Most modern reinforcement learning algorithms optimize a cumulative single-step cost along a trajectory. The optimized motions are often 'unnatural', representing, for example, behaviors with sudden accelerations that waste energy and lack predictability. In this work, we present a novel paradigm of controlling nonlinear systems via the minimization of the Koopman spectrum cost: a cost over the Koopman operator of the controlled dynamics. This induces a broader class of dynamical behaviors that evolve over stable manifolds such as nonlinear oscillators, closed loops, and smooth movements. We demonstrate that some dynamics characterizations that are not possible with a cumulative cost are feasible in this paradigm, which generalizes the classical eigenstructure and pole assignments to nonlinear decision making. Moreover, we present a sample efficient online learning algorithm for our problem that enjoys a sub-linear regret bound under some structural assumptions.

In this paper, we propose a new class of positive definite kernels based on the spectral truncation, which has been discussed in the fields of noncommutative geometry and $C^*$-algebra. We focus on kernels whose inputs and outputs are functions and generalize existing kernels, such as polynomial, product, and separable kernels, by introducing a truncation parameter $n$ that describes the noncommutativity of the products appearing in the kernels. When $n$ goes to infinity, the proposed kernels tend to the existing commutative kernels. If $n$ is finite, they exhibit different behavior, and the noncommutativity induces interactions along the data function domain. We show that the truncation parameter $n$ is a governing factor leading to performance enhancement: by setting an appropriate $n$, we can balance the representation power and the complexity of the representation space. The flexibility of the proposed class of kernels allows us to go beyond previous commutative kernels.

We present a unified constructive universal approximation theorem covering a wide range of learning machines including both shallow and deep neural networks based on the group representation theory. Constructive here means that the distribution of parameters is given in a closed-form expression (called the ridgelet transform). Contrary to the case of shallow models, expressive power analysis of deep models has been conducted in a case-by-case manner. Recently, Sonoda et al. (2023a,b) developed a systematic method to show a constructive approximation theorem from scalar-valued joint-group-invariant feature maps, covering a formal deep network. However, each hidden layer was formalized as an abstract group action, so it was not possible to cover real deep networks defined by composites of nonlinear activation function. In this study, we extend the method for vector-valued joint-group-equivariant feature maps, so to cover such real networks.

This paper presents a novel approach for estimating the Koopman operator defined on a reproducing kernel Hilbert space (RKHS) and its spectra. We propose an estimation method, what we call Jet Dynamic Mode Decomposition (JetDMD), leveraging the intrinsic structure of RKHS and the geometric notion known as jets to enhance the estimation of the Koopman operator. This method refines the traditional Extended Dynamic Mode Decomposition (EDMD) in accuracy, especially in the numerical estimation of eigenvalues. This paper proves JetDMD's superiority through explicit error bounds and convergence rate for special positive definite kernels, offering a solid theoretical foundation for its performance. We also delve into the spectral analysis of the Koopman operator, proposing the notion of extended Koopman operator within a framework of rigged Hilbert space. This notion leads to a deeper understanding of estimated Koopman eigenfunctions and capturing them outside the original function space. Through the theory of rigged Hilbert space, our study provides a principled methodology to analyze the estimated spectrum and eigenfunctions of Koopman operators, and enables eigendecomposition within a rigged RKHS. We also propose a new effective method for reconstructing the dynamical system from temporally-sampled trajectory data of the dynamical system with solid theoretical guarantee. We conduct several numerical simulations using the van der Pol oscillator, the Duffing oscillator, the H\'enon map, and the Lorenz attractor, and illustrate the performance of JetDMD with clear numerical computations of eigenvalues and accurate predictions of the dynamical systems.

To investigate neural network parameters, it is easier to study the distribution of parameters than to study the parameters in each neuron. The ridgelet transform is a pseudo-inverse operator that maps a given function $f$ to the parameter distribution $\gamma$ so that a network $\mathtt{NN}[\gamma]$ reproduces $f$, i.e. $\mathtt{NN}[\gamma]=f$. For depth-2 fully-connected networks on a Euclidean space, the ridgelet transform has been discovered up to the closed-form expression, thus we could describe how the parameters are distributed. However, for a variety of modern neural network architectures, the closed-form expression has not been known. In this paper, we explain a systematic method using Fourier expressions to derive ridgelet transforms for a variety of modern networks such as networks on finite fields $\mathbb{F}_p$, group convolutional networks on abstract Hilbert space $\mathcal{H}$, fully-connected networks on noncompact symmetric spaces $G/K$, and pooling layers, or the $d$-plane ridgelet transform.

Machine learning has a long collaborative tradition with several fields of mathematics, such as statistics, probability and linear algebra. We propose a new direction for machine learning research: $C^*$-algebraic ML $-$ a cross-fertilization between $C^*$-algebra and machine learning. The mathematical concept of $C^*$-algebra is a natural generalization of the space of complex numbers. It enables us to unify existing learning strategies, and construct a new framework for more diverse and information-rich data models. We explain why and how to use $C^*$-algebras in machine learning, and provide technical considerations that go into the design of $C^*$-algebraic learning models in the contexts of kernel methods and neural networks. Furthermore, we discuss open questions and challenges in $C^*$-algebraic ML and give our thoughts for future development and applications.

We identify hidden layers inside a deep neural network (DNN) with group actions on the data domain, and formulate a formal deep network as a dual voice transform with respect to the Koopman operator, a linear representation of the group action. Based on the group theoretic arguments, particularly by using Schur's lemma, we show a simple proof of the universality of DNNs. Keywords: deep neural network, group representation, Koopman operator, Schur's lemma, voice transform

The symmetry and geometry of input data are considered to be encoded in the internal data representation inside the neural network, but the specific encoding rule has been less investigated. In this study, we present a systematic method to induce a generalized neural network and its right inverse operator, called the ridgelet transform, from a joint group invariant function on the data-parameter domain. Since the ridgelet transform is an inverse, (1) it can describe the arrangement of parameters for the network to represent a target function, which is understood as the encoding rule, and (2) it implies the universality of the network. Based on the group representation theory, we present a new simple proof of the universality by using Schur's lemma in a unified manner covering a wide class of networks, for example, the original ridgelet transform, formal deep networks, and the dual voice transform. Since traditional universality theorems were demonstrated based on functional analysis, this study sheds light on the group theoretic aspect of the approximation theory, connecting geometric deep learning to abstract harmonic analysis. Keywords: ridgelet transform, universality, joint group invariant function, Schur's lemma

Reproducing kernel Hilbert $C^*$-module (RKHM) is a generalization of reproducing kernel Hilbert space (RKHS) by means of $C^*$-algebra, and the Perron-Frobenius operator is a linear operator related to the composition of functions. Combining these two concepts, we present deep RKHM, a deep learning framework for kernel methods. We derive a new Rademacher generalization bound in this setting and provide a theoretical interpretation of benign overfitting by means of Perron-Frobenius operators. By virtue of $C^*$-algebra, the dependency of the bound on output dimension is milder than existing bounds. We show that $C^*$-algebra is a suitable tool for deep learning with kernels, enabling us to take advantage of the product structure of operators and to provide a clear connection with convolutional neural networks. Our theoretical analysis provides a new lens through which one can design and analyze deep kernel methods.

Overparametrization has been remarkably successful for deep learning studies. This study investigates an overlooked but important aspect of overparametrized neural networks, that is, the null components in the parameters of neural networks, or the ghosts. Since deep learning is not explicitly regularized, typical deep learning solutions contain null components. In this paper, we present a structure theorem of the null space for a general class of neural networks. Specifically, we show that any null element can be uniquely written by the linear combination of ridgelet transforms. In general, it is quite difficult to fully characterize the null space of an arbitrarily given operator. Therefore, the structure theorem is a great advantage for understanding a complicated landscape of neural network parameters. As applications, we discuss the roles of ghosts on the generalization performance of deep learning.