Minimum volume covering ellipsoids
Authors: Harris Elizabeth 
Published: May 2024

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