Abstract
In this paper we interpret Dykstra's iterative procedure for finding anI-projection onto the intersection of closed, convex sets in terms of itsFenchel dual. Seen in terms of its dual formulation, Dykstra's algorithm isintuitive and can be shown to converge monotonically to the correctsolution. Moreover, we show that it is possible to sharply bound thelocation of the constrained optimal solution.
Similar content being viewed by others
References
Agresti, A. (1990). Categorical Data Analysis, Wiley, New York.
Bhattacharya, B. and Dykstra, R. L. (1995). A general duality approach to I-projections, J. Statist. Plann. Inference, 3, 146–159.
Csiszar, I. (1975). I-divergence geometry of probability distributions and minimization problems, Ann. Probab., 3, 146–159.
Deming, W. E. and Stephan, F. F. (1940). On a least square adjustment of a sampled frequency table when the expected marginal totals are known, Ann. Math. Statist., 11, 427–444.
Dykstra, R. L. (1985a). An iterative procedure for obtaining I-projections onto the intersection of convex sets, Ann. Probab., 13, 975–984.
Dykstra, R. L. (1985b). Computational aspects of I-projections, J. Statist. Comput. Simulation, 21, 265–274.
Dykstra, R. L. and Lemke, J. H. (1988). Duality of I-projections and maximum likelihood estimation for log-linear models under cone constraints, J. Amer. Statist. Assoc., 402, 546–554.
Good, I. J. (1963). Maximum entropy for hypothesis formulation, especially for multidimensional contingency tables, Ann. Math. Statist., 34, 911–934.
Hestenes, M. R. (1975). Optimization Theory, Wiley, New York.
Jaynes, E. T. (1957). Information theory and statistical mechanics, Phys. Rev., 106, 620–630.
Kullback, S. (1959). Information Theory and Statistics, Wiley, New York.
Norušis, M. J. (1988). SPSSX Advanced Statistics Guide, 2nd ed., McGraw-Hill, New York.
Patefield, W. M. (1982). Exact tests for trends in ordered contingency tables, J. Roy. Statist. Soc. Ser. C, 31, 32–43.
Rao, C. R. (1965). Linear Statistical Inference and Its Applications, Wiley, New York.
Rockafellar, R. T. (1970). Convex Analysis, Princeton University Press, New York.
Sanov, I. N. (1957). On the probability of large deviations of random variables, Mat. Sb., 42, 11–44
Winkler, W. (1990). On Dykstra's iterative proportional fitting procedure, Ann. Probab., 18, 1410–1415.
Author information
Authors and Affiliations
About this article
Cite this article
Bhattacharya, B., Dykstra, R.L. A Fenchel Duality Aspect of Iterative I-Projection Procedures. Annals of the Institute of Statistical Mathematics 49, 435–446 (1997). https://doi.org/10.1023/A:1003110627413
Issue Date:
DOI: https://doi.org/10.1023/A:1003110627413