最小体积覆盖椭球

IF 1 4区数学 Q3 MATHEMATICS, APPLIED

ANZIAM Journal Pub Date : 2024-05-04 DOI:10.21914/anziamj.v64.17956

Elizabeth Harris

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Sequences converging to D-optimal designs of experiments. Ann. Statist. 1.2 (1973), pp. 342–352. doi: 10.1214/aos/1176342371\nY.-J. Chen, M.-Y. Ju, and K.-S. Hwang. A virtual torque-based approach to kinematic control of redundant manipulators. IEEE Trans. Indust. Elec. 64.2 (2017), pp. 1728–1736. doi: 10.1109/TIE.2016.2548439\nF. L. Chernousko. Ellipsoidal state estimation for dynamical systems. Nonlin. Anal. 63.5-7 (2005), pp. 872–879. doi: 10.1016/j.na.2005.01.009\nD. Eberly. 3D game engine design: A practical approach to real-time computer graphics. CRC Press, 2007. doi: 10.1201/b18212\nM. Frank and P. Wolfe. An algorithm for quadratic programming. Naval Res. Log. Quart. 3.1-2 (1956), pp. 95–110. doi: 10.1002/nav.3800030109\nR. Harman, L. Filová, and P. Richtárik. A randomized exchange algorithm for computing optimal approximate designs of experiments. J. Am. Stat. Ass. 115.529 (2020), pp. 348–361. doi: 10.1080/01621459.2018.1546588\nR. Harman and L. Pronzato. Improvements on removing nonoptimal support points in D-optimum design algorithms. Stat. Prob. Lett. 77.1 (2007), pp. 90–94. doi: 10.1016/j.spl.2006.05.014\nF. John. Extremum problems with inequalities as subsidiary conditions. Traces and emergence of nonlinear programming. Springer Basel, 2014, pp. 197–215. doi: 10.1007/978-3-0348-0439-4_9\nL. Källberg and D. Andrén. Active set strategies for the computation of minimum-volume enclosing ellipsoids. Tech. rep. Mälardalen University, 2019. url: http://www.es.mdu.se/publications/5680-\nL. G. Khachiyan. Rounding of polytopes in the real number model of computation. Math. Op. Res. 21.2 (1996), pp. 307–320. doi: 10.1287/moor.21.2.307\nJ. Kudela. Minimum-volume covering ellipsoids: Improving the efficiency of the Wolfe–Atwood algorithm for large-scale instances by pooling and batching. MENDEL 25.2 (2019), pp. 19–26. doi: 10.13164/mendel.2019.2.019\nP. Kumar and E. A. Yildirim. Minimum-volume enclosing ellipsoids and core sets. J. Op. Theor. Appl. 126.1 (2005), pp. 1–21. doi: 10.1007/s10957-005-2653-6\nS. Rosa and R. Harman. Computing minimum-volume enclosing ellipsoids for large datasets. Comput. Stat. Data Anal. 171, 107452 (2022). doi: 10.1016/j.csda.2022.107452\nJ. B. Rosen. Pattern separation by convex programming. J. Math. Anal. Appl. 10.1 (1965), pp. 123–134. doi: 10.1016/0022-247X(65)90150-2\nP. J. Rousseeuw and M. Hubert. Robust statistics for outlier detection. WIREs: Data mining and knowledge discovery 1.1 (2011), pp. 73–79. doi: 10.1002/widm.2\nF. Schweppe. Recursive state estimation: Unknown but bounded errors and system inputs. IEEE Trans. Auto. Control 13.1 (1968), pp. 22–28. doi: 10.1109/TAC.1968.1098790\nR. Sibson. Discussion of Dr Wynn’s and of Dr Laycock’s papers. J. Roy. Stat. Soc. B 34.2 (1972), pp. 181–183. doi: 10.1111/j.2517-6161.1972.tb00898.x\nS. D. Silvey. Optimal design: An introduction to the theory for parameter estimation. Ettore Majorana International Science Series. London, Chapman and Hall, 1980. doi: 10.1007/978-94-009-5912-5\nP. Sun and R. M. Freund. Computation of minimum-volume covering ellipsoids. Op. Res. 52.5 (2004), pp. 690–706. doi: 10.1287/opre.1040.0115\nD. M. Titterington. Estimation of correlation coefficients by ellipsoidal trimming. J. Roy. Stat. Soc.: C 27.3 (1978), pp. 227–234. doi: 10.2307/2347157\nD. M. Titterington. Optimal design: Some geometrical aspects of D-optimality. Biometrika 62.2 (1975), pp. 313–320. doi: 10.2307/2335366\nM. J. Todd. Minimum-volume ellipsoids: Theory and algorithms. MOS-SIAM Series on Optimization. SIAM, 2016. doi: 10.1137/1.9781611974386\nP. Wolfe. Convergence theory in nonlinear programming. ed. by J. Abadie. Integer and Nonlinear Programming. 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Appl. 10.1 (1965), pp. 123–134. doi: 10.1016/0022-247X(65)90150-2\\nP. J. Rousseeuw and M. Hubert. Robust statistics for outlier detection. WIREs: Data mining and knowledge discovery 1.1 (2011), pp. 73–79. doi: 10.1002/widm.2\\nF. Schweppe. Recursive state estimation: Unknown but bounded errors and system inputs. IEEE Trans. Auto. Control 13.1 (1968), pp. 22–28. doi: 10.1109/TAC.1968.1098790\\nR. Sibson. Discussion of Dr Wynn’s and of Dr Laycock’s papers. J. Roy. Stat. Soc. B 34.2 (1972), pp. 181–183. doi: 10.1111/j.2517-6161.1972.tb00898.x\\nS. D. Silvey. Optimal design: An introduction to the theory for parameter estimation. Ettore Majorana International Science Series. London, Chapman and Hall, 1980. doi: 10.1007/978-94-009-5912-5\\nP. Sun and R. M. Freund. Computation of minimum-volume covering ellipsoids. Op. Res. 52.5 (2004), pp. 690–706. doi: 10.1287/opre.1040.0115\\nD. M. Titterington. Estimation of correlation coefficients by ellipsoidal trimming. J. Roy. Stat. 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North Holland, Amsterdam, 1970, pp. 1–36. doi: 10.1007/BF00932858\\n\",\"PeriodicalId\":50745,\"journal\":{\"name\":\"ANZIAM Journal\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-05-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ANZIAM Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.21914/anziamj.v64.17956\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ANZIAM Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.21914/anziamj.v64.17956","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

摘要

我们提出了一种自适应批处理策略的新初始化方法，用于计算由 n 个点组成的集合的ε-近似最小体积覆盖椭圆（MVCE）。我们将重点放在中等大小的数据集上（维度 d = 100，n = 1 000 000）。自适应批处理策略以优化-删除-适应的循环方式工作：我们使用较少数量的点求解 MVCE 问题，删除保证不在 MVCE 边界上的点，然后仔细选择新的一批点。我们提出了一种新的初始化方法，即选择与一些最高杠杆分数相对应的点。我们用数值示例表明，与随机选择点进行初始化相比，这种新的初始化往往能缩短计算时间并减少循环总数。L. Atwood.收敛于 D 最佳实验设计的序列。Ann.Statist.1.2 (1973), pp.Ju, and K.-S.Hwang.基于虚拟转矩的冗余机械手运动控制方法。IEEE Trans.Indust.64.2 (2017)，pp. 1728-1736. doi: 10.1109/TIE.2016.2548439F.L. Chernousko.动力学系统的椭圆状态估计。Nonlin.Anal.63.5-7 (2005), pp.Eberly.三维游戏引擎设计：CRC Press, 2007.CRC Press, 2007. Doi: 10.1201/b18212M.Frank and P. Wolfe.二次编程算法。Naval Res.Log.Quart.3.1-2 (1956), pp.Harman, L. Filová, and P. Richtárik.计算最优近似实验设计的随机交换算法》。J. Am.Stat.J. Am. Stat.115.529（2020），第 348-361 页。DOI：10.1080/01621459.2018.1546588R。Harman and L. Pronzato.在 D-最优设计算法中去除非最优支持点的改进。Stat.Prob.Doi: 10.1016/j.spl.2006.05.014F.John.以不等式为附属条件的极值问题.Traces and emergence of nonlinear programming.Springer Basel, 2014, pp.Källberg and D. Andrén.计算最小体积包围椭球体的有源集策略。Tech.Mälardalen University, 2019. url: http://www.es.mdu.se/publications/5680-L.G. Khachiyan.实数计算模型中的多边形舍入。Math. Op.Di: 10.1287/moor.21.2.307J.Kudela.最小体积覆盖椭球：通过集合和批处理提高大规模实例的 Wolfe-Atwood 算法效率.DOI: 10.13164/mendel.2019.2.019P.Kumar and E. A. Yildirim.最小体积包围椭球和核心集.J. Op.126.1 (2005), pp.Rosa and R. Harman.计算大型数据集的最小体积包围椭圆。Comput.Stat.Data Anal.DOI：10.1016/j.csda.2022.107452J.B. Rosen.凸编程的模式分离.J. Math.Anal.DOI: 10.1016/0022-247X(65)90150-2P.J. Rousseeuw 和 M. Hubert.离群点检测的稳健统计。WIREs：doi: 10.1002/widm.2F.Schweppe.递归状态估计：未知但有界的误差和系统输入。IEEE Trans.Auto.doi: 10.1109/TAC.1968.1098790R.Sibson.讨论 Wynn 博士和 Laycock 博士的论文。J. Roy.Stat.B 34.2 (1972), pp.D. Silvey.最优设计：参数估计理论导论》。Ettore Majorana 国际科学丛书。伦敦，查普曼和霍尔，1980 年。DOI：10.1007/978-94-009-5912-5P.Sun and R. M. Freund.最小体积覆盖椭球的计算.doi: 10.1287/opre.1040.0115D.M. Titterington.Estimation of correlation coefficients by ellipsoidal trimming.J. Roy.Stat.Soc.: C 27.3 (1978), pp.M. Titterington.最优设计：最优设计：D-最优性的一些几何方面。Biometrika 62.2 (1975), pp.J. Todd.最小体积椭圆体：Theory and algorithms.MOS-SIAM Series on Optimization.SIAM, 2016. doi: 10.1137/1.9781611974386P.Wolfe.非线性编程中的收敛理论》。J. Abadie 编。整数与非线性编程》。1-36. doi: 10.1007/BF00932858

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Minimum volume covering ellipsoids

We present a new initialisation of an adaptive batch strategy to compute the ε-approximate minimum volume covering ellipsoid (MVCE) for a set of n points. We focus on moderately sized datasets (up to dimension d = 100 and n = 1 000 000). The adaptive batch strategy works in an optimisation-deletion-adaptation cycle: we solve the MVCE problem using a smaller number of points, we delete points from consideration that are guaranteed to not lie on the boundary of the MVCE, and then carefully select a new batch of points. We propose a new initialisation, which involves selecting the points corresponding to some highest leverage scores. We show using numerical examples that this new initialisation tends to improve computation time as well as reduce the total number of cycles, as compared with initialising with a random selection of points. References C. L. Atwood. Sequences converging to D-optimal designs of experiments. Ann. Statist. 1.2 (1973), pp. 342–352. doi: 10.1214/aos/1176342371 Y.-J. Chen, M.-Y. Ju, and K.-S. Hwang. A virtual torque-based approach to kinematic control of redundant manipulators. IEEE Trans. Indust. Elec. 64.2 (2017), pp. 1728–1736. doi: 10.1109/TIE.2016.2548439 F. L. Chernousko. Ellipsoidal state estimation for dynamical systems. Nonlin. Anal. 63.5-7 (2005), pp. 872–879. doi: 10.1016/j.na.2005.01.009 D. Eberly. 3D game engine design: A practical approach to real-time computer graphics. CRC Press, 2007. doi: 10.1201/b18212 M. Frank and P. Wolfe. An algorithm for quadratic programming. Naval Res. Log. Quart. 3.1-2 (1956), pp. 95–110. doi: 10.1002/nav.3800030109 R. Harman, L. Filová, and P. Richtárik. A randomized exchange algorithm for computing optimal approximate designs of experiments. J. Am. Stat. Ass. 115.529 (2020), pp. 348–361. doi: 10.1080/01621459.2018.1546588 R. Harman and L. Pronzato. Improvements on removing nonoptimal support points in D-optimum design algorithms. Stat. Prob. Lett. 77.1 (2007), pp. 90–94. doi: 10.1016/j.spl.2006.05.014 F. John. Extremum problems with inequalities as subsidiary conditions. Traces and emergence of nonlinear programming. Springer Basel, 2014, pp. 197–215. doi: 10.1007/978-3-0348-0439-4_9 L. Källberg and D. Andrén. Active set strategies for the computation of minimum-volume enclosing ellipsoids. Tech. rep. Mälardalen University, 2019. url: http://www.es.mdu.se/publications/5680- L. G. Khachiyan. Rounding of polytopes in the real number model of computation. Math. Op. Res. 21.2 (1996), pp. 307–320. doi: 10.1287/moor.21.2.307 J. Kudela. Minimum-volume covering ellipsoids: Improving the efficiency of the Wolfe–Atwood algorithm for large-scale instances by pooling and batching. MENDEL 25.2 (2019), pp. 19–26. doi: 10.13164/mendel.2019.2.019 P. Kumar and E. A. Yildirim. Minimum-volume enclosing ellipsoids and core sets. J. Op. Theor. Appl. 126.1 (2005), pp. 1–21. doi: 10.1007/s10957-005-2653-6 S. Rosa and R. Harman. Computing minimum-volume enclosing ellipsoids for large datasets. Comput. Stat. Data Anal. 171, 107452 (2022). doi: 10.1016/j.csda.2022.107452 J. B. Rosen. Pattern separation by convex programming. J. Math. Anal. Appl. 10.1 (1965), pp. 123–134. doi: 10.1016/0022-247X(65)90150-2 P. J. Rousseeuw and M. Hubert. Robust statistics for outlier detection. WIREs: Data mining and knowledge discovery 1.1 (2011), pp. 73–79. doi: 10.1002/widm.2 F. Schweppe. Recursive state estimation: Unknown but bounded errors and system inputs. IEEE Trans. Auto. Control 13.1 (1968), pp. 22–28. doi: 10.1109/TAC.1968.1098790 R. Sibson. Discussion of Dr Wynn’s and of Dr Laycock’s papers. J. Roy. Stat. Soc. B 34.2 (1972), pp. 181–183. doi: 10.1111/j.2517-6161.1972.tb00898.x S. D. Silvey. Optimal design: An introduction to the theory for parameter estimation. Ettore Majorana International Science Series. London, Chapman and Hall, 1980. doi: 10.1007/978-94-009-5912-5 P. Sun and R. M. Freund. Computation of minimum-volume covering ellipsoids. Op. Res. 52.5 (2004), pp. 690–706. doi: 10.1287/opre.1040.0115 D. M. Titterington. Estimation of correlation coefficients by ellipsoidal trimming. J. Roy. Stat. Soc.: C 27.3 (1978), pp. 227–234. doi: 10.2307/2347157 D. M. Titterington. Optimal design: Some geometrical aspects of D-optimality. Biometrika 62.2 (1975), pp. 313–320. doi: 10.2307/2335366 M. J. Todd. Minimum-volume ellipsoids: Theory and algorithms. MOS-SIAM Series on Optimization. SIAM, 2016. doi: 10.1137/1.9781611974386 P. Wolfe. Convergence theory in nonlinear programming. ed. by J. Abadie. Integer and Nonlinear Programming. North Holland, Amsterdam, 1970, pp. 1–36. doi: 10.1007/BF00932858

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来源期刊

ANZIAM Journal 数学-应用数学

CiteScore

1.30

自引率

11.10%

发文量

审稿时长

1 months

期刊介绍： The ANZIAM Journal considers papers in any field of applied mathematics and related mathematical sciences with the aim of rapid publication in print and electronic formats. Novel applications of mathematics in real situations are especially welcomed. All papers should include some indication of applicability, and an introduction that can be understood by non-specialist readers from the whole applied mathematical community.