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Joel Rosenfeld
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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This article addresses several longstanding misconceptions concerning Koopman operators, including the existence of lattices of eigenfunctions, common eigenfunctions between Koopman operators, and boundedness and compactness of Koopman operators, among others. Counterexamples are provided for each misconception. This manuscript also proves that the Gaussian RBFs native space only supports bounded Koopman operator corresponding to affine dynamics, which shows that the assumption of boundedness is very limiting. A framework for DMD is presented that requires only densely defined Koopman operators over reproducing kernel Hilbert spaces, and the effectiveness of this approach is demonstrated through reconstruction examples.
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A majority of methods from dynamical systems analysis, especially those in applied settings, rely on Poincares geometric picture that focuses on dynamics of states. While this picture has fueled our field for a century, it has shown difficulties in handling high-dimensional, ill-described, and uncertain systems, which are more and more common in engineered systems design and analysis of big data measurements. This overview article presents an alternative framework for dynamical systems, based on the dynamics of observables picture. The central object is the Koopman operator: an infinite-dimensional, linear operator that is nonetheless capable of capturing the full nonlinear dynamics. The first goal of this paper is to make it clear how methods that appeared in different papers and contexts all relate to each other through spectral properties of the Koopman operator. The second goal is to present these methods in a concise manner in an effort to make the framework accessible to researchers who would like to apply them, but also, expand and improve them. Finally, we aim to provide a road map through the literature where each of the topics was described in detail. We describe three main concepts: Koopman mode analysis, Koopman eigenquotients, and continuous indicators of ergodicity. For each concept we provide a summary of theoretical concepts required to define and study them, numerical methods that have been developed for their analysis, and, when possible, applications that made use of them. The Koopman framework is showing potential for crossing over from academic and theoretical use to industrial practice. Therefore, the paper highlights its strengths, in applied and numerical contexts. Additionally, we point out areas where an additional research push is needed before the approach is adopted as an off-the-shelf framework for analysis and design.
Let $A$ and $B$ be positive semidefinite matrices. The limit of the expression $Z_p:=(A^{p/2}B^pA^{p/2})^{1/p}$ as $p$ tends to $0$ is given by the well known Lie-Trotter-Kato formula. A similar formula holds for the limit of $G_p:=(A^p,#,B^p)^{2/p}$ as $p$ tends to $0$, where $X,#,Y$ is the geometric mean of $X$ and $Y$. In this paper we study the complementary limit of $Z_p$ and $G_p$ as $p$ tends to $infty$, with the ultimate goal of finding an explicit formula, which we call the anti Lie-Trotter formula. We show that the limit of $Z_p$ exists and find an explicit formula in a special case. The limit of $G_p$ is shown for $2times2$ matrices only.
Large algebraic structures are found inside the space of sequences of continuous functions on a compact interval having the property that, the series defined by each sequence converges absolutely and uniformly on the interval but the series of the upper bounds diverges. So showing that there exist many examples satisfying the conclusion but not the hypothesis of the Weierstrass M-test.
We show that for a connected Lie group $G$, its Fourier algebra $A(G)$ is weakly amenable only if $G$ is abelian. Our main new idea is to show that weak amenability of $A(G)$ implies that the anti-diagonal, $check{Delta}_G={(g,g^{-1}):gin G}$, is a set of local synthesis for $A(Gtimes G)$. We then show that this cannot happen if $G$ is non-abelian. We conclude for a locally compact group $G$, that $A(G)$ can be weakly amenable only if it contains no closed connected non-abelian Lie subgroups. In particular, for a Lie group $G$, $A(G)$ is weakly amenable if and only if its connected component of the identity $G_e$ is abelian.
We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks. Precisely, we consider feedforward networks with a complex activation function $sigma : mathbb{C} to mathbb{C}$ in which each neuron performs the operation $mathbb{C}^N to mathbb{C}, z mapsto sigma(b + w^T z)$ with weights $w in mathbb{C}^N$ and a bias $b in mathbb{C}$, and with $sigma$ applied componentwise. We completely characterize those activation functions $sigma$ for which the associated complex networks have the universal approximation property, meaning that they can uniformly approximate any continuous function on any compact subset of $mathbb{C}^d$ arbitrarily well. Unlike the classical case of real networks, the set of good activation functions which give rise to networks with the universal approximation property differs significantly depending on whether one considers deep networks or shallow networks: For deep networks with at least two hidden layers, the universal approximation property holds as long as $sigma$ is neither a polynomial, a holomorphic function, or an antiholomorphic function. Shallow networks, on the other hand, are universal if and only if the real part or the imaginary part of $sigma$ is not a polyharmonic function.
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