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Projections in L2 and L1 normed spaces and SVDs

Vartan Choulakian

Abstract

We compare the essential properties of projections in the L2 and L1 normed spaces by two methods: Projection operators and by minimization of the distance. In Euclidean geometry the orthogonality (L2- conjugacy) plays central role; while in L1 normed space L1- conjugacy (the sign function) plays central role. Furthermore, this fact appears in the Pythagorean Theorem and its Taxicab analogue. We also compare three singular value decompositions: SVD, Taxicab SVD and L1min-SVD.

Projections in L2 and L1 normed spaces and SVDs

Abstract

We compare the essential properties of projections in the L2 and L1 normed spaces by two methods: Projection operators and by minimization of the distance. In Euclidean geometry the orthogonality (L2- conjugacy) plays central role; while in L1 normed space L1- conjugacy (the sign function) plays central role. Furthermore, this fact appears in the Pythagorean Theorem and its Taxicab analogue. We also compare three singular value decompositions: SVD, Taxicab SVD and L1min-SVD.

Paper Structure

This paper contains 13 sections, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on $l_{p}$- normed spaces for $p=1$ and 2
  3. Pythagorean Theorem
  4. Taxicab Pythagorean Theorem
  5. Corollary 1
  6. Corollary 2
  7. Example
  8. Solution by Taxicab Pythagorean Theorem
  9. Modification
  10. SVD, Taxicab SVD and l$_{1}\min-$ SVD
  11. SVD and Taxicab SVD
  12. SVD and l$_{1}\min-$ SVD
  13. Conclusion