TOTAL LEAST SQUARES PROBLEMS - Neural-based Orthogonal Data Fitting: The EXIN Neural Networks - page 3

• Acoustic radiation problems, geology, inverse scattering, geophysical tomo-
graphic imaging, and so on See, respectively, [79,58,171,104].
• Environmental sciences . This includes the fitting of Horton's infiltration
curve using real data from field experiments and linear and nonlinear
rainfall-runoff modeling using real data from watersheds in Louisiana and
Belgium [159].
• Astronomy . This includes the determination of preliminary orbit in celestial
mechanics [19], its differential correction, and the estimation of various
parameters of galactic kinematics [12].

1.3 PRELIMINARIES

The TLS method solves a set of m linear equations in n × d unknowns X rep-
resented in matrix form by

AX ≈ B (1.1)

where A is the m × n data matrix and B is the m × d observation matrix .If
m > n , the system is overdetermined .If d > 1, the problem is multidimensional .If
n > 1, the problem is multidivariate . X is the n × d minimum norm least squares
solution and X is the minimum norm total least squares solution of eq. (1.1).
Singular value decomposition (SVD) of the matrix A ( m > n )ineq.(1.1)is
denoted by

U V T A = (1.2)

where
U = U 1 ; U 2 , U 1 = u 1 , ... , u n , U 2 = u n + 1 , ... , u m ,
u i

m , U T U = U U T

∈ = I m

V = v 1 , v n , v i

n , V T V = V V T

... , ∈ = I n

= diag σ 1 , ... , σ n ∈

m × n , σ 1 ≥···≥ σ n ≥ 0

and the SVD of the m × ( n + d ) matrix [ A ; B ] ( m > n ) in eq. (1.1) is denoted by

[ A ; B ] = U V T

(1.3)

where

, u n ] , U 2 = u n + 1 , , u m , U =

[ U 1 ; U 2 ] , U 1 = [ u 1 , ... ...

m , U T U

UU T u i ∈ = = I m