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Connect the dots: how many random points can a regular curve pass through?
Ery Arias-Castro, David L. Donoho, Xiaoming Huo, and Craig A. Tovey
Source: Adv. in Appl. Probab. Volume 37, Number 3
(2005), 571-603.
Abstract
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.
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Keywords: Curve detection; filament detection; epsilon-entropy; configuration function; concentration of measure; longest increasing subsequence; traveling salesman problem; pattern recognition
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aap/1127483737
Digital Object Identifier: doi:10.1239/aap/1127483737
Mathematical Reviews number (MathSciNet): MR2156550
Zentralblatt MATH identifier: 1081.60006
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.
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.
Mathematical Reviews (MathSciNet): MR2275244
Zentralblatt MATH: 1091.62095
Digital Object Identifier: doi:10.1214/009053605000000787
Project Euclid: euclid.aos/1146576265
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.
Mathematical Reviews (MathSciNet): MR2246369
Digital Object Identifier: doi:10.1109/TIT.2005.850056
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/$\sim$baik/.
Mathematical Reviews (MathSciNet): MR1989002
Zentralblatt MATH: 1040.60080
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.
Mathematical Reviews (MathSciNet): MR1682248
Digital Object Identifier: doi:10.1090/S0894-0347-99-00307-0
JSTOR: links.jstor.org
Zentralblatt MATH: 0932.05001
Beck, J. and Chen, W. W. L. (1987). Irregularities of Distribution (Cambridge Tracts Math. 89). Cambridge University Press.
Mathematical Reviews (MathSciNet): MR903025
Bronšteĭn, E. M. (1976). $\varepsilon$-entropy of convex sets and functions. Sibirsk. Mat. Ž. 17, 508--514, 715.
Mathematical Reviews (MathSciNet): MR415155
Clements, G. F. (1963). Entropies of sets of functions of bounded variation. Canad. J. Math. 15, 422--432.
Mathematical Reviews (MathSciNet): MR158969
DeVore, R. A. and Lorentz, G. G. (1993). Constructive Approximation. Springer, New York.
Mathematical Reviews (MathSciNet): MR1261635
Zentralblatt MATH: 0797.41016
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.
Donoho, D. L. (1999). Wedgelets: nearly minimax estimation of edges. Ann. Statist. 27, 859--897.
Mathematical Reviews (MathSciNet): MR1724034
Digital Object Identifier: doi:10.1214/aos/1018031261
Project Euclid: euclid.aos/1018031261
Zentralblatt MATH: 0957.62029
Donoho, D. L. and Huo, X. (2002). Beamlets and multiscale image analysis. In Multiscale and Multiresolution Methods (Lecture Notes Comput. Sci. Eng. 20), eds T. Barth, T. Chan and R. Haimes, Springer, Berlin,
pp. 149--196.
pp. 149--196.
Edmunds, D. E. and Triebel, H. (1996). Function Spaces, Entropy Numbers, and Differential Operators. Cambridge University Press.
Mathematical Reviews (MathSciNet): MR1410258
Zentralblatt MATH: 0865.46020
Federer, H. (1969). Geometric Measure Theory. Springer, New York.
Mathematical Reviews (MathSciNet): MR257325
Zentralblatt MATH: 0176.00801
Few, L. (1955). The shortest path and the shortest road through $n$ points. Mathematika 2, 141--144.
Mathematical Reviews (MathSciNet): MR78715
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.
Frieze, A. M. (1991). On the length of the longest monotone subsequence in a random permutation. Ann. Appl. Prob. 1, 301--305.
Mathematical Reviews (MathSciNet): MR1102322
JSTOR: links.jstor.org
Groeneboom, P. (2001). Ulam's problem and Hammersley's process. Ann. Prob. 29, 683--690.
Mathematical Reviews (MathSciNet): MR1849174
Digital Object Identifier: doi:10.1214/aop/1008956689
Project Euclid: euclid.aop/1008956689
Zentralblatt MATH: 1013.60003
Huo, X., Donoho, D. L., Tovey, C. A. and Arias-Castro, E. (2004). Dynamic programming methods for `connecting the dots'. Submitted.
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.
Mathematical Reviews (MathSciNet): MR1381947
Zentralblatt MATH: 0845.90123
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.
Kolmogorov, A. N. and Tikhomirov, V. M. (1959). $\varepsilon$-entropy and $\varepsilon$-capacity of sets in function spaces. Uspekhi Mat. Nauk 14, 3--86 (in Russian). English translation: Trans. Amer. Math. Soc. 17, 277--364.
Mathematical Reviews (MathSciNet): MR112032
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.
Lane, E. (2003). Board not ready to connect the dots. Probe into shuttle tragedy still working out the details. NewYork Newsday.
Logan, B. F. and Shepp, L. A. (1977). A variational problem for random Young tableaux. Adv. Math. 26, 206--222.
Mathematical Reviews (MathSciNet): MR1417317
Digital Object Identifier: doi:10.1016/0001-8708(77)90030-5
Zentralblatt MATH: 0363.62068
Matoušek, J. (1999). Geometric Discrepancy. An Illustrated Guide (Algorithms and Combinatorics 18). Springer, Berlin.
Mathematical Reviews (MathSciNet): MR1697825
Meyer, Y. (1992). Wavelets and Operators. Cambridge University Press.
Mathematical Reviews (MathSciNet): MR1228209
Zentralblatt MATH: 0776.42019
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.
In Analysis, Geometry, Number Theory: The Mathematics of Leon Ehrenpreis (Philadelphia, PA, 1998; Contemp. Math. 251), American Mathematical Society, Providence, RI, pp. 439--451.
Mathematical Reviews (MathSciNet): MR1771285
Zentralblatt MATH: 0966.60010
Schmidt, W. M. (1969). Irregularities of distribution. III. Pacific J. Math. 29, 225--234.
Mathematical Reviews (MathSciNet): MR245531
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/.
Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. John Wiley, New York.
Mathematical Reviews (MathSciNet): MR838963
Soshnikov, A. (1999). Universality at the edge of the spectrum in Wigner random matrices. Commun. Math. Phys. 207, 697--733.
Mathematical Reviews (MathSciNet): MR1727234
Digital Object Identifier: doi:10.1007/s002200050743
Zentralblatt MATH: 1062.82502
Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Inst. Études Sci. Pub. Math. 81, 73--205.
Mathematical Reviews (MathSciNet): MR1361756
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.
Mathematical Reviews (MathSciNet): MR1821139
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.
Mathematical Reviews (MathSciNet): MR480398
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