2000 character limit reached
An SVD-like Decomposition of Bounded-Input Bounded-Output Functions (2404.00112v1)
Published 29 Mar 2024 in math.OC, cs.SY, and eess.SY
Abstract: The Singular Value Decomposition (SVD) of linear functions facilitates the calculation of their 2-induced norm and row and null spaces, haLLMarks of linear control theory. In this work, we present a function representation that, similar to SVD, provides an upper bound on the 2-induced norm of bounded-input bounded-output functions, as well as facilitates the computation of generalizations of the notions of row and null spaces. Borrowing from the notion of "lifting" in Koopman operator theory, we construct a finite-dimensional lifting of inputs that relaxes the unitary property of the right-most matrix in traditional SVD, $V*$, to be an injective, norm-preserving mapping to a slightly higher-dimensional space.
- Functional analysis. Courier Corporation, 2000.
- Polynomial decomposition algorithms. Journal of Symbolic Computation, 1(2):159–168, 1985.
- Frédéric Brechenmacher. Histoire du théorème de Jordan de la décomposition matricielle (1870-1930). Formes de représentation et méthodes de décomposition. PhD thesis, Ecole des Hautes Etudes en Sciences Sociales (EHESS), 2006.
- Modern koopman theory for dynamical systems. SIAM Review, 64(2):229–340, 2022.
- M. D. Buhmann. Radial basis functions. Acta Numerica, 9:1–38, 2000.
- Generalized svd reduced-order observers for nonlinear systems. In 2020 American Control Conference (ACC), pages 3473–3478, 2020.
- The approximation of one matrix by another of lower rank. Psychometrika, 1(3):211–218, 1936.
- JBJ Fourier. Mémoire sur la propagation de la chaleur dans les corps solides,(nepublikováno) pro institute de france. Paris, podáno, 21, 1807.
- Michael J Greenacre. Theory and applications of correspondence analysis. 1984.
- Matrix analysis. Cambridge university press, 2012.
- Extended dynamic mode decomposition with invertible dictionary learning. Neural Networks, 173:106177, 2024.
- A class of logistic functions for approximating state-inclusive koopman operators. In 2018 Annual American Control Conference (ACC), pages 4803–4810, 2018.
- Koopman operator in systems and control. Springer, 2020.
- Koopman operator learning using invertible neural networks. Journal of Computational Physics, 501:112795, 2024.
- Igor Mezić. Analysis of fluid flows via spectral properties of the koopman operator. Annual review of fluid mechanics, 45:357–378, 2013.
- Wavelet analysis and applications. Springer Science & Business Media, 2007.
- Walter Rudin. Real and complex analysis. 1987. Cited on, 156:16, 1987.
- Nonlinear svd with asymmetric kernels: feature learning and asymmetric nystr\\\backslash\” om method. arXiv preprint arXiv:2306.07040, 2023.
- A non-linear generalization of singular value decomposition and its application to cryptanalysis. arXiv preprint arXiv:0711.4910, 2007.
- Charles F. Van Loan. Generalizing the singular value decomposition. SIAM Journal on Numerical Analysis, 13(1):76–83, 1976.
- A data–driven approximation of the koopman operator: Extending dynamic mode decomposition. Journal of Nonlinear Science, 25:1307–1346, 2015.
- Learning deep neural network representations for koopman operators of nonlinear dynamical systems, 2017.
- Machine learning by function decomposition. In ICML, pages 421–429. Citeseer, 1997.