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The L1 Method

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Scientific Data Analysis

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Abstract

In Chapter 4 we pointed out that the L 1 criterion, i.e., minimize the sum of the absolute values of the residualsis,

$$ {{\left\| r \right\|}_{1}} = {{\left\| {A\cdot X - d} \right\|}_{1}} = \min , $$
(6.1)

because of its robustness, extremely useful for data reduction. Where does this robustness come from? The answer to this question will be given in several stages.

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References

  • Abdelmalek, N.N. (1971). Linear L1 Approximation for a Discrete Point Set and L1 Solutions of Overdetermined Linear Equations, J. ACM, 18, p. 41.

    Article  MathSciNet  MATH  Google Scholar 

  • Abdelmalek, N.N. (1975). An Efficient Method for the Discrete Linear L1 Approximation Problem, Math. Comp., 29, p. 844.

    MathSciNet  MATH  Google Scholar 

  • Armstrong, R.D. and Godfrey, J.P. (1979). Two Linear Programming Algorithms for the Linear Discrete L1 Norm Problem, Math. Comp., 33, p. 289.

    MathSciNet  MATH  Google Scholar 

  • Barrodale, L and Roberts, F.D.K. (1973). An Improved Algorithm for Discrete L 1 Linear Approximation, SI AM J. Numer. Anal, 10, p. 839.

    Article  MathSciNet  MATH  Google Scholar 

  • Barrodale, L and Roberts, F.D.K. (1974). Algorithm 478: Solution of an Overdetermined System of Equations in the L1 Norm, Commun. ACM, 17, p. 319.

    Article  Google Scholar 

  • Bartels, RH., Conn, AR., and Sinclair, JW. (1978). Minimization Techniques for Piecewise Differentiable Functions: The L 1 Solution to an Overdetermined Linear System, SIAM J. Numer. Anal, 15, p. 224.

    Article  MathSciNet  MATH  Google Scholar 

  • Branham, R.L., Jr. (1986). Error Estimates with L1 Solutions, Celest. Mech., 39, p. 239.

    Article  MathSciNet  Google Scholar 

  • Forsythe, G., Malcolm, M.A., and Moler, C.B. (1977). Computer Methods for Mathematical Computations (Prentice-Hall, Englewood Cliffs, N.J.).

    MATH  Google Scholar 

  • Franklin, J. (1987). Convergence in Karmarkar’s Algorithm for Linear Programming, SIAM J. Num. Anal, 24, p. 928.

    Article  MathSciNet  MATH  Google Scholar 

  • Gass, S.L (1975). Linear Programming: Methods and Applications ( McGraw-Hill, New York).

    MATH  Google Scholar 

  • Gauss, K.F. (1963). Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections ( Dover, New York).

    MATH  Google Scholar 

  • Karmarkar, N. (1984). A New Polynomial-Time Algorithm for Linear Programming, Combinatorica, 4, p. 373.

    Article  MathSciNet  MATH  Google Scholar 

  • Patterson, E.M. (1969). Topology ( Interscience, New York).

    Google Scholar 

  • Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T. (1986). Numerical Recipes: The Art of Scientific Computing ( Cambridge University Press, Cambridge).

    Google Scholar 

  • Tewarson, R.P. (1973). Sparse Matrices (Academic Press, Orlando, Florida).

    MATH  Google Scholar 

  • Zelen, M. and Severo, NC. (1965). Probability Functions. In Abramowitz, M. and Stegun, LA. (eds.) Handbook of Mathematical Functions (Dover, New York)

    Google Scholar 

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Authors and Affiliations

  1. Jefe Area Matematicas y Del Centro Regional de Investigaciones Científicas y Tecnológicas, Casilla de Correo 131, 5500, Mendoza, Argentina

    Richard L. Branham Jr.

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  1. Richard L. Branham Jr.
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© 1990 Springer-Verlag New York Inc.

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Branham, R.L. (1990). The L1 Method. In: Scientific Data Analysis. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3362-6_6

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  • DOI: https://doi.org/10.1007/978-1-4612-3362-6_6

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7981-5

  • Online ISBN: 978-1-4612-3362-6

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