of the
underlying low-rank tensor approximations and sophisticated contractions of
core tensors, tensor networks have the ability to perform distributed
computations on otherwise prohibitively large volumes of data/parameters,
thereby alleviating or even eliminating the curse of dimensionality. The
usefulness of this concept is illustrated over a number of applied areas,
including generalized regression and classification (support tensor machines,
canonical correlation analysis, higher order partial least squares),
generalized eigenvalue decomposition, Riemannian optimization, and in the
optimization of deep neural networks. Part 1 and Part 2 of this work can be
used either as stand-alone separate texts, or indeed as a conjoint
comprehensive review of the exciting field of low-rank tensor networks and
tensor decompositions.Comment: 232 page
Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives
Part 2 of this monograph builds on the introduction to tensor networks and
their operations presented in Part 1. It focuses on tensor network models for
super-compressed higher-order representation of data/parameters and related
cost functions, while providing an outline of their applications in machine
learning and data analytics. A particular emphasis is on the tensor train (TT)
and Hierarchical Tucker (HT) decompositions, and their physically meaningful
interpretations which reflect the scalability of the tensor network approach.
Through a graphical approach, we also elucidate how, by virtue of the
underlying low-rank tensor approximations and sophisticated contractions of
core tensors, tensor networks have the ability to perform distributed
computations on otherwise prohibitively large volumes of data/parameters,
thereby alleviating or even eliminating the curse of dimensionality. The
usefulness of this concept is illustrated over a number of applied areas,
including generalized regression and classification (support tensor machines,
canonical correlation analysis, higher order partial least squares),
generalized eigenvalue decomposition, Riemannian optimization, and in the
optimization of deep neural networks. Part 1 and Part 2 of this work can be
used either as stand-alone separate texts, or indeed as a conjoint
comprehensive review of the exciting field of low-rank tensor networks and
tensor decompositions.Comment: 232 page
Control-Relevant System Identification using Nonlinear Volterra and Volterra-Laguerre Models
One of the key impediments to the wide-spread use of nonlinear control in industry is the availability of suitable nonlinear models. Empirical models, which are obtained from only the process input-output data, present a convenient alternative to the more involved fundamental models. An important advantage of the empirical models is that their structure can be chosen so as to facilitate the controller design problem. Many of the widely used empirical model structures are linear, and in some cases this basic model formulation may not be able to adequately capture the nonlinear process dynamics. One of the commonly used nonlinear dynamic empirical model structures is the Volterra model, and this work develops a systematic approach to the identification of third-order Volterra and Volterra-Laguerre models from process input-output data.First, plant-friendly input sequences are designed that exploit the Volterra model structure and use the prediction error variance (PEV) expression as a metric of model fidelity. Second, explicit estimator equations are derived for the linear, nonlinear diagonal, and higher-order sub-diagonal kernels using the tailored input sequences. Improvements in the sequence design are also presented which lead to a significant reduction in the amount of data required for identification. Finally, the third-order off-diagonal kernels are estimated using a cross-correlation approach. As an application of this technique, an isothermal polymerization reactor case study is considered.In order to overcome the noise sensitivity and highly parameterized nature of Volterra models, they are projected onto an orthonormal Laguerre basis. Two important variables that need to be selected for the projection are the Laguerre pole and the number of Laguerre filters. The Akaike Information Criterion (AIC) is used as a criterion to determine projected model quality. AIC includes contributions from both model size and model quality, with the latter characterized by the sum-squared error between the Volterra and the Volterra-Laguerre model outputs. Reduced Volterra-Laguerre models were also identified, and the control-relevance of identified Volterra-Laguerre models was evaluated in closed-loop using the model predictive control framework. Thus, this work presents a complete treatment of the problem of identifying nonlinear control-relevant Volterra and Volterra-Laguerre models from input-output data
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