Abstract
We consider the problem of computing the (two-sided) Hausdorff distance between the unit \(\ell _{p_{1}}\) and \(\ell _{p_{2}}\) norm balls in finite dimensional Euclidean space for \(1 \leq p_{1} < p_{2} \leq \infty \), and derive a closed-form formula for the same. We also derive a closed-form formula for the Hausdorff distance between the \(k_{1}\) and \(k_{2}\) unit \(D\)-norm balls, which are certain polyhedral norm balls in \(d\) dimensions for \(1 \leq k_{1} < k_{2} \leq d\). When two different \(\ell _{p}\) norm balls are transformed via a common linear map, we obtain several estimates for the Hausdorff distance between the resulting convex sets. These estimates upper bound the Hausdorff distance or its expectation, depending on whether the linear map is arbitrary or random. We then generalize the developments for the Hausdorff distance between two set-valued integrals obtained by applying a parametric family of linear maps to different \(\ell _{p}\) unit norm balls, and then taking the Minkowski sums of the resulting sets in a limiting sense. To illustrate an application, we show that the problem of computing the Hausdorff distance between the reach sets of a linear dynamical system with different unit norm ball-valued input uncertainties, reduces to this set-valued integral setting.
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References
Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 151. Cambridge university press (2014)
Hausdorff, F.: Grundzüge der Mengenlehre, vol. 7. von Veit (1914)
Hildenbrand, W.: Core and equilibria of a large economy. (psme-5). In: Core and Equilibria of a Large Economy.(PSME-5). Princeton university press (2015)
Stoyan, D., Kendall, W.S., Chiu, S.N., Mecke, J.: Stochastic Geometry and Its Applications. Wiley, New York (2013)
Serra, J.: Hausdorff distances and interpolations. Comput. Imaging Vision 12, 107–114 (1998)
Huttenlocher, D.P., Klanderman, G.A., Rucklidge, W.J.: Comparing images using the Hausdorff distance. IEEE Trans. Pattern Anal. Mach. Intell. 15(9), 850–863 (1993)
Jesorsky, O., Kirchberg, K.J., Frischholz, R.W.: Robust face detection using the Hausdorff distance. In: International Conference on Audio- and Video-Based Biometric Person Authentication, pp. 90–95. Springer, Berlin (2001)
De Blasi, F.: On the differentiability of multifunctions. Pac. J. Math. 66(1), 67–81 (1976)
Serry, M., Reissig, G.: Overapproximating reachable tubes of linear time-varying systems. IEEE Trans. Autom. Control 67(1), 443–450 (2021)
Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I: Fundamentals, vol. 305. Springer, Berlin (2013)
Shisha, O., Mond, B.: Differences of means. Bull. Am. Math. Soc. 73(3), 328–333 (1967)
Pecsvaradi, T., Narendra, K.S.: Reachable sets for linear dynamical systems. Inf. Control 19(4), 319–344 (1971)
Witsenhausen, H.: A remark on reachable sets of linear systems. IEEE Trans. Autom. Control 17(4), 547–547 (1972)
Chutinan, A., Krogh, B.H.: Verification of polyhedral-invariant hybrid automata using polygonal flow pipe approximations. In: Hybrid Systems: Computation and Control: Second International Workshop, HSCC’99 Berg en Dal, The Netherlands, March 29–31, 1999 Proceedings 2, pp. 76–90. Springer, Berlin (1999)
Kurzhanski, A., Vályi, I.: Ellipsoidal Calculus for Estimation and Control. Springer, Berlin (1997)
Varaiya, P.: Reach set computation using optimal control. In: Verification of Digital and Hybrid Systems, pp. 323–331 (2000)
Le Guernic, C., Girard, A.: Reachability analysis of linear systems using support functions. Nonlinear Anal. Hybrid Syst. 4(2), 250–262 (2010)
Althoff, M., Frehse, G., Girard, A.: Set propagation techniques for reachability analysis. Annu. Rev. Control Robot. Auton. Syst. 4, 369–395 (2021)
Haddad, S., Halder, A.: The curious case of integrator reach sets, part i: Basic theory. IEEE Trans. Autom. Control (2023)
Haddad, S., Halder, A.: Anytime ellipsoidal over-approximation of forward reach sets of uncertain linear systems. In: Proceedings of the Workshop on Computation-Aware Algorithmic Design for Cyber-Physical Systems, pp. 20–25 (2021)
Atallah, M.J.: A linear time algorithm for the Hausdorff distance between convex polygons. Inf. Process. Lett. 17(4), 207–209 (1983)
Belogay, E., Cabrelli, C., Molter, U., Shonkwiler, R.: Calculating the Hausdorff distance between curves. Inf. Process. Lett. 64(1) (1997)
Aspert, N., Santa-Cruz, D., Ebrahimi, T.: Mesh: measuring errors between surfaces using the Hausdorff distance. In: Proceedings. IEEE International Conference on Multimedia and Expo, vol. 1, pp. 705–708. IEEE (2002)
Taha, A.A., Hanbury, A.: An efficient algorithm for calculating the exact Hausdorff distance. IEEE Trans. Pattern Anal. Mach. Intell. 37(11), 2153–2163 (2015)
Goffin, J.-L., Hoffman, A.J.: On the relationship between the Hausdorff distance and matrix distances of ellipsoids. Linear Algebra Appl. 52, 301–313 (1983)
Alt, H., Behrends, B., Blömer, J.: Approximate matching of polygonal shapes. Ann. Math. Artif. Intell. 13(3), 251–265 (1995)
Alt, H., Braß, P., Godau, M., Knauer, C., Wenk, C.: Computing the Hausdorff distance of geometric patterns and shapes. In: Discrete and Computational Geometry: The Goodman-Pollack Festschrift, pp. 65–76 (2003)
König, S.: Computational aspects of the Hausdorff distance in unbounded dimension. J. Comput. Geom. 5(1), 250–274 (2014)
Jungeblut, P., Kleist, L., Miltzow, T.: The complexity of the Hausdorff distance. arXiv preprint. arXiv:2112.04343 (2021)
Marošević, T.: The Hausdorff distance between some sets of points. Math. Commun. 23(2), 247–257 (2018)
Grothendieck, A.: Résumé de la Théorie Métrique des Produits Tensoriels topologiques. Soc. de Matemática de São Paulo (1956)
Alon, N., Naor, A.: Approximating the cut-norm via Grothendieck’s inequality. In: Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, pp. 72–80 (2004)
Kindler, G., Naor, A., Schechtman, G.: The UGC hardness threshold of the \({L}_{p}\) Grothendieck problem. Math. Oper. Res. 35(2), 267–283 (2010)
Rahal, S., Li, Z.: Norm induced polyhedral uncertainty sets for robust linear optimization. Optim. Eng., 1–37 (2021)
Bertsimas, D., Pachamanova, D., Sim, M.: Robust linear optimization under general norms. Oper. Res. Lett. 32(6), 510–516 (2004)
DCCP Python package, GitHub repository. https://github.com/cvxgrp/dccp
Shen, X., Diamond, S., Gu, Y., Boyd, S.: Disciplined convex-concave programming. In: 2016 IEEE 55th Conference on Decision and Control (CDC), pp. 1009–1014. IEEE (2016)
Hendrickx, J.M., Olshevsky, A.: Matrix p-norms are NP-hard to approximate if \(p\neq 1,2,\infty \). SIAM J. Matrix Anal. Appl. 31(5), 2802–2812 (2010)
Steinberg, D.: Computation of matrix norms with applications to robust optimization. Research thesis, Technion-Israel University of Technology (2005)
Bhaskara, A., Vijayaraghavan, A.: Approximating matrix p-norms. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 497–511. SIAM, Philadelphia (2011)
Barak, B., Brandao, F.G., Harrow, A.W., Kelner, J., Steurer, D., Zhou, Y.: Hypercontractivity, sum-of-squares proofs, and their applications. In: Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing, pp. 307–326 (2012)
Bhattiprolu, V., Ghosh, M., Guruswami, V., Lee, E., Tulsiani, M.: Approximability of \(p\rightarrow q\) matrix norms: generalized Krivine rounding and hypercontractive hardness. In: Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1358–1368. SIAM, Philadelphia (2019)
Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math. 97(4), 1061–1083 (1975)
Saloff-Coste, L.: Lectures on finite Markov chains. In: Lectures on Probability Theory and Statistics, Saint-Flour, 1996. Lecture Notes in Math, vol. 1665, pp. 301–413 (1997)
Biswal, P.: Hypercontractivity and its applications. arXiv preprint. arXiv:1101.2913 (2011)
Wang, C., Zheng, D.-S., Chen, G.-L., Zhao, S.-Q.: Structures of p-isometric matrices and rectangular matrices with minimum p-norm condition number. Linear Algebra Appl. 184, 261–278 (1993)
Li, C.-K., So, W.: Isometries of \(\ell _{p}\) norm. Am. Math. Mon. 101(5), 452–453 (1994)
Glueck, J.: What are the matrices preserving the \(\ell ^{1}\)-norm? MathOverflow. https://mathoverflow.net/q/288084. (version: 2017-12-09), https://mathoverflow.net/q/288084
Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 2.1. http://cvxr.com/cvx (2014)
Bennett, G., Goodman, V., Newman, C.: Norms of random matrices. Pac. J. Math. 59(2), 359–365 (1975)
Guédon, O., Hinrichs, A., Litvak, A.E., Prochno, J.: On the expectation of operator norms of random matrices. In: Geometric Aspects of Functional Analysis, pp. 151–162. Springer, Berlin (2017)
Aumann, R.J.: Integrals of set-valued functions. J. Math. Anal. Appl. 12(1), 1–12 (1965)
Liapounoff, A.: Sur les fonctions-vecteurs completement additives. Izv. Ross. Akad. Nauk Ser. Mat. 4(6), 465–478 (1940)
Halmos, P.R.: The range of a vector measure. Bull. Am. Math. Soc. 54(4), 416–421 (1948)
Guseinov, K.G., Ozer, O., Akyar, E., Ushakov, V.: The approximation of reachable sets of control systems with integral constraint on controls. NoDEA Nonlinear Differ. Equ. Appl. 14(1), 57–73 (2007)
Dueri, D., Raković, S.V., Açıkmeşe, B.: Consistently improving approximations for constrained controllability and reachability. In: 2016 European Control Conference (ECC), pp. 1623–1629. IEEE (2016)
Halder, A.: Smallest ellipsoid containing \(p \)-sum of ellipsoids with application to reachability analysis. IEEE Trans. Autom. Control 66(6), 2512–2525 (2020)
Brockett, R.W.: Finite Dimensional Linear Systems. Wiley, New York (1970)
Haddad, S., Halder, A.: The convex geometry of integrator reach sets. In: 2020 American Control Conference (ACC), pp. 4466–4471. IEEE (2020)
Haddad, S., Halder, A.: Certifying the intersection of reach sets of integrator agents with set-valued input uncertainties. IEEE Control Syst. Lett. 6, 2852–2857 (2022)
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Haddad, S., Halder, A. A Note on the Hausdorff Distance Between Norm Balls and Their Linear Maps. Set-Valued Var. Anal 31, 30 (2023). https://doi.org/10.1007/s11228-023-00692-1
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DOI: https://doi.org/10.1007/s11228-023-00692-1