3,421 research outputs found
Estimation of constant and time-varying dynamic parameters of HIV infection in a nonlinear differential equation model
Modeling viral dynamics in HIV/AIDS studies has resulted in a deep
understanding of pathogenesis of HIV infection from which novel antiviral
treatment guidance and strategies have been derived. Viral dynamics models
based on nonlinear differential equations have been proposed and well developed
over the past few decades. However, it is quite challenging to use experimental
or clinical data to estimate the unknown parameters (both constant and
time-varying parameters) in complex nonlinear differential equation models.
Therefore, investigators usually fix some parameter values, from the literature
or by experience, to obtain only parameter estimates of interest from clinical
or experimental data. However, when such prior information is not available, it
is desirable to determine all the parameter estimates from data. In this paper
we intend to combine the newly developed approaches, a multi-stage
smoothing-based (MSSB) method and the spline-enhanced nonlinear least squares
(SNLS) approach, to estimate all HIV viral dynamic parameters in a nonlinear
differential equation model. In particular, to the best of our knowledge, this
is the first attempt to propose a comparatively thorough procedure, accounting
for both efficiency and accuracy, to rigorously estimate all key kinetic
parameters in a nonlinear differential equation model of HIV dynamics from
clinical data. These parameters include the proliferation rate and death rate
of uninfected HIV-targeted cells, the average number of virions produced by an
infected cell, and the infection rate which is related to the antiviral
treatment effect and is time-varying. To validate the estimation methods, we
verified the identifiability of the HIV viral dynamic model and performed
simulation studies.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS290 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Applying Bayesian Machine Learning Methods to Theoretical Surface Science
Machine learning is a rapidly evolving field in computer science with increasingly many applications to other domains. In this thesis, I present a Bayesian machine learning approach to solving a problem in theoretical surface science: calculating the preferred active site on a catalyst surface for a given adsorbate molecule. I formulate the problem as a low-dimensional objective function. I show how the objective function can be approximated into a certain confidence interval using just one iteration of the self-consistent field (SCF) loop in density functional theory (DFT). I then use Bayesian optimization to perform a global search for the solution. My approach outperforms the current state-of-the-art method, constrained minima hopping, for CO on ferric oxide by a factor of 75 to 1. This thesis is the first documented application of Bayesian optimization to surface science
Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs
Laplacian mixture models identify overlapping regions of influence in
unlabeled graph and network data in a scalable and computationally efficient
way, yielding useful low-dimensional representations. By combining Laplacian
eigenspace and finite mixture modeling methods, they provide probabilistic or
fuzzy dimensionality reductions or domain decompositions for a variety of input
data types, including mixture distributions, feature vectors, and graphs or
networks. Provable optimal recovery using the algorithm is analytically shown
for a nontrivial class of cluster graphs. Heuristic approximations for scalable
high-performance implementations are described and empirically tested.
Connections to PageRank and community detection in network analysis demonstrate
the wide applicability of this approach. The origins of fuzzy spectral methods,
beginning with generalized heat or diffusion equations in physics, are reviewed
and summarized. Comparisons to other dimensionality reduction and clustering
methods for challenging unsupervised machine learning problems are also
discussed.Comment: 13 figures, 35 reference
Learning nonparametric ordinary differential equations from noisy data
Learning nonparametric systems of Ordinary Differential Equations (ODEs) dot
x = f(t,x) from noisy data is an emerging machine learning topic. We use the
well-developed theory of Reproducing Kernel Hilbert Spaces (RKHS) to define
candidates for f for which the solution of the ODE exists and is unique.
Learning f consists of solving a constrained optimization problem in an RKHS.
We propose a penalty method that iteratively uses the Representer theorem and
Euler approximations to provide a numerical solution. We prove a generalization
bound for the L2 distance between x and its estimator and provide experimental
comparisons with the state-of-the-art.Comment: 25 pages, 6 figure
Learning Nonparametric Ordinary Differential Equations from Noisy Data
Learning nonparametric systems of Ordinary Differential Equations (ODEs) from noisy data is an emerging machine learning topic. We use the well-developed theory of Reproducing Kernel Hilbert Spaces (RKHS) to define candidates for for which the solution of the ODE exists and is unique. Learning consists of solving a constrained optimization problem in an RKHS. We propose a penalty method that iteratively uses the Representer theorem and Euler approximations to provide a numerical solution. We prove a generalization bound for the distance between and its estimator. Experiments are provided for the FitzHugh-Nagumo oscillator, the Lorenz system, and for predicting the Amyloid level in the cortex of aging subjects. In all cases, we show competitive results compared with the state-of-the-art
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
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