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Learning optimal smooth invariant subspaces for data approximation

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2024年第2期538卷

作者： Barbieri, D. Cabrelli, C. Hernandez, E. Molter, U. Univ Autonoma Madrid Madrid 28049 Spain Univ Buenos Aires RA-1428 Buenos Aires Argentina Consejo Nacl Invest Cient & Tecn IMAS RA-1428 Buenos Aires Argentina

In this article, we consider the problem of approximating a finite set of data (usually huge in applications) by invariant subspaces generated by a small set of smooth functions. The invariance is either by translations under a full-rank lattice or through the action of crystallographic groups. Smoothness is ensured by stipulating that the generators belong to a Paley-Wiener space, which is selected in an optimal way based on the characteristics of the given data. To complete our investigation, we analyze the fundamental role played by the lattice in the process of approximation. (c) 2024 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).

关键词： Invariant subspaces data approximation Paley-Wiener spaces Optimal subspaces

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Functional data approximation on bounded domains using polygonal finite elements

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COMPUTER AIDED GEOMETRIC DESIGN 2018年 63卷 149-163页

作者： Cao, Juan Xiao, Yanyang Chen, Zhonggui Wang, Wenping Bajaj, Chandrajit [a]School of Mathematical Sciences Xiamen University Xiamen 361005 China [b]Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computation Xiamen University Xiamen 361005 China [c]Department of Computer Science Xiamen University Xiamen 361005 China [d]Department of Computer Science The University of Hong Kong 999077 China [e]Department of Computer Science University of Texas at Austin Austin TX 78712 USA

We construct and analyze piecewise approximations of functional data on arbitrary 2D bounded domains using generalized barycentric finite elements, and particularly quadratic serendipity elements for planar polygons. We compare approximation qualities (precision/convergence) of these partition-of-unity finite elements through numerical experiments, using Wachspress coordinates, natural neighbor coordinates, Poisson coordinates, mean value coordinates, and quadratic serendipity bases over polygonal meshes on the domain. For a convex n-sided polygon, the quadratic serendipity elements have 2n basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, rather than the usual n(n + 1)/2 basis functions to achieve quadratic convergence. Two greedy algorithms are proposed to generate Voronoi meshes for adaptive functional/scattered data approximations. Experimental results show space/accuracy advantages for these quadratic serendipity finite elements on polygonal domains versus traditional finite elements over simplicial meshes. Polygonal meshes and parameter coefficients of the quadratic serendipity finite elements obtained by our greedy algorithms can be further refined using an L-2-optimization to improve the piecewise functional approximation. We conduct several experiments to demonstrate the efficacy of our algorithm for modeling features/discontinuities in functional data/image approximation. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Polygonal elements Barycentric coordinates data approximation Voronoi tessellation

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data approximation for Time Series data in Wireless Sensor Networks

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INTERNATIONAL JOURNAL OF data WAREHOUSING AND MINING 2016年第3期12卷 1-13页

作者： Xu, Xiaobin Beijing Univ Posts & Telecommun Sch Software Engn Beijing Peoples R China

data prediction approaches are proposed in many fields to approximate time series data with a tolerable error. These approaches typically build prediction functions based on assumptions of the data variation. Nonetheless, if the variation of real-world time series data does not follow the assumption, the performance of data prediction will be limited. This paper presents a lightweight data approximation approach for time series data. This approach utilizes binary codes to represent original values, directly shortening their lengths in the cost of data precision. Then the author implements this approach in the WSN scenario. Two types of application layer messages and their transmission scheme are presented. These approaches are employed in WSN applications to: (1) report abnormal conditions in time, and (2) reduce data transmissions independently of data variations. Series of simulations are built on the basis of five real datasets. Simulation results based on five real datasets validate the performances of the proposed approach.

关键词： Accuracy data approximation Prediction Time Series data

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Inverse-free recursive multiresolution algorithms for a data approximation problem

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2016年第4期72卷 1177-1187页

作者： Fortes, M. A. Raydan, M. Sajo-Castelli, A. M. Univ Granada Dept Matemat Aplicada E-18071 Granada Spain Univ Simon Bolivar Dept Comp Cient & Estadist Ap 89000 Caracas 1080A Venezuela

We present inverse-free recursive multiresolution algorithms for data approximation problems based on energy functionals minimization. During the multiresolution process a linear system needs to be solved at each different resolution level, which can be solved with direct or iterative methods. Numerical results are reported, using the sparse Cholesky factorization, for two applications: one concerning the localization of regions in which the energy of a given surface is mostly concentrated, and another one regarding noise reduction of a given dataset. In addition, for large-scale data approximation problems that require a very fine resolution, we discuss the use of the Preconditioned Conjugate Gradient (PCG) iterative method coupled with a specialized monolithic preconditioner, for which one preconditioner is built for the highest resolution level and then the corresponding blocks of that preconditioner are used as preconditioners for the forthcoming lower levels. (C) 2016 Elsevier Ltd. All rights reserved.

关键词： Inverse-free methods Multiresolution algorithms data approximation Powell-Sabin element Noise reduction Energy localization

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Global-Support Rational Curve Method for data approximation with Bat Algorithm 11th

Global-Support Rational Curve Method for Data Approximation ...

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11th IFIP WG 12.5 International Conference on Artificial Intelligence Applications and Innovations (AIAI)

作者： Iglesias, Andres Galvez, Akemi Collantes, Marta Univ Cantabria Dept Appl Math & Computat Sci Avda Castros S-N E-39005 Santander Spain Toho Univ Fac Sci Dept Informat Sci Funabashi Chiba 2748510 Japan

ISBN: (纸本)9783319238685;9783319238678

The problem of obtaining an approximating curve from a given set of data points appears recurrently in several applied and industrial domains, such as CAD/CAM, computer graphics and animation, medicine, and many others. Although polynomial blending functions are usually applied to tackle this issue, some shapes cannot yet be adequately approximated by using the polynomial scheme. In this paper we address this limitation by applying rational global-support blending functions, particularly rational Bezier curves. Our method is based on a nature-inspired meta-heuristic called bat algorithm, which has been recently introduced to solve difficult optimization problems. To check the performance of our approach, it has been applied to some illustrative examples of 2D and 3D curves. Our results show that the method performs very well, being able to yield a satisfactory approximating curve with a high degree of accuracy.

关键词： data approximation rational curve bat algorithm meta-heuristic technique Bezier curve

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Least squares monotonic unimodal approximations to successively updated data and an application to a Covid-19 outbreak

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OPTIMIZATION METHODS & SOFTWARE 2024年第6期39卷 1464-1494页

作者： Demetriou, I. C. Natl & Kapodistrian Univ Athens Dept Econ 1 Sofokleous & Aristidou St Athens 10559 Greece

approximations to noisy data are constructed to obtain unimodality of successively updated data. Precisely, a method is developed that calculates least squares monotonic increasing - decreasing approximations to n data, for successive values of n. Having fixed n, the statement of the increasing--decreasing constraints in terms of first differences of the data subject to one sign change gives rise to a discrete search calculation, because the position of the sign change is also an unknown of the optimization problem. Although this problem may have many local minima for each value of n, the method is made quite efficient by taking advantage of two properties of the optimization calculation. The first property, for fixed n, provides necessary and sufficient conditions that reduce the optimal selection of the position to solving two sequences of monotonic approximation problems to subranges of data in the least complexity $ \mathcal {O}(n) $ O(n). The sufficiency conditions is a new result. The second property shows that the optimal position increases as n increases. Hence, if this position of a fit to the first n-1 data is available, then in order to calculate the required position of a fit to the first n data there is no need to consider data that are before the current position. A Fortran program has been written and some timings of numerical results are given for up to $ n=30{,}000 $ n=30,000 data and various levels of noise. They seem to be proportional to n. The method is applied to data of daily Covid-19 deaths of the United Kingdom for a period that exhibits a major outbreak, and obtains insights of the evolution of the process as new data enter the calculation. The results reveal certain features of the calculation that may be helpful to supporting policy making, when used in modelling epidemic processes. Further, a procedure is proposed where our method initializes the nonlinear least squares calculation of Richards epidemiological model, and the numerical res

关键词： Covid-19 outbreak data approximation first differences least squares monotonic nonlinear least squares Richards epidemiological model turning point unimodal regression

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Review and computational comparison of adaptive least-squares finite element schemes

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2024年 172卷 1-15页

作者： Bringmann, Philipp TU Wien Inst Anal & Sci Comp Wiedner Hauptstr 8-10 A-1040 Vienna Austria

The convergence analysis for least-squares finite element methods led to various adaptive mesh-refinement strategies: Collective marking algorithms driven by the built-in a posteriori error estimator or an alternative explicit residual-based error estimator as well as a separate marking strategy based on the alternative error estimator and an optimal data approximation algorithm. This paper reviews and discusses available convergence results. In addition, all three strategies are investigated empirically for a set of benchmarks examples of second order elliptic partial differential equations in two spatial dimensions. Particular interest is on the choice of the marking and refinement parameters and the approximation of the given data. The numerical experiments are reproducible using the author's software package octAFEM available on the platform Code Ocean.

关键词： Least-squares finite element method Adaptive mesh refinement Alternative a posteriori error estimation Separate marking data approximation Numerical experiments

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A Computational Template for Three-Dimensional Modeling of the Vascular Scaffold of the Human Thyroid Gland

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TISSUE ENGINEERING PART A 2023年第1-2期29卷 47-57页

作者： Spaletta, Giulia Sofroniou, Mark Barbaro, Fulvio di Conza, Giusy Mosca, Salvatore Toni, Roberto Univ Bologna Dept Stat Sci Bologna Italy Wolfram Res R&D Dept Champaign IL USA Univ Parma Dept Med & Surg DIMEC Unit Biomed Biotechnol & Translat Sci S BI BI T Lab Regenerat Morphol & Bioartificial Struct Re Mo Parma Italy Univ Parma Museum & Hist Lib Biomed BIOMED Parma Italy Univ Vita Salute San Raffaele Fellow Program Orthopaed & Traumatol Course Disorders Locomotor Syst Milan Italy Med Ctr Galliera Endocrinol Diabetol Nutr Disorders Outpatient Clin OSTEONET Unit Osteoporosis Nutr Endocrinol & Innov San Venanzio Di Galliera Bologna Italy Acad Sci Inst Bologna Sect Med Sci 4 Bologna Italy Tufts Univ Tufts Med Ctr Dept Med Div Endocrinol Diabet & MetabSch Med Boston MA USA Maggiore Hosp DIMEC SBIBIT Unit Lab Regenerat Morphol & Bioartificial Struct Re Mo Via Gramsci 14 I-43124 Parma Italy

We recently designed an innovative scaffold-bioreactor unit for the bioengineering of a three-dimensional (3D) bioartificial human thyroid gland or its miniaturized replica as a part of a microfluidic chip test system. This device is based on the evidence that the 3D geometry of the intraglandular stromal/vascular scaffold (SVS;i.e., the fibrous and vascular matrix) of mammalian viscera plays a key role in guiding growth and differentiation of in vitro seeded cells. Therefore, we initiated a research program focused on computer-aided reconstruction of the 2nd to 4th order intralobar arterial network (IAN) of the human thyroid gland as a reliable surrogate for its 3D SVS, to be used as an input for rapid prototyping of a biomaterial replica. To this end, we developed a computational template that works within the Mathematica environment, giving rise to a quasi-fractal growth of the IAN distribution, constrained within an approximation of the thyroid lobe shape as a closed surface. Starting from edge detection of planar images of real human thyroid lobes acquired by in vivo real-time ultrasonography, we performed data approximation of the lobar profiles based on splines and Bezier curves, providing 3D lobar shapes as geometric boundaries for vessel growth by a diffusion-limited aggregation model. Our numerical procedures allowed for a robust connection between development of lobar arterial trees and thyroid lobe shape, led to a vascular self-similarity consistent with that of a cadaveric lobar arterial cast, and reproduced arterial vessels in a proportion not statistically different from that described for the real human thyroid gland. We conclude that our algorithmic template offers a reliable reproduction of the extremely complex IAN of the adult human thyroid lobe, potentially useful as a computational guidance for bioprinting of thyroid lobe matrix replicas. In addition, due to the simplicity and limited number of morphometrical parameters required by our system,

关键词： thyroid computational modeling data approximation diffusion-limited aggregation scaffold bioartificial organ

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IoTSAX: A Dynamic Abstractive Entity Summarization Approach With approximation and Embedding-Based Reasoning Rules in Publish/Subscribe Systems

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IEEE INTERNET OF THINGS JOURNAL 2022年第3期9卷 1830-1847页

作者： Pavlopoulou, Niki Curry, Edward Natl Univ Ireland Galway Insight Ctr Data Analyt Galway H91 AEX4 Ireland

Users are interested in entity information and may use paradigms like publish/subscribe systems in Internet of Things (IoT), where entity-centric data comes from multiple sources. The IoT application demands contributed to the data integration and semantic interoperability need while maintaining high usability and low processing system resources usage. Existing approaches dealt with semantic interoperability by using/extending ontologies with strict schemata/semantics and difficult to be updated. Other approaches tackled high usability by providing sensor data abstractions that infer high-level knowledge, but they are either complex for real-time processing or have semantic/syntactic coupling making them nonflexible. Therefore, our key question is: can a dynamic entity summarization system that provides high-level abstractions while ensuring semantic/syntactic decoupling, and low use of processing system resources, be defined? In this article, we address this question by proposing the abstractive publish/subscribe summarization system that provides abstractive summaries of numerical IoT data for user-friendly subscriptions by using IoTSAX, an enhanced symbolic aggregate approximation (SAX) methodology for dynamic IoT environments, and approximate rule-based reasoning by using embedding models. Our results for two use cases, medicine and smart cities, show that although abstraction can be 2 to 3 times slower in latency, it achieves up to 98% reduction in the notifications' number. IoTSAX outperforms SAX in approximation error up to 2 to 3 times more and in compression space-saving percentage when data redundancy occurs, while it has a similar or better latency and throughput performance. Finally, concept hierarchy-based embedding models can achieve F-scores of up to 0.87 for approximate rule-based reasoning.

关键词： Sensors Semantics Internet of Things Cognition Interoperability Ontologies Sensor systems Abstractive entity summarization approximate reasoning data approximation Internet of Things (IoT) publish subscribe

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Experimental Study on Machine Learning with approximation to data Streams 6

Experimental Study on Machine Learning with Approximation to...

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6th International Conference on Internet of Things - Systems, Management and Security (IOTSMS)

作者： Fu, Zhang Jiang, Jiani Ericsson Res Kista Sweden Royal Inst Technol KTH Stockholm Sweden

ISBN: (纸本)9781728129495

Realtime transferring of data streams enables many data analytics and machine learning applications in the areas of e.g. massive IoT and industrial automation. Big data volume of those streams is a significant burden or overhead not only to the transportation network, but also to the corresponding application servers. Therefore, researchers and scientists focus on reducing the amount of data needed to be transferred via data compressions and approximations. However, how to do data compression and approximation is highly dependent on the corresponding applications. In this paper, we did a study on the impact of data approximation to the machine learning applications. In particular, from the experimental perspective, we show the tradeoff among the approximation error bound, compression ratio and the prediction accuracy of multiple machine learning methods. We believe that, with proper choice, data approximation can dramatically reduce the amount of data transferred with limited impact on the machine learning applications.

关键词： Big data Streams Machine Learning data approximation data Transportation

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