Discrete & Computational Geometry

, Volume 56, Issue 4, pp 910–939

# Two Proofs for Shallow Packings

• Kunal Dutta
• Esther Ezra
• Arijit Ghosh
Article

## Abstract

We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let $$\mathcal {V}$$ be a finite set system defined over an n-point set X; we view $$\mathcal {V}$$ as a set of indicator vectors over the n-dimensional unit cube. A $$\delta$$-separated set of $$\mathcal {V}$$ is a subcollection $$\mathcal {W}$$, s.t. the Hamming distance between each pair $$\mathbf{u}, \mathbf{v}\in \mathcal {W}$$ is greater than $$\delta$$, where $$\delta > 0$$ is an integer parameter. The $$\delta$$-packing number is then defined as the cardinality of a largest $$\delta$$-separated subcollection of $$\mathcal {V}$$. Haussler showed an asymptotically tight bound of $$\Theta ((n/\delta )^d)$$ on the $$\delta$$-packing number if $$\mathcal {V}$$ has VC-dimension (or primal shatter dimension) d. We refine this bound for the scenario where, for any subset, $$X' \subseteq X$$ of size $$m \le n$$ and for any parameter $$1 \le k \le m$$, the number of vectors of length at most k in the restriction of $$\mathcal {V}$$ to $$X'$$ is only $$O(m^{d_1} k^{d-d_1})$$, for a fixed integer $$d > 0$$ and a real parameter $$1 \le d_1 \le d$$ (this generalizes the standard notion of bounded primal shatter dimension when $$d_1 = d$$). In this case when $$\mathcal {V}$$ is “k-shallow” (all vector lengths are at most k), we show that its $$\delta$$-packing number is $$O(n^{d_1} k^{d-d_1}/\delta ^d)$$, matching Haussler’s bound for the special cases where $$d_1=d$$ or $$k=n$$. We present two proofs, the first is an extension of Haussler’s approach, and the second extends the proof of Chazelle, originally presented as a simplification for Haussler’s proof.

## Keywords

Packing lemma and shallow packing lemma Set systems of finite VC–dimension Primal shatter function Clarkson–Shor property

## Mathematics Subject Classification

05B40 52C15 52C17 52C45 68R05

## Notes

### Acknowledgments

We authors would like to thank two anonymous referees for their useful comments. The second author wishes to thank Boris Aronov, Sariel Har-Peled, Aryeh Kontorovich, and Wolfgang Mulzer for useful discussions and suggestions. Last but not least, the second author thanks Ramon Van Handel, for various discussions and for spotting an error in an earlier version of this paper. Work on this paper by Kunal Dutta and Arijit Ghosh has been supported by the Indo-German Max-Planck Center for Computer Science (IMPECS). Work on this paper by Esther Ezra has been supported by NSF Grants CCF-11-17336, CCF-12-16689, and NSF CAREER CCF-15-53354. A preliminary version of this paper appeared in Proc. Sympos. Computational Geometry, 2015, pp. 96–110  [8]

## References

1. 1.
Alon, N., Spencer, J.: The Probabilistic Method, 3rd edn. Wiley, New York (2008)
2. 2.
Auger, A., Doerr, B.: Theory of Randomized Search Heuristics: Foundations and Recent Developments. World Scientific, Singapore (2011)
3. 3.
Bshouty, N.H., Li, Y., Long, P.M.: Using the doubling dimension to analyze the generalization of learning algorithms. J. Comput. Syst. Sci. 75(6), 323–335 (2009)
4. 4.
Chazelle, B.: A note on Haussler’s packing lemma (1992) (unpublished manuscript)Google Scholar
5. 5.
Chazelle, B., Welzl, E.: Quasi-optimal range searching in spaces of finite VC-dimension. Discrete Comput. Geom. 4(5), 467–489 (1989)
6. 6.
Clarkson, K.L., Shor, P.W.: Applications of random sampling in computational geometry, II. Discrete Comput. Geom. 4(5), 387–421 (1989)
7. 7.
Dudley, R.M.: Central limit theorems for empirical measures. Ann. Probab. 6(6), 899–1049 (1978)
8. 8.
Dutta, K., Ezra, E., Ghosh. A.: Two proofs for shallow packings. In: Proceedings of the 31st International Symposium on Computational Geometry, pp. 96–110 (2015)Google Scholar
9. 9.
Ezra, E.: A size-sensitive discrepancy bound for set systems of bounded primal shatter dimension. SIAM J. Comput. 45(1):84–101 (2016) (A preliminary version appeard in Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1378–1388. SIAM (2014))Google Scholar
10. 10.
Gottlieb, L.-A., Kontorovich, A., Mossel, E.: VC bounds on the cardinality of nearly orthogonal function classes. Discrete Math. 312(10), 1766–1775 (2012)
11. 11.
Haussler, D.: Decision theoretic generalizations of the PAC model for neural net and other learning applications. Inf. Comput. 100(1), 78–150 (1992)
12. 12.
Haussler, D.: Sphere packing numbers for subsets of the Boolean n-cube with bounded Vapnik–Chervonenkis dimension. J. Comb. Theory, Ser. A 69(2), 217–232 (1995)
13. 13.
Haussler, D., Littlestone, N., Warmuth, M.K.: Predicting 0, 1-functions on randomly drawn points. Inf. Comput. 115(2), 248–292 (1994)
14. 14.
Har-Peled, S.: Geometric Approximation Algorithms. American Mathematical Society, Boston (2011)
15. 15.
Har-Peled, S., Sharir, M.: Relative $$(\varepsilon, \rho )$$-approximations in geometry. Discrete Comput. Geom. 45(3), 462–496 (2011)
16. 16.
Haussler, D., Welzl, E.: Epsilon-nets and simplex range queries. Discrete Comput. Geom. 2, 127–151 (1987)
17. 17.
Li, Y., Long, P.M., Srinivasan, A.: Improved bounds on the sample complexity of learning. J. Comput. Syst. Sci. 62(3), 516–527 (2001)
18. 18.
Lovett, S., Meka, R.: Constructive discrepancy minimization by walking on the edges. In: Proceedings of the IEEE 53rd Annual Symposium on Foundations of Computer Science (FOCS ’12) pp. 61–67. IEEE Computer Society, Washington, DC (2012)Google Scholar
19. 19.
Matoušek, J.: Reporting points in halfspaces. Comput. Geom. 2(3), 169–186 (1992)
20. 20.
Matousek, J.: Tight upper bounds for the discrepancy of half-spaces. Discrete Comput. Geom. 13, 593–601 (1995)
21. 21.
Matousek, J.: Geometric Discrepancy: An Illustrated Guide. Algorithms and Combinatorics. Springer, Berlin (1999)
22. 22.
Matousek, J.: Lectures on Discrete Geometry. Springer, Secaucus (2002)
23. 23.
Mulzer, W.: Chernoff Bounds, Personal note. http://page.mi.fu-berlin.de/mulzer/notes/misc/chernoff.pdf
24. 24.
Mustafa, N.H.: A simple proof of the shallow packing lemma. Discrete Comput. Geom. 55(3), 739–743 (2016)
25. 25.
Panconesi, A., Srinivasan, A.: Randomized distributed edge coloring via an extension of the Chernoff–Hoeffding bounds. SIAM J. Comput. 26, 350–368 (1997)
26. 26.
Pollard, D.: Convergence of Stochastic Processes. Springer, New York (1984)
27. 27.
Sauer, N.: On the density of families of sets. J. Comb. Theory, Ser. A 13(1), 145–147 (1972)
28. 28.
Sharir, M., Agarwal, P.K.: Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, New York (1995)
29. 29.
Shelah, S.: A combinatorial problem, stability and order for models and theories in infinitary languages. Pac. J. Math. 41, 247–261 (1972)Google Scholar
30. 30.
Vapnik, V., Chervonenkis, A.: On the uniform convergence of relative frequencies of events to their probabilities. Theory Prob. Appl. 16(2), 264–280 (1971)
31. 31.
Welzl, E.: On spanning trees with low crossing numbers. In: Data Structures and Efficient Algorithms, Final Report on the DFG Special Joint Initiative, pp. 233–249. Springer, London (1992)Google Scholar

## Authors and Affiliations

• Kunal Dutta
• 1
• Esther Ezra
• 2
• Arijit Ghosh
• 3
1. 1.DataShape, INRIA Sophia Antipolis – MéditerranéeValbonneFrance
2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
3. 3.ACM Unit, Indian Statistical InstituteKolkataIndia