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Y , namely the behavior of canonical bases, which are also called orthonormal bases ex and ey , or Cartan frames of reference fex1 ; : : : ; exm j0g spanning y y the parameter space X and fe1 ; : : : ; en j0g spanning the observation space Y, here a 7 ! ex and b 7 ! ey . 10, the matrix of the metric ex e0 x D Im and e0 y ey D In takes the canonical form (“modular”). 99, left). 99, right) are achieved by the changes of bases (“CBS”) of type left ex 7 ! a, a 7 ! 101, left), and of type right ey 7 !

28) applies. 32). 34). A/ D 1, a one-dimensional subspace of X D R2 . For instance, if we introduce the parameter x3 D , the other coordinates of the parameter space X D R2 amount to x1 D 2x3 D 2 ; x2 D 3x3 D 3. A/ is a parameterized straight line L10 through the origin O illustrated by Fig. 3. The parameter space X D Rr (here r = 2) is sliced by the subspace, the 12 1 The First Problem of Algebraic Regression Fig. A/ D d (here a straight line L10 through the origin O). In the following section, let us consider the interpretation of MINOS by the three fundamental partitionings: algebraic (rank partitioning), geometric (slicing), and set-theoretical (fibering).

3 v232 3 v212 4 v2 5 D 22 102700 v232 2 2 2 2/ 2 2 2/ 2 2 2/ 2 2 2 D D 5 9 32 49 260 211 260 1 2 2 9 2/ C . 154) (subject to v213 C v223 C v233 D 1), Ä 23 35 Ä v13 v23 2v13 C 3v23 C 5v33 D 0 3v13 C 5v23 C 9v33 D 0 Ä Ä Ä Ä 5 5 3 5 v D v33 , 13 D v33 v23 9 3 2 9 2 2 9 2 1 ;v D ;v D : 7 23 14 33 14 There are four combinatorial solutions to generate square roots. 159) Here we have chosen the one with the positive sign exclusively. 162) 1-3 Case Study Let us continue this chapter by considering some important case studies.