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Gotowa bibliografia na temat „Smoothed functional algorithms”

Autor: Grafiati

Data publikacji: 6 września 2023

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Artykuły w czasopismach na temat "Smoothed functional algorithms"

Bhatnagar, Shalabh. "Adaptive Newton-based multivariate smoothed functional algorithms for simulation optimization". ACM Transactions on Modeling and Computer Simulation 18, nr 1 (grudzień 2007): 1–35. http://dx.doi.org/10.1145/1315575.1315577.

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Bhatnagar, Shalabh, i Vivek S. Borkar. "Multiscale Chaotic SPSA and Smoothed Functional Algorithms for Simulation Optimization". SIMULATION 79, nr 10 (październik 2003): 568–80. http://dx.doi.org/10.1177/0037549703039988.

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Ghoshdastidar, Debarghya, Ambedkar Dukkipati i Shalabh Bhatnagar. "Smoothed Functional Algorithms for Stochastic Optimization Using q -Gaussian Distributions". ACM Transactions on Modeling and Computer Simulation 24, nr 3 (2.05.2014): 1–26. http://dx.doi.org/10.1145/2628434.

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Ghoshdastidar, Debarghya, Ambedkar Dukkipati i Shalabh Bhatnagar. "Newton-based stochastic optimization using q-Gaussian smoothed functional algorithms". Automatica 50, nr 10 (październik 2014): 2606–14. http://dx.doi.org/10.1016/j.automatica.2014.08.021.

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Prasad, H. L., L. A. Prashanth, Shalabh Bhatnagar i Nirmit Desai. "Adaptive Smoothed Functional Algorithms for Optimal Staffing Levels in Service Systems". Service Science 5, nr 1 (marzec 2013): 29–55. http://dx.doi.org/10.1287/serv.1120.0035.

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Lakshmanan, K., i Shalabh Bhatnagar. "Quasi-Newton smoothed functional algorithms for unconstrained and constrained simulation optimization". Computational Optimization and Applications 66, nr 3 (15.09.2016): 533–56. http://dx.doi.org/10.1007/s10589-016-9875-4.

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NOROUZZADEH, P., B. RAHMANI i M. S. NOROUZZADEH. "FORECASTING SMOOTHED NON-STATIONARY TIME SERIES USING GENETIC ALGORITHMS". International Journal of Modern Physics C 18, nr 06 (czerwiec 2007): 1071–86. http://dx.doi.org/10.1142/s0129183107011133.

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We introduce kernel smoothing method to extract the global trend of a time series and remove short time scales variations and fluctuations from it. A multifractal detrended fluctuation analysis (MF-DFA) shows that the multifractality nature of TEPIX returns time series is due to both fatness of the probability density function of returns and long range correlations between them. MF-DFA results help us to understand how genetic algorithm and kernel smoothing methods act. Then we utilize a recently developed genetic algorithm for carrying out successful forecasts of the trend in financial time series and deriving a functional form of Tehran price index (TEPIX) that best approximates the time variability of it. The final model is mainly dominated by a linear relationship with the most recent past value, while contributions from nonlinear terms to the total forecasting performance are rather small.

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Kostjukov, V. A., M. Y. Medvedev i V. Kh Pshikhopov. "Algorithms for Planning Smoothed Individual Trajectories of Ground Robots". Mekhatronika, Avtomatizatsiya, Upravlenie 23, nr 11 (3.11.2022): 585–95. http://dx.doi.org/10.17587/mau.23.585-595.

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The article is devoted to the development of an algorithm for constructing the trajectory of a robotic platform moving in an environment with obstacles. This algorithm is based on the application of a special local optimization procedure at each planning step and allows us to obtain feasible program trajectories without increasing the computational complexity of algorithms compared to existing methods. The algorithm is based on the application of the improved method of potential fields and subsequent smoothing of the resulting trajectory. The improving of the potential field method consists in a new way of detecting and avoiding local minima. When a local minimum is detected, it is added to the map as an additional obstacle, which makes it possible to avoid it during further trajectory planning. To circumvent obstacles that can be approximated by polygons, the method of the effective point to the obstacle is proposed, which is the equivalent of the latter in relation to the current location of the moving robotic platform when using this planning method. A two-stage technique for smoothing piecewise linear trajectories is proposed. It is assumed that there is some initial suboptimal curve found by any planning method. This curve is optimized using a functional that includes the length of the trajectory and the deviation of the optimized curve from the original curve. At the second stage, the linear segments of the planned straight line are conjugated with second-order curves. As a result, the planned trajectory of motion is a quadratic-linear curve with a smooth function of the trajectory velocity. At the same time, the proposed method of coupling rectilinear sections of the trajectory does not require sudden changes in speed when passing turns. Simulation results confirming the effectiveness of the proposed method of planning the trajectories of robots are considered and discussed.

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Vijayan, Nithia, i Prashanth L.A. "Smoothed functional-based gradient algorithms for off-policy reinforcement learning: A non-asymptotic viewpoint". Systems & Control Letters 155 (wrzesień 2021): 104988. http://dx.doi.org/10.1016/j.sysconle.2021.104988.

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Cheng, Chuen-Sheng, Pei-Wen Chen i Yu-Tang Wu. "Phase I Analysis of Nonlinear Profiles Using Anomaly Detection Techniques". Applied Sciences 13, nr 4 (7.02.2023): 2147. http://dx.doi.org/10.3390/app13042147.

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In various industries, the process or product quality is evaluated by a functional relationship between a dependent variable y and one or a few input variables x, expressed as y=fx. This relationship is called a profile in the literature. Recently, profile monitoring has received a lot of research attention. In this study, we formulated profile monitoring as an anomaly-detection problem and proposed an outlier-detection procedure for phase I nonlinear profile analysis. The developed procedure consists of three key processes. First, we obtained smoothed nonlinear profiles using the spline smoothing method. Second, we proposed a method for estimating the proportion of outliers in the dataset. A distance-based decision function was developed to identify potential outliers and provide a rough estimate of the contamination rate. Finally, PCA was used as a dimensionality reduction method. An outlier-detection algorithm was then employed to identify outlying profiles based on the estimated contamination rate. The algorithms considered in this study included Local Outlier Factor (LOF), Elliptic Envelope (EE), and Isolation Forest (IF). The proposed procedure was evaluated using a nonlinear profile that has been studied by various researchers. We compared various competing methods based on commonly used metrics such as type I error, type II error, and F2 score. Based on the evaluation metrics, our experimental results indicate that the performance of the proposed method is better than other existing methods. When considering the smallest and hardest-to-detect variation, the LOF algorithm, with the contamination rate determined by the method proposed in this study, achieved type I errors, type II errors, and F2 scores of 0.049, 0.001, and 0.951, respectively, while the performance metrics of the current best method were 0.081, 0.015, and 0.899, respectively.

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Rozprawy doktorskie na temat "Smoothed functional algorithms"

Sällberg, Gustav, i Pontus Söderbäck. "Thesis - Optimizing Smooth Local Volatility Surfaces with Power Utility Functions". Thesis, Linköpings universitet, Produktionsekonomi, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-120090.

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The master thesis is focused on how a local volatility surfaces can be extracted by optimization with respectto smoothness and price error. The pricing is based on utility based pricing, and developed to be set in arisk neutral pricing setting. The pricing is done in a discrete multinomial recombining tree, where the timeand price increments optionally can be equidistant. An interpolation algorithm is used if the option that shallbe priced is not matched in the tree discretization. Power utility functions are utilized, where the log-utilitypreference is especially studied, which coincides with the (Kelly) portfolio that systematically outperforms anyother portfolio. A fine resolution of the discretization is generally a property that is sought after, thus a seriesof derivations for the implementation are done to restrict the computational encumbrance and thus allow finer discretization. The thesis is mainly focused on the derivation of the method rather than finding optimal parameters thatgenerate the local volatility surfaces. The method has shown that smooth surfaces can be extracted, whichconsider market prices. However, due to lacking available interest and dividend data, the pricing error increasessymmetrically for longer option maturities. However, the method shows exponential convergence and robustnessto different initial (flat) volatilities for the optimization initiation. Given an optimal smooth local volatility surface, an arbitrary payoff function can then be used to price thecorresponding option, which could be path-dependent, such as barrier options. However, only vanilla optionswill be considered in this thesis. Finally, we find that the developed

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Barajas, Leandro G. "Process Control in High-Noise Environments Using A Limited Number Of Measurements". Diss., Georgia Institute of Technology, 2003. http://hdl.handle.net/1853/7741.

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The topic of this dissertation is the derivation, development, and evaluation of novel hybrid algorithms for process control that use a limited number of measurements and that are suitable to operate in the presence of large amounts of process noise. As an initial step, affine and neural network statistical process models are developed in order to simulate the steady-state system behavior. Such models are vitally important in the evaluation, testing, and improvement of all other process controllers referred to in this work. Afterwards, fuzzy logic controller rules are assimilated into a mathematical characterization of a model that includes the modes and mode transition rules that define a hybrid hierarchical process control. The main processing entity in such framework is a closed-loop control algorithm that performs global and then local optimizations in order to asymptotically reach minimum bias error; this is done while requiring a minimum number of iterations in order to promptly reach a desired operational window. The results of this research are applied to surface mount technology manufacturing-lines yield optimization. This work achieves a practical degree of control over the solder-paste volume deposition in the Stencil Printing Process (SPP). Results show that it is possible to change the operating point of the process by modifying certain machine parameters and even compensate for the difference in height due to change in print direction.

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Vestin, Albin, i Gustav Strandberg. "Evaluation of Target Tracking Using Multiple Sensors and Non-Causal Algorithms". Thesis, Linköpings universitet, Reglerteknik, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-160020.

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Today, the main research field for the automotive industry is to find solutions for active safety. In order to perceive the surrounding environment, tracking nearby traffic objects plays an important role. Validation of the tracking performance is often done in staged traffic scenarios, where additional sensors, mounted on the vehicles, are used to obtain their true positions and velocities. The difficulty of evaluating the tracking performance complicates its development. An alternative approach studied in this thesis, is to record sequences and use non-causal algorithms, such as smoothing, instead of filtering to estimate the true target states. With this method, validation data for online, causal, target tracking algorithms can be obtained for all traffic scenarios without the need of extra sensors. We investigate how non-causal algorithms affects the target tracking performance using multiple sensors and dynamic models of different complexity. This is done to evaluate real-time methods against estimates obtained from non-causal filtering. Two different measurement units, a monocular camera and a LIDAR sensor, and two dynamic models are evaluated and compared using both causal and non-causal methods. The system is tested in two single object scenarios where ground truth is available and in three multi object scenarios without ground truth. Results from the two single object scenarios shows that tracking using only a monocular camera performs poorly since it is unable to measure the distance to objects. Here, a complementary LIDAR sensor improves the tracking performance significantly. The dynamic models are shown to have a small impact on the tracking performance, while the non-causal application gives a distinct improvement when tracking objects at large distances. Since the sequence can be reversed, the non-causal estimates are propagated from more certain states when the target is closer to the ego vehicle. For multiple object tracking, we find that correct associations between measurements and tracks are crucial for improving the tracking performance with non-causal algorithms.

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Lakshmanan, K. "Online Learning and Simulation Based Algorithms for Stochastic Optimization". Thesis, 2012. http://hdl.handle.net/2005/3245.

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In many optimization problems, the relationship between the objective and parameters is not known. The objective function itself may be stochastic such as a long-run average over some random cost samples. In such cases finding the gradient of the objective is not possible. It is in this setting that stochastic approximation algorithms are used. These algorithms use some estimates of the gradient and are stochastic in nature. Amongst gradient estimation techniques, Simultaneous Perturbation Stochastic Approximation (SPSA) and Smoothed Functional(SF) scheme are widely used. In this thesis we have proposed a novel multi-time scale quasi-Newton based smoothed functional (QN-SF) algorithm for unconstrained as well as constrained optimization. The algorithm uses the smoothed functional scheme for estimating the gradient and the quasi-Newton method to solve the optimization problem. The algorithm is shown to converge with probability one. We have also provided here experimental results on the problem of optimal routing in a multi-stage network of queues. Policies like Join the Shortest Queue or Least Work Left assume knowledge of the queue length values that can change rapidly or hard to estimate. If the only information available is the expected end-to-end delay as with our case, such policies cannot be used. The QN-SF based probabilistic routing algorithm uses only the total end-to-end delay for tuning the probabilities. We observe from the experiments that the QN-SF algorithm has better performance than the gradient and Jacobi versions of Newton based smoothed functional algorithms. Next we consider constrained routing in a similar queueing network. We extend the QN-SF algorithm to this case. We study the convergence behavior of the algorithm and observe that the constraints are satisfied at the point of convergence. We provide experimental results for the constrained routing setup as well. Next we study reinforcement learning algorithms which are useful for solving Markov Decision Process(MDP) when the precise information on transition probabilities is not known. When the state, and action sets are very large, it is not possible to store all the state-action tuples. In such cases, function approximators like neural networks have been used. The popular Q-learning algorithm is known to diverge when used with linear function approximation due to the ’off-policy’ problem. Hence developing stable learning algorithms when used with function approximation is an important problem. We present in this thesis a variant of Q-learning with linear function approximation that is based on two-timescale stochastic approximation. The Q-value parameters for a given policy in our algorithm are updated on the slower timescale while the policy parameters themselves are updated on the faster scale. We perform a gradient search in the space of policy parameters. Since the objective function and hence the gradient are not analytically known, we employ the efficient one-simulation simultaneous perturbation stochastic approximation(SPSA) gradient estimates that employ Hadamard matrix based deterministic perturbations. Our algorithm has the advantage that, unlike Q-learning, it does not suffer from high oscillations due to the off-policy problem when using function approximators. Whereas it is difficult to prove convergence of regular Q-learning with linear function approximation because of the off-policy problem, we prove that our algorithm which is on-policy is convergent. Numerical results on a multi-stage stochastic shortest path problem show that our algorithm exhibits significantly better performance and is more robust as compared to Q-learning. Future work would be to compare it with other policy-based reinforcement learning algorithms. Finally, we develop an online actor-critic reinforcement learning algorithm with function approximation for a problem of control under inequality constraints. We consider the long-run average cost Markov decision process(MDP) framework in which both the objective and the constraint functions are suitable policy-dependent long-run averages of certain sample path functions. The Lagrange multiplier method is used to handle the inequality constraints. We prove the asymptotic almost sure convergence of our algorithm to a locally optimal solution. We also provide the results of numerical experiments on a problem of routing in a multistage queueing network with constraints on long-run average queue lengths. We observe that our algorithm exhibits good performance on this setting and converges to a feasible point.

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Książki na temat "Smoothed functional algorithms"

Moerder, Daniel D. Constrained minimization of smooth functions using a genetic algorithm. Hampton: National Aeronautics and Space Administration, Langley Research Center, 1994.

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N, Pamadi Bandu, i Langley Research Center, red. Constrained minimization of smooth functions using a genetic algorithm. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1994.

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Fitting Smooth Functions to Data. American Mathematical Society, 2020.

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Części książek na temat "Smoothed functional algorithms"

Bhatnagar, S., H. Prasad i L. Prashanth. "Smoothed Functional Gradient Schemes". W Stochastic Recursive Algorithms for Optimization, 77–102. London: Springer London, 2013. http://dx.doi.org/10.1007/978-1-4471-4285-0_6.

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Bhatnagar, S., H. Prasad i L. Prashanth. "Newton-Based Smoothed Functional Algorithms". W Stochastic Recursive Algorithms for Optimization, 133–48. London: Springer London, 2013. http://dx.doi.org/10.1007/978-1-4471-4285-0_8.

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Lakshmanan, K., i Shalabh Bhatnagar. "Smoothed Functional and Quasi-Newton Algorithms for Routing in Multi-stage Queueing Network with Constraints". W Distributed Computing and Internet Technology, 175–86. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-19056-8_12.

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Bläser, Markus, Bodo Manthey i B. V. Raghavendra Rao. "Smoothed Analysis of Partitioning Algorithms for Euclidean Functionals". W Lecture Notes in Computer Science, 110–21. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-22300-6_10.

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J. Zaslavski, Alexander. "Gradient Algorithm with a Smooth Objective Function". W Convex Optimization with Computational Errors, 127–50. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-37822-6_4.

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Zaslavski, Alexander J. "Gradient Algorithm with a Smooth Objective Function". W Springer Optimization and Its Applications, 59–72. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-30921-7_4.

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Mok, RenHao, i Mohd Ashraf Ahmad. "Power Production Optimization of Model-Free Wind Farm Using Smoothed Functional Algorithm". W Lecture Notes in Electrical Engineering, 679–89. Singapore: Springer Singapore, 2022. http://dx.doi.org/10.1007/978-981-16-8690-0_60.

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Giesl, Peter. "Towards Calculating the Basin of Attraction of Non-Smooth Dynamical Systems Using Radial Basis Functions". W Approximation Algorithms for Complex Systems, 205–25. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-16876-5_9.

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Cruttwell, Geoffrey S. H., Bruno Gavranović, Neil Ghani, Paul Wilson i Fabio Zanasi. "Categorical Foundations of Gradient-Based Learning". W Programming Languages and Systems, 1–28. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-030-99336-8_1.

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AbstractWe propose a categorical semantics of gradient-based machine learning algorithms in terms of lenses, parametric maps, and reverse derivative categories. This foundation provides a powerful explanatory and unifying framework: it encompasses a variety of gradient descent algorithms such as ADAM, AdaGrad, and Nesterov momentum, as well as a variety of loss functions such as MSE and Softmax cross-entropy, shedding new light on their similarities and differences. Our approach to gradient-based learning has examples generalising beyond the familiar continuous domains (modelled in categories of smooth maps) and can be realized in the discrete setting of boolean circuits. Finally, we demonstrate the practical significance of our framework with an implementation in Python.

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Bredies, Kristian. "Recovering Piecewise Smooth Multichannel Images by Minimization of Convex Functionals with Total Generalized Variation Penalty". W Efficient Algorithms for Global Optimization Methods in Computer Vision, 44–77. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014. http://dx.doi.org/10.1007/978-3-642-54774-4_3.

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Streszczenia konferencji na temat "Smoothed functional algorithms"

Ghoshdastidar, Debarghya, Ambedkar Dukkipati i Shalabh Bhatnagar. "q-Gaussian based Smoothed Functional algorithms for stochastic optimization". W 2012 IEEE International Symposium on Information Theory - ISIT. IEEE, 2012. http://dx.doi.org/10.1109/isit.2012.6283013.

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Segeth, Karel. "Multivariate smooth interpolation that employs polyharmonic functions". W Programs and Algorithms of Numerical Mathematics 19. Institute of Mathematics, Czech Academy of Sciences, 2019. http://dx.doi.org/10.21136/panm.2018.15.

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Khamisov, O. V. "Optimization with quadratic support functions in nonconvex smooth optimization". W NUMERICAL COMPUTATIONS: THEORY AND ALGORITHMS (NUMTA–2016): Proceedings of the 2nd International Conference “Numerical Computations: Theory and Algorithms”. Author(s), 2016. http://dx.doi.org/10.1063/1.4965331.

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Mohanty, Amit, i Bin Yao. "Indirect Adaptive Robust Control of Uncertain Systems With Unknown Asymmetric Input Deadband Using a Smooth Inverse". W ASME 2009 Dynamic Systems and Control Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/dscc2009-2771.

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In this paper, we present an indirect adaptive robust controller (IARC) for output tracking of a class of uncertain nonlinear systems with unknown input asymmetric deadband in presence of uncertain nonlinearities and parametric uncertainties. Most of the parameter adaptation algorithms, such as, gradient-type and least squares-type require that the unknown parameters of a system appear in affine with known regressor functions globally. However, deadband nonlinearity can not be represented in those global linear parametric form. Therefore, the existing parameter estimation algorithms for deadband focus on some approximate linear parametric model. Hence, even in absence of any other uncertain nonlinearities and disturbances, these algorithms can never achieve asymptotic tracking. Departing from those approximate deadband estimation, we design an indirect parameter estimation algorithm with online condition monitoring. This parameter estimation algorithm in conjunction with a well-designed robust controller and a deadband inverse function can be used to obtain asymptotic tracking without restoring to discontinuous control law. With this strong result in our repertoire, we proceed to design a smooth deadband inverse (SDI) function to avoid certain problems during implementation, e.g, control input chattering and significant appearance of high-frequency dynamics. The effect of such an approximation on the L2-norm of output tracking error is analytically determined. We also show that while operating away from the deadband, the proposed controller even with an SDI can achieve asymptotic tracking. In presence of disturbances and other uncertain nonlinearities, the proposed IARC controller attains guaranteed transient performance and final tracking accuracy.

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Juan Geng, Lai-Sheng Wang, Ai-Min Fu i Qi-Qing Song. "A smoothed rank function algorithm based Hyperbolic Tangent function for matrix completion". W 2012 International Conference on Machine Learning and Cybernetics (ICMLC). IEEE, 2012. http://dx.doi.org/10.1109/icmlc.2012.6359558.

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Zhifu Cui, Hang Zhang i Wei Lu. "An improved smoothed l0-norm algorithm based on multiparameter approximation function". W 2010 12th IEEE International Conference on Communication Technology (ICCT). IEEE, 2010. http://dx.doi.org/10.1109/icct.2010.5688553.

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Shadloo, Mostafa Safdari, Amir Zainali i Mehmet Yildiz. "Fluid-Structure Interaction Simulation by Smoothed Particle Hydrodynamics". W ASME 2010 3rd Joint US-European Fluids Engineering Summer Meeting collocated with 8th International Conference on Nanochannels, Microchannels, and Minichannels. ASMEDC, 2010. http://dx.doi.org/10.1115/fedsm-icnmm2010-31137.

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In this article, a modified SPH algorithm is proposed to solve Fluid-Structure Interaction (FSI) problems including fluid flow in interaction with compatible structures under a large deformation. To validate the current algorithm against available data in literature, we consider two important benchmark cases; namely, an oscillating elastic beam and dam breaking problems. The proposed algorithm is based on the elimination of the intermediate data transfer steps between the fluid and the solid structures, whereby resulting in an easy and time-saving simulation method. With the test application studied, we were able to prove the ability of the modified SPH method for solving of fluid and solid domains monolithically without the need to define an interfacial boundary condition or any additional steps to simulate the deformation of an elastic dam. Numerical results suggest that upon choosing correct SPH parameters such as smoothing function, and lengths, as well as coefficients for artificial viscosity and artificial stress, one can obtain results in satisfactorily agreement with numerical findings of earlier works.

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Borup, Liana, i Alan Parkinson. "Comparison of Four Non-Derivative Optimization Methods on Two Problems Containing Heuristic and Analytic Knowledge". W ASME 1992 Design Technical Conferences. American Society of Mechanical Engineers, 1992. http://dx.doi.org/10.1115/detc1992-0114.

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Abstract Hybrid engineering models — models that combine both analytic and heuristic knowledge — can address a wide range of design issues. However, models that combine these two types of information are often non-smooth and thus cannot be optimized by traditional derivative-based optimization algorithms. This paper is a study of four non-derivative optimization methods: Simulated Annealing, Genetic Algorithms, Flexible Polyhedron Search, and Function Approximation. Each algorithm was tested on two engineering problems that combine heuristic and analytic information. The algorithms were then evaluated with respect to ease of implementation, robustness, and efficiency. Although each algorithm has advantages/disadvantages that may make it preferable for specific problems, in this study the Flexible Polyhedron Search was the best algorithm both in terms of ease of implementation and efficiency. This method found good designs in a relatively few number of design analyses. It was not, however, as robust as Simulated Annealing.

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Venkataraman, P. "Determining the Ordinary Differential Equation From Noisy Data". W ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2011. http://dx.doi.org/10.1115/detc2011-47658.

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A challenging inverse problem is to identify the smooth function and the differential equation it represents from uncertain data. This paper extends the procedure previously developed for smooth data. The approach involves two steps. In the first step the data is smoothed using a recursive Bezier filter. For smooth data a single application of the filter is sufficient. The final set of data points provides a smooth estimate of the solution. More importantly, it will also identify smooth derivatives of the function away from the edges of the domain. In the second step the values of the function and its derivatives are used to establish a specific form of the differential equation from a particular class of the same. Since the function and its derivatives are known, the only unknowns are parameters describing the structure of the differential equations. These parameters are of two kinds: the exponents of the derivatives and the coefficients of the terms in the differential equations. These parameters can be determined by defining an optimization problem based on the residuals in a reduced domain. To avoid the trivial solution a discrete global search is used to identify these parameters. An example involving a third order constant coefficient linear differential equation is presented. A basic simulated annealing algorithm is used for the global search. Once the differential form is established, the unknown initial and boundary conditions can be obtained by backward and forward numerical integration from the reduced region.

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Xiao, Yichi, Zhe Li, Tianbao Yang i Lijun Zhang. "SVD-free Convex-Concave Approaches for Nuclear Norm Regularization". W Twenty-Sixth International Joint Conference on Artificial Intelligence. California: International Joint Conferences on Artificial Intelligence Organization, 2017. http://dx.doi.org/10.24963/ijcai.2017/436.

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Minimizing a convex function of matrices regularized by the nuclear norm arises in many applications such as collaborative filtering and multi-task learning. In this paper, we study the general setting where the convex function could be non-smooth. When the size of the data matrix, denoted by m x n, is very large, existing optimization methods are inefficient because in each iteration, they need to perform a singular value decomposition (SVD) which takes O(m^2 n) time. To reduce the computation cost, we exploit the dual characterization of the nuclear norm to introduce a convex-concave optimization problem and design a subgradient-based algorithm without performing SVD. In each iteration, the proposed algorithm only computes the largest singular vector, reducing the time complexity from O(m^2 n) to O(mn). To the best of our knowledge, this is the first SVD-free convex optimization approach for nuclear-norm regularized problems that does not rely on the smoothness assumption. Theoretical analysis shows that the proposed algorithm converges at an optimal O(1/\sqrt{T}) rate where T is the number of iterations. We also extend our algorithm to the stochastic case where only stochastic subgradients of the convex function are available and a special case that contains an additional non-smooth regularizer (e.g., L1 norm regularizer). We conduct experiments on robust low-rank matrix approximation and link prediction to demonstrate the efficiency of our algorithms.

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