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Connect the dots: how many random points can a regular curve pass through?

Part of: Inference from stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Ery Arias-Castro ,
David L. Donoho ,
Xiaoming Huo  and
Craig A. Tovey
Ery Arias-Castro*
Affiliation:
Stanford University
David L. Donoho*
Affiliation:
Stanford University
Xiaoming Huo*
Affiliation:
Georgia Institute of Technology
Craig A. Tovey*
Affiliation:
Georgia Institute of Technology
*
∗ Postal address: Department of Statistics, Stanford University, Stanford, CA 94305, USA.
∗ Postal address: Department of Statistics, Stanford University, Stanford, CA 94305, USA.
∗∗ Postal address: School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA.
∗∗ Postal address: School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA.
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Abstract

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Given a class Γ of curves in [0, 1]2, we ask: in a cloud of n uniform random points, how many points can lie on some curve γ ∈ Γ? Classes studied here include curves of length less than or equal to L, Lipschitz graphs, monotone graphs, twice-differentiable curves, and graphs of smooth functions with m-bounded derivatives. We find, for example, that there are twice-differentiable curves containing as many as OP(n1/3) uniform random points, but not essentially more than this. More generally, we consider point clouds in higher-dimensional cubes [0, 1]d and regular hypersurfaces of specified codimension, finding, for example, that twice-differentiable k-dimensional hypersurfaces in Rd may contain as many as OP(nk/(2d-k)) uniform random points. We also consider other notions of ‘incidence’, such as curves passing through given location/direction pairs, and find, for example, that twice-differentiable curves in R2 may pass through at most OP(n1/4) uniform random location/direction pairs. Idealized applications in image processing and perceptual psychophysics are described and several open mathematical questions are identified for the attention of the probability community.

Keywords

Curve detectionfilament detectionε-entropyconfiguration functionconcentration of measurelongest increasing subsequencetraveling salesman problempattern recognition

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 62M40: Random fields; image analysis
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 37 , Issue 3 , September 2005 , pp. 571 - 603
Copyright
Copyright © Applied Probability Trust 2005 

References

Abramowicz, H., Horn, D., Naftali, U. and Sahar-Pikielny, C. (1996). An orientation-selective neural network and its application to cosmic muon identification. Nucl. Instr. Meth. Phys. Res. A 378, 305–311.Google Scholar
Arias-Castro, E., Donoho, D. L. and Huo, X. (2005). Adaptive multiscale detection of filamentary structures embedded in a background of uniform random points. To appear in Ann. Statist. Google Scholar
Arias-Castro, E., Donoho, D. L. and Huo, X. (2005). Near-optimal detection of geometric objects by fast multiscale methods. IEEE Trans. Inf. Theory 51, 2402–2425.Google Scholar
Baik, J. (2003). Limiting distribution of last passage percolation models. In Proc. 14th Internat. Congr. Math. Phys. Available at http://www.math.lsa.umich.edu/∼baik/.Google Scholar
Baik, J., Deift, P. and Johansson, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12, 1119–1178.CrossRefGoogle Scholar
Beck, J. and Chen, W. W. L. (1987). Irregularities of Distribution (Cambridge Tracts Math. 89). Cambridge University Press.Google Scholar
Bronšteı˘n, E. M. (1976). ε-entropy of convex sets and functions. Sibirsk. Mat. Ž. 17, 508–514, 715.Google Scholar
Clements, G. F. (1963). Entropies of sets of functions of bounded variation. Canad. J. Math. 15, 422–432.Google Scholar
DeVore, R. A. and Lorentz, G. G. (1993). Constructive Approximation. Springer, New York.Google Scholar
Diamond, J. and Kiely, K. (2002). Administration, agencies failed to connect the dots. USA Today, 17 May 2002. Available at http://www.usatoday.com/news/washington/2002/05/17/failure-usatcov.htm.Google Scholar
Donoho, D. L. (1999). Wedgelets: nearly minimax estimation of edges. Ann. Statist. 27, 859–897.CrossRefGoogle Scholar
Donoho, D. L. and Huo, X. (2002). Beamlets and multiscale image analysis. In Multiscale and Multiresolution Methods (Lecture Notes Comput. Sci. Eng. 20), eds Barth, T., Chan, T. and Haimes, R., Springer, Berlin, pp. 149–196.Google Scholar
Edmunds, D. E. and Triebel, H. (1996). Function Spaces, Entropy Numbers, and Differential Operators. Cambridge University Press.CrossRefGoogle Scholar
Federer, H. (1969). Geometric Measure Theory. Springer, New York.Google Scholar
Few, L. (1955). The shortest path and the shortest road through n points. Mathematika 2, 141–144.CrossRefGoogle Scholar
Field, D. J., Hayes, A. and Hess, R. F. (1993). Contour integration by the human visual system: evidence for a local ‘association field’. Vision Res. 33, 173–193.CrossRefGoogle ScholarPubMed
Frieze, A. M. (1991). On the length of the longest monotone subsequence in a random permutation. Ann. Appl. Prob. 1, 301–305.Google Scholar
Groeneboom, P. (2001). Ulam's problem and Hammersley's process. Ann. Prob. 29, 683–690.Google Scholar
Huo, X., Donoho, D. L., Tovey, C. A. and Arias-Castro, E. (2004). Dynamic programming methods for ‘connecting the dots’. Submitted.Google Scholar
Johnson, D. S., McGeoch, L. A. and Rothberg, E. E. (1996). Asymptotic experimental analysis for the Held–Karp traveling salesman bound. In Proc. 7th Annual ACM/SIAM Symp. Discrete Algorithms (Atlanta, GA), ACM, New York, pp. 341–350.Google Scholar
Kahn, J. M., Katz, R. H. and Pister, K. S. J. (2000). Emerging challenges: mobile networking for smart dust. J. Commun. Networks 2, 188–196.CrossRefGoogle Scholar
Kolmogorov, A. N. and Tikhomirov, V. M. (1959). ε-entropy and ε-capacity of sets in function spaces. Uspekhi Mat. Nauk 14, 3–86 (in Russian). English translation: Trans. Amer. Math. Soc. 17, 277-364.Google Scholar
Kovacs, I. and Julesz, B. (1993). A closed curve is much more than an incomplete one – effect of closure in figure–ground segmentation. Proc. Nat. Acad. Sci. USA 90, 7495–7497.Google Scholar
Lane, E. (2003). Board not ready to connect the dots. Probe into shuttle tragedy still working out the details. NewYork Newsday.Google Scholar
Logan, B. F. and Shepp, L. A. (1977). A variational problem for random Young tableaux. Adv. Math. 26, 206–222.Google Scholar
Matoušek, J. (1999). Geometric Discrepancy. An Illustrated Guide (Algorithms and Combinatorics 18). Springer, Berlin.Google Scholar
Meyer, Y. (1992). Wavelets and Operators. Cambridge University Press.Google Scholar
Odlyzko, A. M. and Rains, E. M. (2000). On longest increasing subsequences in random permutations. In Analysis, Geometry, Number Theory: The Mathematics of Leon Ehrenpreis (Philadelphia, PA, 1998; Contemp. Math. 251), American Mathematical Society, Providence, RI, pp. 439–451.Google Scholar
Schmidt, W. M. (1969). Irregularities of distribution. III. Pacific J. Math. 29, 225–234.Google Scholar
Schneider, B. (2002). Connecting the dots. CNN.com, Inside Politics, 17 May 2002. Available at http://www.cnn.com/2002/ALLPOLITICS/05/17/pol.play.dots/.Google Scholar
Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. John Wiley, New York.Google Scholar
Soshnikov, A. (1999). Universality at the edge of the spectrum in Wigner random matrices. Commun. Math. Phys. 207, 697–733.Google Scholar
Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Inst. Études Sci. Pub. Math. 81, 73–205.Google Scholar
Tracy, C. A. and Widom, H. (2001). On the distributions of the lengths of the longest monotone subsequences in random words. Prob. Theory Relat. Fields 119, 350–380.Google Scholar
Vershik, A. M. and Kerov, S. V. (1977). Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux. Dokl. Akad. Nauk. SSSR 233, 1024–1027 (in Russian). English translation: Soviet Math. Dokl. 18, 527-531.Google Scholar