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PAC Learning with Generalized Samples and an Applicaiton to Stochastic Geometry
September 1993 (vol. 15 no. 9)
pp. 933-942

An extension of the standard probably approximately correct (PAC) learning model that allows the use of generalized samples is introduced. A generalized sample is viewed as a pair consisting of a functional on the concept class together with the value obtained by the functional operating on the unknown concept. It appears that this model can be applied to a number of problems in signal processing and geometric reconstruction to provide sample size bounds under a PAC criterion. A specific application of the generalized model to a problem of curve reconstruction is considered, and some connections with a result from stochastic geometry are discussed.

[1] N. Abe and M. K. Warmuth, "On the computational complexity of approximating distributions by probabilistic automata," inProc. Third Ann. Workshop Comput. Learning Theory, 1990, pp. 52-66.
[2] A. D. Alexandrov and Yu. G. Reshetnyak,General Theory of Irregular Curves, Mathematics and Its Applications (Soviet series). Boston: Kluwer, 1989, vol. 29.
[3] A. J. Baddeley, "Stochastic geometry and image analysis," inCWI Monographs. Amsterdam: North Holland, 1986.
[4] G. M. Benedek and A. Itai, "Learnability by fixed distributions," inProc. First Workshop Computat. Learning Theory, 1988, pp. 80-90.
[5] A. Blumer, A. Ehrenfeucht, D. Haussler, and M. Warmuth, "Classifying learnable geometric concepts with the Vapnik-Chervonenkis dimension," inProc. 18th ACM Symp. Theory Comput.(Berkeley, CA), 1986, pp. 273-282.
[6] R. M. Dudley, "Central limit theorems for empirical measures,"Ann. Probab., vol. 6, pp. 899-929, 1978.
[7] D. Haussler, "Decision theoretic generalizations of the PAC learning model for neural net and other learning applications,"Inform. Comput., vol. 100, pp. 78-150, 1992.
[8] W. C. Karl, "Reconstructing objects from projections," Ph.D. thesis, Dept. of EECS, Mass. Inst. of Technol., Feb. 1991.
[9] S. R. Kulkarni, "On metric entropy, Vapnik-Chervonenkis dimension and learnability for a class of distributions," Cent. for Intell. Contr. Syst. Rep. CICS-P-160, Mass. Inst. of Technol., 1989.
[10] S. R. Kulkarni, "Problems of computational and information complexity in machine vision and learning," Ph.D. thesis, Dept. of Elect. Eng. Comput. Sci., Mass. Inst. of Technol., June 1991.
[11] A. S. Lele, S. R. Kulkarni, and A. S. Willsky, "Convex set estimation from support line measurements and applications to target reconstruction from laser radar data," inSPIE Proc. Laser Radar V, 1990, pp. 58-82, vol. 1222 (submitted toJ. Opt. Soc. Amer.).
[12] P. A. P. Moran, "Measuring the length of a curve,"Biometrika, vol. 53, pp. 359-364, 1966.
[13] D. Pollard,Convergence of Stochastic Processes. New York: Springer-Verlag, 1984.
[14] J. L. Prince and A. S. Willsky, "Estimating convex sets from noisy support line measurements,"IEEE Trans. Patt. Anal. Machine Intell., vol. 12, pp. 377-389, 1990.
[15] T. J. Richardson, forthcoming, 1992.
[16] L. A. Santalo, "Integral geometry and geometric probability,"Encyclopedia of Mathematics and its Applications. Reading, MA: Addison-Wesley, 1976, vol. 1.
[17] S. Sherman, "A comparison of linear measures in the plane,"Duke Math. J., vol. 9, pp. 1-9, 1942.
[18] S. S. Skiena, "Geometric probing," Ph.D. thesis, Dept. of Comput. Sci., Univ. of Illinois, Urbana-Champaign, Rep. no. UIUCDCS-R-88-1425, Apr. 1988.
[19] H. Steinhaus, "Length, shape, and area,"Colloquium Mathematicum, vol. 3, pp. 1-13, 1954.
[20] L. G. Valiant, "A theory of the learnable,"Comm. ACM, vol. 27, pp. 1134-1142, Nov. 1984.
[21] V. N. Vapnik and A. Ya. Chervonenkis, "On the uniform convergence of relative frequencies to their probabilities,"Theory Probab. Applications, vol. 16, no. 2, pp. 264-280, 1971.
[22] V. N. Vapnik and A. Ya. Chervonenkis, "On the uniform convergence of relative frequencies to their probabilities,"Theory Probab. Applications, vol. 16, no. 2, pp. 264-280, 1971.
[23] V. N. Vapnik,Estimation of Dependencies Based on Empirical Data. Berlin: Springer-Verlag, 1982.
[24] G. Wahba, "Spline models for observational data," inSeries in Applied Mathematics. New York: SIAM, 1990, vol. 59.

Index Terms:
probably approximately correct learning; PAC learning; generalized samples; stochastic geometry; signal processing; geometric reconstruction; sample size bounds; curve reconstruction; geometry; learning systems; signal processing; stochastic processes
Citation:
S.R. Kulkarni, S.K. Mitter, J.N. Tsitsiklis, O. Zeitouni, "PAC Learning with Generalized Samples and an Applicaiton to Stochastic Geometry," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 15, no. 9, pp. 933-942, Sept. 1993, doi:10.1109/34.232080
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