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Yuka Hashimoto

Publications and source records attributed to Yuka Hashimoto.

At least 19 recordsLinked to original sources
arXiv2026

Unified generalization analysis for physics informed neural networks

Physics-Informed Neural Networks (PINNs) and their variational counterparts (VPINNs) are neural networks that incorporate physical laws, making them useful for scientific problems. Existing generalization analyses for PINNs and VPINNs remain limited, often requiring restrictive assumptions such as stability conditions or linear ellipticity. In this paper, we derive generalization bounds for neural networks that involve differentiation with respect to input variables, covering PINNs and VPINNs under a unified framework. We apply Taylor expansion to represent nonlinear differential operators as linear operators on a high-dimensional space, enabling the use of Koopman-based analysis and showing that high-rank networks can generalize well even in settings involving differential operators. We also show that the nonlinearity of the differential operator exponentially enlarges the bound, highlighting its significant impact on generalization.

cs.LG↗
arXiv2025

Why High-rank Neural Networks Generalize?: An Algebraic Framework with RKHSs

We derive a new Rademacher complexity bound for deep neural networks using Koopman operators, group representations, and reproducing kernel Hilbert spaces (RKHSs). The proposed bound describes why the models with high-rank weight matrices generalize well. Although there are existing bounds that attempt to describe this phenomenon, these existing bounds can be applied to limited types of models. We introduce an algebraic representation of neural networks and a kernel function to construct an RKHS to derive a bound for a wider range of realistic models. This work paves the way for the Koopman-based theory for Rademacher complexity bounds to be valid for more practical situations.

cs.LG↗
arXiv2025

Position: Quantum Kernel Machines Should Move Beyond Scalar-Valued Kernels to Realize Their Potential

Quantum kernel functions built using quantum-mechanical principles and have emerged as a centerpiece of quantum machine learning. The initial enthusiasm for quantum kernel machines has been tempered by recent studies suggesting that quantum kernels could not offer significant computational or statistical advantages when learning from classical data. However, most of the research in this area has been devoted to scalar-valued kernels in standard classification or regression settings for which classical kernel methods are efficient and effective, leaving very little room for improvement with quantum kernels. In this position paper, we argue that progress in this field requires moving beyond scalar-valued kernels toward more expressive kernel frameworks. Scalar-valued kernels lack the degrees of freedom necessary to fully exploit intrinsically quantum resources such as entanglement and are not rich enough to deal with complex learning tasks where classical learning methods struggle. Building on recent advances in operator-valued kernel learning and $C^*$-algebraic kernel representations, we propose a roadmap for designing quantum kernels capable of leveraging entanglement and non-commutative structures to tackle complex structured prediction problems. To support this viewpoint, we present an initial proof-of-concept illustrating how quantum operator-valued kernel formulations can reveal structural dependencies that remain difficult to access for scalar-valued kernel methods. This shift in focus could open a pathway toward a new generation of quantum kernel machines and a more faithful exploration of their potential advantages.

quant-ph↗
arXiv2025

Why and When Deep is Better than Shallow: Implementation-Agnostic State-Transition Model of Deep Learning

Why and when does depth improve generalization? We study this question in an implementation-agnostic state-transition model, where a depth-$k$ predictor is a readout class $H$ composed with the word ball $B(k,F)$ generated by hidden state transitions. Generalization bounds separate implementation error, approximation error, and statistical complexity, and upper bound the depth-dependent variance term by a Dudley entropy integral over $B(k,F)$, with a conditional lower-bound diagnostic under readout separation. We identify geometric and semigroup mechanisms that keep this entropy contribution saturated or polynomial, and contrast them with separation mechanisms that recover the classical exponential-growth obstruction. Coupling these variance upper bounds with approximation rates gives typical depth trade-off patterns, clarifying that depth is statistically favorable when approximation improves rapidly while the transition semigroup remains geometrically tame.

cs.LG↗
arXiv2025

A Data-Driven Framework for Koopman Semigroup Estimation in Stochastic Dynamical Systems

We present Stochastic Dynamic Mode Decomposition (SDMD), a novel data-driven framework for approximating the Koopman semigroup in stochastic dynamical systems. Unlike existing methods, SDMD explicitly incorporates sampling time into its approximation, ensuring numerical stability and precision. By directly approximating the Koopman semigroup instead of the generator, SDMD avoids computationally expensive matrix exponential computations, which offers a more efficient and practical pathway for analyzing stochastic dynamics. The framework further integrates neural networks to automate basis selection, which reduces the reliance on manual intervention while maintaining computational efficiency. Rigorous theoretical guarantees, including convergence in the large data limit, zero-limit of sampling time, and large dictionary size, establish the method's reliability. Numerical experiments on canonical stochastic systems validate SDMD's effectiveness in approximating eigenvalues and eigenfunctions of the stochastic Koopman operator.

math.DS↗
arXiv2024

Deep Koopman-layered Model with Universal Property Based on Toeplitz Matrices

We propose deep Koopman-layered models with learnable parameters in the form of Toeplitz matrices for analyzing the transition of the dynamics of time-series data. The proposed model has both theoretical solidness and flexibility. By virtue of the universal property of Toeplitz matrices and the reproducing property underlying the model, we show its universality and generalization property. In addition, the flexibility of the proposed model enables the model to fit time-series data coming from nonautonomous dynamical systems. When training the model, we apply Krylov subspace methods for efficient computations, which establish a new connection between Koopman operators and numerical linear algebra. We also empirically demonstrate that the proposed model outperforms existing methods on eigenvalue estimation of multiple Koopman operators for nonautonomous systems.

cs.LG↗
arXiv2024

Spectral Truncation Kernels: Noncommutativity in $C^*$-algebraic Kernel Machines

A central question in vector- and function-valued learning is how to design kernels that capture both local and non-local interactions while remaining computationally tractable. Existing operator-valued kernels offer only partial answers: separable kernels are efficient but fail to model interactions across the function domain, while commutative kernels capture only pointwise structure. To address this, we propose spectral truncation kernels, a new class of positive definite kernels for vector- and function-valued learning based on spectral truncation and $C^*$-algebra. By allowing noncommutative products in the kernel construction, the proposed kernels induce interactions across the data function domain and fill the gap between existing separable and commutative kernels. In addition, by using the $C^*$-algebraic framework, we reduce the computational cost compared to the existing vector-valued RKHS framework with operator-valued kernels.

stat.ML↗
arXiv2024

Deep Ridgelet Transform and Unified Universality Theorem for Deep and Shallow Joint-Group-Equivariant Machines

We present a constructive universal approximation theorem for learning machines equipped with joint-group-equivariant feature maps, called the joint-equivariant machines, based on the group representation theory. ``Constructive'' here indicates that the distribution of parameters is given in a closed-form expression known as the ridgelet transform. Joint-group-equivariance encompasses a broad class of feature maps that generalize classical group-equivariance. Particularly, fully-connected networks are not group-equivariant but are joint-group-equivariant. Our main theorem also unifies the universal approximation theorems for both shallow and deep networks. Until this study, the universality of deep networks has been shown in a different manner from the universality of shallow networks, but our results discuss them on common ground. Now we can understand the approximation schemes of various learning machines in a unified manner. As applications, we show the constructive universal approximation properties of four examples: depth-$n$ joint-equivariant machine, depth-$n$ fully-connected network, depth-$n$ group-convolutional network, and a new depth-$2$ network with quadratic forms whose universality has not been known.

cs.LG↗
arXiv2024

Quantum Circuit $C^*$-algebra Net

This paper introduces quantum circuit $C^*$-algebra net, which provides a connection between $C^*$-algebra nets proposed in classical machine learning and quantum circuits. Using $C^*$-algebra, a generalization of the space of complex numbers, we can represent quantum gates as weight parameters of a neural network. By introducing additional parameters, we can induce interaction among multiple circuits constructed by quantum gates. This interaction enables the circuits to share information among them, which contributes to improved generalization performance in machine learning tasks. As an application, we propose to use the quantum circuit $C^*$-algebra net to encode classical data into quantum states, which enables us to integrate classical data into quantum algorithms. Numerical results demonstrate that the interaction among circuits improves performance significantly in image classification, and encoded data by the quantum circuit $C^*$-algebra net are useful for downstream quantum machine learning tasks.

cs.LG↗
arXiv2024

Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces

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 Extended Dynamic Mode Decomposition (JetEDMD), 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 JetEDMD's superiority through explicit error bounds and convergence rate for special positive definite kernels, offering a solid theoretical foundation for its performance. We also investigate the spectral analysis of the Koopman operator, proposing the notion of an extended Koopman operator within a framework of a 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 JetEDMD with clear numerical computations of eigenvalues and accurate predictions of the dynamical systems.

math.DS↗
arXiv2024

$C^*$-Algebraic Machine Learning: Moving in a New Direction

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.

cs.LG↗
arXiv2023

Koopman spectral analysis of skew-product dynamics on Hilbert $C^*$-modules

We introduce a linear operator on a Hilbert $C^*$-module for analyzing skew-product dynamical systems. The operator is defined by composition and multiplication. We show that it admits a decomposition in the Hilbert $C^*$-module, called eigenoperator decomposition, that generalizes the concept of the eigenvalue decomposition. This decomposition reconstructs the Koopman operator of the system in a manner that represents the continuous spectrum through eigenoperators. In addition, it is related to the notions of cocycle and Oseledets subspaces and it is useful for characterizing coherent structures under skew-product dynamics. We present numerical applications to simple systems on two-dimensional domains.

math.DS↗
arXiv2023

Deep Learning with Kernels through RKHM and the Perron-Frobenius Operator

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.

stat.ML↗
arXiv2023

Koopman-based generalization bound: New aspect for full-rank weights

We propose a new bound for generalization of neural networks using Koopman operators. Whereas most of existing works focus on low-rank weight matrices, we focus on full-rank weight matrices. Our bound is tighter than existing norm-based bounds when the condition numbers of weight matrices are small. Especially, it is completely independent of the width of the network if the weight matrices are orthogonal. Our bound does not contradict to the existing bounds but is a complement to the existing bounds. As supported by several existing empirical results, low-rankness is not the only reason for generalization. Furthermore, our bound can be combined with the existing bounds to obtain a tighter bound. Our result sheds new light on understanding generalization of neural networks with full-rank weight matrices, and it provides a connection between operator-theoretic analysis and generalization of neural networks.

cs.LG↗
arXiv2023

Noncommutative $C^*$-algebra Net: Learning Neural Networks with Powerful Product Structure in $C^*$-algebra

We propose a new generalization of neural network parameter spaces with noncommutative $C^*$-algebra, which possesses a rich noncommutative structure of products. We show that this noncommutative structure induces powerful effects in learning neural networks. Our framework has a wide range of applications, such as learning multiple related neural networks simultaneously with interactions and learning equivariant features with respect to group actions. Numerical experiments illustrate the validity of our framework and its potential power.

math.OA↗
arXiv2022

Learning in RKHM: a $C^*$-Algebraic Twist for Kernel Machines

Supervised learning in reproducing kernel Hilbert space (RKHS) and vector-valued RKHS (vvRKHS) has been investigated for more than 30 years. In this paper, we provide a new twist to this rich literature by generalizing supervised learning in RKHS and vvRKHS to reproducing kernel Hilbert $C^*$-module (RKHM), and show how to construct effective positive-definite kernels by considering the perspective of $C^*$-algebra. Unlike the cases of RKHS and vvRKHS, we can use $C^*$-algebras to enlarge representation spaces. This enables us to construct RKHMs whose representation power goes beyond RKHSs, vvRKHSs, and existing methods such as convolutional neural networks. Our framework is suitable, for example, for effectively analyzing image data by allowing the interaction of Fourier components.

stat.ML↗
arXiv2022

Operator-theoretic Analysis of Mutual Interactions in Synchronized Dynamics

Analyzing synchronized nonlinear oscillators is one of the most important and attractive topics in nonlinear science. By understanding the interactions between the oscillators, we can figure out the synchronization process. A promising approach to the analysis of interacting oscillators in nonlinear science is the application of the phase model. In this paper, we propose a data-driven approach to extract mutual interactions of synchronized oscillators based on the phase model. Recently, applying machine learning techniques to estimate models in physics has been actively investigated. We propose an operator-theoretic approach to estimate the phase model of interacting oscillators. We reduce the estimation problem to a multiparameter eigenvalue problem of the Koopman operator, a linear operator that describes a dynamical system. By reducing the problem to a linear algebraic problem, we can theoretically show that the proposed approach is stable with respect to perturbations in the given data.

nlin.AO↗