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A Comparison between Fixed-Basis and Variable-Basis Schemes for Function Approximation and Functional Optimization

Giorgio Gnecco
2012 Journal of Applied Mathematics  
Fixed-basis and variable-basis approximation schemes are compared for the problems of function approximation and functional optimization (also known as infinite programming).  ...  to achieve a desired error in function approximation or approximate optimization.  ...  Acknowledgment The author was partially supported by a PRIN grant from the Italian Ministry for University and Research, project "Adaptive State Estimation and Optimal Control."  ... 
doi:10.1155/2012/806945 fatcat:wn6y6fhqi5e6letwabsbmzxdzq

On a Variational Norm Tailored to Variable-Basis Approximation Schemes

Giorgio Gnecco, Marcello Sanguineti
2011 IEEE Transactions on Information Theory  
Index Terms-Approximation schemes, convex hulls, infinite-dimensional optimization, upper and lower bounds, variation with respect to a set, 1 -norm.  ...  A variational norm associated with sets of computational units and used in function approximation, learning from data, and infinite-dimensional optimization is investigated.  ...  Kainen for fruitful discussions and to one Reviewer for his/her detailed comments and suggestions, which allowed to improve and generalize some results contained in the first manuscript and to derive Propositions  ... 
doi:10.1109/tit.2010.2090198 fatcat:ne4qy6pv4bcxlofcjplfoctqqa

Dynamic Programming and Value-Function Approximation in Sequential Decision Problems: Error Analysis and Numerical Results

Mauro Gaggero, Giorgio Gnecco, Marcello Sanguineti
2012 Journal of Optimization Theory and Applications  
These properties are exploited to approximate such functions by means of certain nonlinear approximation schemes, which include splines of suitable order and Gaussian radial-basis networks with variable  ...  Numerical comparisons with classical linear approximators are presented.  ...  approximations Fixed-basis schemes Variable-basis schemes Number of discretization points of X t L L Number of optimizations L + 1 (Eqs.  ... 
doi:10.1007/s10957-012-0118-2 fatcat:3bkvo2oyy5cotmtcjresukrit4

Real-time cardiac MRI using low-rank and sparsity penalties

Sajan Goud, Yue Hu, Mathews Jacob
2010 2010 IEEE International Symposium on Biomedical Imaging: From Nano to Macro  
Experiments on numerical phantoms show a significant reduction in artifacts at high acceleration factors, in comparison to current schemes.  ...  We exploit the fact that the spatio-temporal data can be represented as the linear combination of a few temporal basis functions.  ...  The comparisons between the proposed method and classical schemes were done over a wide range of net acceleration factors, (A).  ... 
doi:10.1109/isbi.2010.5490154 dblp:conf/isbi/GoudHJ10 fatcat:zszvovkebnclhfhyrvrdvdef5q

Comparison of worst case errors in linear and neural network approximation

V. Kurkova, M. Sanguineti
2002 IEEE Transactions on Information Theory  
A theoretical framework for such a description is developed in the context of nonlinear approximation by fixed versus variable basis functions.  ...  Comparisons of approximation rates are formulated in terms of certain norms tailored to sets of basis functions. The results are applied to perceptron networks.  ...  Lugosi (Pompeu Fabra University) for his many useful comments. They also wish to thank Prof. P. C. Kainen (Georgetown University) and Prof. S. Giulini (University of Genoa) for helpful discussions.  ... 
doi:10.1109/18.971754 fatcat:amye6ekr4nfq5kexzshq4adq2i

Blind Compressive Sensing Dynamic MRI

S. G. Lingala, M. Jacob
2013 IEEE Transactions on Medical Imaging  
This scheme models the dynamic signal as a sparse linear combination of temporal basis functions, chosen from a large dictionary.  ...  We formulate the reconstruction as a constrained optimization problem; the objective function is the linear combination of a data consistency term and sparsity promoting l1 prior of the coefficients.  ...  At each step, we solve for a specific variable, assuming the other variables to be fixed; we systematically cycle through these subproblems until convergence. The subproblems are specified below.  ... 
doi:10.1109/tmi.2013.2255133 pmid:23542951 pmcid:PMC3902976 fatcat:xqxr277qurdtlnvxdmoh5ybjae

X-Y separable pyramid steerable scalable kernels

Shy, Perona
1994 Proceedings of IEEE Conference on Computer Vision and Pattern Recognition CVPR-94  
We present a method for generating compact steerable, filter kernel approximations, in which the basis kemels are x-y separable.  ...  This "pseudo-SVD " impmves upon a p w w u s scheme due to Pemna in that it reduces convolution time and storage requirements.  ...  Acknowledgements The authors wish to acknowledge the following people: Mike Burl for his suggestions and work regarding the pyramid scheme; Thomas h u n g for debugging the matlab code and helping with  ... 
doi:10.1109/cvpr.1994.323835 dblp:conf/cvpr/ShyP94 fatcat:5adzzyrrx5g6xbhuoe6tu44nqa

An Online Kernel-Based Clustering Approach for Value Function Approximation [chapter]

Nikolaos Tziortziotis, Konstantinos Blekas
2012 Lecture Notes in Computer Science  
Value function approximation is a critical task in solving Markov decision processes and accurately representing reinforcement learning agents.  ...  By considering the value function as a linear combination of the constructed basis functions, the weights are simultaneously optimized in a temporal-difference framework in order to minimize the Bellman  ...  The proposed method has been tested to several known simulated environments where we have made comparisons with a recent value function approximation approach that uses fixed Fourier basis functions.  ... 
doi:10.1007/978-3-642-30448-4_23 fatcat:4vgvowvhkfbvfjdtxlbmqwp36m

Blind compressed sensing with sparse dictionaries for accelerated dynamic MRI

Sajan Goud Lingala, Mathews Jacob
2013 2013 IEEE 10th International Symposium on Biomedical Imaging  
Several algorithms that model the voxel time series as a sparse linear combination of basis functions in a fixed dictionary were introduced to recover dynamic MRI data from under sampled Fourier measurements  ...  We have recently demonstrated that the joint estimation of dictionary basis and the sparse coefficients from the k-space data results in improved reconstructions.  ...  At each step, we solve for a specific variable, assuming the other variables to be fixed; we systematically cycle through these subproblems until convergence.  ... 
doi:10.1109/isbi.2013.6556398 pmid:24691250 pmcid:PMC3969877 dblp:conf/isbi/LingalaJ13 fatcat:3o5oi5mrlrhflim5u63e6kt2um

Accelerated whole-brain multi-parameter mapping using blind compressed sensing

Sampada Bhave, Sajan Goud Lingala, Casey P. Johnson, Vincent A. Magnotta, Mathews Jacob
2015 Magnetic Resonance in Medicine  
BCS was observed to be more robust to patient-specific motion as compared to other CS schemes and resulted in minimal degradation of parameter maps in the presence of motion.  ...  Purpose: To introduce a blind compressed sensing (BCS) framework to accelerate multi-parameter MR mapping, and demonstrate its feasibility in high-resolution, whole-brain T 1 ρ and T 2 mapping.  ...  We use a variable splitting and augmented Lagrangian optimization scheme to enforce the constraint in Eq. [4] .  ... 
doi:10.1002/mrm.25722 pmid:25850952 pmcid:PMC4598248 fatcat:mwytgsi7xbbszhn7bwhl2g7yra

Elucidating the mechanism responsible for anomalous thermal expansion in a metal–organic framework

Dewald P. van Heerden, Catharine Esterhuysen, Leonard J. Barbour
2016 Dalton Transactions  
A mechanistic model is developed to reproduce concerted changes in the internal coordinates of the coordination helix of a MOF and evaluated using DFT.  ...  The variable was stepped between 0.00 and 0.25 Å in 0.05 Å increments (yielding 5.146 Å ≤ c' ≤ 5.396 Å).  ...  Hydrogen atom positions, including those of the truncated ligand, were optimized at the various levels of theory (a specific density functional and basis set combination) for the = 0.0 Å model.  ... 
doi:10.1039/c5dt01927c pmid:26171815 fatcat:yg3sea633jaifh5zt7iwb47d7u

UAV Trajectory and Communication Co-design: Flexible Path Discretization and Path Compression [article]

Yijun Guo, Changsheng You, Changchuan Yin, Rui Zhang
2020 arXiv   pre-print
First, we propose a flexible path discretization scheme that optimizes only a number of selected waypoints (designable waypoints) along the UAV path for complexity reduction, while all the designable and  ...  To tackle the challenge of infinite design variables arising from the continuous-time UAV trajectory optimization, a commonly adopted approach is by approximating the UAV trajectory with piecewise-linear  ...  Fig. 4 : 4 Comparison of path approximation for PC based on different basis paths. basis paths of the latter scheme are equally important and thus the approximation accuracy is very sensitive to the number  ... 
arXiv:2010.07068v1 fatcat:2zk2kgs2jnfcllp4hloxreox3m

Variable neural networks for adaptive control of nonlinear systems

G.P. Liu, V. Kadirkamanathan, S.A. Billings
1999 IEEE Transactions on Systems Man and Cybernetics Part C (Applications and Reviews)  
Based on the Gaussian radial basis function (GRBF) variable neural network, an adaptive control scheme is presented.  ...  A novel neural network architecture, referred to as a variable neural network, is proposed and shown to be useful in approximating the unknown nonlinearities of dynamical systems.  ...  The fixed neural network usually needs a large number of basis functions in most cases even for a simple problem.  ... 
doi:10.1109/5326.740668 fatcat:bdbkmaef2bh6xkj7txw3lkdq6a

Quantifying Registration Uncertainty With Sparse Bayesian Modelling

Loic Le Folgoc, Herve Delingette, Antonio Criminisi, Nicholas Ayache
2017 IEEE Transactions on Medical Imaging  
Bayesian modelling has proven powerful to automate the tuning of registration hyperparameters, such as the trade-off between the data and regularization functionals.  ...  In this paper we are interested in the theoretical and empirical quality of uncertainty estimates derived under this approximate scheme vs. under the exact model.  ...  Fig. 8 (second row) and Fig. 9 (middle column) report the estimates of the mean and uncertainty for the Fixed Basis MCMC scheme.  ... 
doi:10.1109/tmi.2016.2623608 pmid:27831863 fatcat:ovykhsxc3fcqnaz55ln6csyxly

Multi-objective optimal control approach for static voltage stability of power system considering interval uncertainty of the wind farm output

Yuerong Yang, Shunjiang Lin, Qiong Wang, Yuquan Xie, Mingbo Liu
2020 IEEE Access  
A parametric approximation (PA) method is used to obtain the approximate functional relationship between the optimal objective function values and the decision variables of the inner-layer and mid-layer  ...  J 2 J P a point on the utopian linē J 1 ,J 2 normalized J 1 and J 2 λ o , λ o dependent variables before and after removing the cross-term s required s-th orthogonal basis k sj , l sj coefficients of  ...  A PA method is used to first obtain the approximate functional relationship between the optimal objective function values and decision variables of the inner-layer and mid-layer optimization models, and  ... 
doi:10.1109/access.2020.3002931 fatcat:3fs34zhp2vaslmtfobcqlcpgra
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