# An Infinite Dimensional Variational Problem Arising in Estimation Theory

• Anthony M. Bloch
• Christopher I. Byrnes
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 29)

## Abstract

In this paper we derive the existence of and a parameterization for the local and global minima for the (total) least squares estimation of linear models describing an infinite sequence X of data in a (separable) Hilbert space. By definition, this problem is an infinite-dimensional nonlinear variational problem; e.g. for line fitting this is the problem of finding the minima of the (least squares) distance function
$${f_X}:\mathbb{C}{\mathbb{P}^{\infty }} \to \mathbb{R}$$
(*)
on infinite projective space.

While the importance of such variational problems in estimation theory and in approximation theory (cf. [14],[16] ) is clear, we want to draw attention to the impact of such problems on the emerging theory of nonlinear infinite-dimensional control. The application to such diverse problems of estimation and control as nonlinear filtering and the attitude control of flexible spacecraft of an infinite-dimensional realization theory, replete with controllability and observability criteria, is widely appreciated. Of course, both the nonlinear theory in finite dimensions and the linear theory in infinite dimensions are highly developed. Yet there is presently no infinite-dimensional analogue of some of the best understood, relatively simple, nonlinear situations, e.g. controllability of left-invariant systems on homogeneous spaces(Brockett [6]) or observability of Morse-Smale systems (Aeyels [1]).

Returning to (*), we note that the solution obtained in finite dimensions by Byrnes and Willems [7] used a blend of the two techniques mentioned above; viz. by viewing ℂ ℙn (or Grass (d,n)) as a homogeneous space, they studied the Morse theory of fX using the Lie theory of coadjoint orbits. In infinite dimensions this is far more delicate since the Banach-Lie theory of general adjoint orbits is much more technical and since fX does not satisfy Smale’s condition (C) for his infinite-dimensional Morse Theory. Among the technical contributions contained in this paper, we show that certain workable analogues of these finite-dimensional situations exist by analyzing (*) in terms of a generalization of Smale’s Morse Theory, ideally suited to the critical point analysis of linear functionals restricted to “finite rank orbits” of the infinite unitary group.

## Keywords

Morse Theory Infinite Dimension Critical Manifold Hamiltonian Vector Field Adjoint Orbit
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
D.J. Aeyels, ‘Global Observability of Morse-Smale Systems’, J. of Diff. Eqns. 45 (1982) 1–15.
2. 2.
M.F. Atiyah, ‘Convexity and Commuting Hamiltonians’, Bull. Lond. Math. Soc. 14 (1982) 1–15.
3. 3.
G. Birkhoff, ‘Analytic Groups’, Trans. A.M.S. 43 (1983) 61–107.
4. 4.
A.M. Bloch, Total Least Squares Estimation and Completely Integrable Hamiltonian Systems, Ph.D. Thesis, Harvard (in preparation).Google Scholar
5. 5.
A.M. Bloch and C.I. Byrnes, ‘Morse Theory on Trace Class Orbits’, to appear.Google Scholar
6. 6.
R.W. Brockett, ‘System Theory on Group Manifolds and Coset Spaces’, SIAM J. Control 10 (1972) 265–284.
7. 7.
C.I. Byrnes and J.C. Willems, ‘Least Squares Estimation, Linear Programming and Momentum’, to appear.Google Scholar
8. 8.
G.H. Golub and C.F. van Loan, ‘An Analysis of the Total Least Squares Problem’, SIAM J. Num. Analy. 17 No. 6, (1980) 883–893.
9. 9.
P. de la Harpe, Classical Banach-Lie Algebras and Banach-Lie Groups of Operators in Hilbert Space, Lecture Notes in Math 285, Springer-Verlag, Berline 1972.
10. 10.
A. Horn, ‘Doubly stochastic matrices and the diagonal of a rotation matrix’, Amer. J. Math. 76 (1956) 620–630.
11. 11.
B. Maissen, ‘Lie-Gruppen mit Banachräumen als Parameterräume’, Acta Math 108 (1962) 229–270.
12. 12.
J. Milnor, Morse Theory, Annals of Mathematics Studies Number 51, Princeton, New Jersey 1963.
13. 13.
R.S. Palais, Morse Theory on Hilbert Manifolds, Topology 2 (1963), 299–340.
14. 14.
I. Schur, ‘Uber eine Klasse von Mittelbildungen mit Anwendungen auf der Determinantentheorie’, Sitzungberichte der Berliner Mathematischen Gesellschaft 22 (1903) 9–20.Google Scholar
15. 15.
S. Smale, ‘Morse Theory and a nonlinear generalization of the Dirichlet problem’, Ann, of Math. 80 (1964) 382–396.
16. 16.
S. Watanabe, Karhunen-Loeve Expansion and Factor Analysis, Theoretical Remarks and Applications, Transactions of the 4th Prague Conference on Information Theory.Google Scholar

© D. Reidel Publishing Company 1986

## Authors and Affiliations

• Anthony M. Bloch
• 1
• Christopher I. Byrnes
• 2
• 3
1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
2. 2.Department of MathematicsArizona State UniversityTempeUSA
3. 3.Department of Electrical and Computer EngineeringArizona State UniversityTempeUSA