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Masahiro Ikeda

Publications and source records attributed to Masahiro Ikeda.

At least 37 records · Page 2Linked to original sources
arXiv2025

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↗
arXiv2025

Existence of solutions to the semilinear damped wave equation with non-$L^2$ slowly decaying data : polynomial nonlinearity case

We study the Cauchy problem of the semilinear damped wave equation with polynomial nonlinearity, and establish the local and global existence of the solution for slowly decaying initial data not belonging to $L^2(\mathbb{R}^n)$ in general. Our approach is based on the $L^p$-$L^q$ estimates of linear solutions and the fractional Leibniz rule in suitable homogeneous Besov spaces.

math.AP↗
arXiv2025

Global dynamics for the energy-critical nonlinear heat equation

We examine the energy-critical nonlinear heat equation in critical spaces for any dimension greater or equal than three. The aim of this paper is two-fold. First, we establish a necessary and sufficient condition on initial data at or below the ground state that dichotomizes the behavior of solutions. Specifically, this criterion determines whether the solution will either exist globally with energy decaying to zero over time or blow up in finite time. Secondly, we derive the decay rate for solutions that exist globally. These results offer a comprehensive characterization of solution behavior for energy-critical conditions in higher-dimensional settings

math.AP↗
arXiv2025

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↗
arXiv2024

Optimal $L^2$-blowup estimates of the Fractional Wave Equation

This article deals with the behavior in time of the solution to the Cauchy problem for a fractional wave equation with a weighted $L^1$ initial data. Initially, we establish the global existence of the solution using Fourier methods and provide upper bounds for the $L^2$ norm and the $H^s$ norm of the solution for any dimension $n\in \mathbb{N}$ and $s\in (0,1)$. However, when $n=1$ and $s \in [\frac{1}{2},1)$, %we have to assume that the initial velocity satisfies we have to impose a stronger assumption $\int_{\mathbb{R}}u_1(x)dx=0$. To remove this stronger assumption, we further use the Fourier splitting method, which yields the optimal blow-up rate for the $L^2$ norm of the solutions. Specifically, when $n=1$, the optimal blow-up rate is $t^{1-\frac{1}{2s}}$ for $s \in (\frac{1}{2},1)$ and $\sqrt{\log t}$ for $s = \frac{1}{2}$.

math.AP↗
arXiv2024

Boundedness of composition operators from Lorentz spaces to Orlicz spaces

The boundedness (continuity) of composition operators from some function space to another one is significant, though there are few results about this problem. Thus, in this study, we provide necessary and sufficient conditions on the boundedness of composition operators from Lorentz spaces to Orlicz spaces. We also give a counter example of a mapping which implies unboundedness of the composition operators from a Lebesgue space $L^p$ to another Lebesgue space $L^q$ with $p>q$. We emphasize that the measure spaces associated with the Lorentz space may be different from those associated with the Orlicz spaces. We give more examples and counterexamples of the composed mappings in the conditions satisfying our main results.

math.FA↗
arXiv2024

Weighted Trudinger-Moser inequalities in the subcritical Sobolev spaces and their applications

We study boundedness, optimality and attainability of Trudinger-Moser type maximization problems in the radial and the subcritical homogeneous Sobolev spaces $\dot{W}^{1,p}_{0, \text{rad}}(B_R^N)\,(p<N)$. Our results give a revision of an error in \cite[Theorem C]{HL}. Also, our inequality converges to the original Trudinger-Moser inequality as $p \nearrow N$ including optimal exponent and concentration limit. Finally, we consider an application of our inequality to elliptic problems with exponential nonlinearity.

math.AP↗
arXiv2024

Upper estimates of the lifespan for the fractional wave equations with time-dependent damping and a power nonlinearity of subcritical and critical Fujita exponent

In this paper, we study the Cauchy problem of the fractional wave equation with time-dependent damping and the source nonlinearity $f(u)\approx |u|^p$: $$ \begin{cases} \partial_t^2u(t,x)+(-Δ)^{σ/2} u(t,x)+b(t) \partial_t u(t,x) =f(u(t,x)),\ &(t,x)\ \in [0,T)\times \mathbb{R}^N,\\ u(0,x)=u_0(x),\ \partial_tu(0,x)=u_1(x),\ &x\ \in\ \mathbb{R}^N, \end{cases} $$ where $b(t)\approx (1+t)^{-β}$. In the subcritical and critical cases $1<p\leq p_c:=1+\frac σN$, we derive the upper estimates of the lifespan for fractional Laplacian with $0<σ<2$ and time-dependent damping $β\in [-1, 1)$ by the framework of ordinary differential inequality. The blow-up results, with the global existence in the supercritical case $p_c<p<\frac{N}{N-σ}$ obtained in [19], shows that the critical exponent for the fractional wave quation is $p_c=1+\fracσ{N}$ for $0<σ<2$. Moreover, together with the lower estimate of lifespan derived in [19], we could conclude that the estimate in this paper is sharp. Note that the our result of the critical case is completely new even in the classical case $b(t)=1$. We also consider the case of $β=1$, and obtain the upper estimate of the lifespan.

math.AP↗
arXiv2024

Optimal control problem of evolution equation governed by hypergraph Laplacian

In this paper, we consider an optimal control problem of an ordinary differential inclusion governed by the hypergraph Laplacian, which is defined as a subdifferential of a convex function and then is a set-valued operator. We can assure the existence of optimal control for a suitable cost function by using methods of a priori estimates established in the previous studies. However, due to the multivaluedness of the hypergraph Laplacian, it seems to be difficult to derive the necessary optimality condition for this problem. To cope with this difficulty, we introduce an approximation operator based on the approximation method of the hypergraph, so-called ``clique expansion.'' We first consider the optimality condition of the approximation problem with the clique expansion of the hypergraph Laplacian and next discuss the convergence to the original problem. In appendix, we state some basic properties of the clique expansion of the hypergraph Laplacian for future works.

math.OC↗
arXiv2024

Dynamic Structure Estimation from Bandit Feedback using Nonvanishing Exponential Sums

This work tackles the dynamic structure estimation problems for periodically behaved discrete dynamical system in the Euclidean space. We assume the observations become sequentially available in a form of bandit feedback contaminated by a sub-Gaussian noise. Under such fairly general assumptions on the noise distribution, we carefully identify a set of recoverable information of periodic structures. Our main results are the (computation and sample) efficient algorithms that exploit asymptotic behaviors of exponential sums to effectively average out the noise effect while preventing the information to be estimated from vanishing. In particular, the novel use of the Weyl sum, a variant of exponential sums, allows us to extract spectrum information for linear systems. We provide sample complexity bounds for our algorithms, and we experimentally validate our theoretical claims on simulations of toy examples, including Cellular Automata.

cs.DM↗
arXiv2024

Koopman Spectrum Nonlinear Regulators and Efficient Online Learning

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.

cs.LG↗
arXiv2024

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↗
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↗
arXiv2024

A unified Fourier slice method to derive ridgelet transform for a variety of depth-2 neural networks

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 $γ$ so that a network $\mathtt{NN}[γ]$ reproduces $f$, i.e. $\mathtt{NN}[γ]=f$. 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.

cs.LG↗
arXiv2024

Unconditional uniqueness and non-uniqueness for Hardy-Hénon parabolic equations

We study the problems of uniqueness for Hardy-Hénon parabolic equations, which are semilinear heat equations with the singular potential (Hardy type) or the increasing potential (Hénon type) in the nonlinear term. To deal with the Hardy-Hénon type nonlinearities, we employ weighted Lorentz spaces as solution spaces. We prove unconditional uniqueness and non-uniqueness, and we establish uniqueness criterion for Hardy-Hénon parabolic equations in the weighted Lorentz spaces. The results extend the previous works on the Fujita equation and Hardy equations in Lebesgue spaces.

math.AP↗
arXiv2024

Koopman and Perron-Frobenius Operators on reproducing kernel Banach spaces

Koopman and Perron-Frobenius operators for dynamical systems have been getting popular in a number of fields in science these days. Properties of the Koopman operator essentially depend on the choice of function spaces where it acts. Particularly the case of reproducing kernel Hilbert spaces (RKHSs) draws more and more attention in data science. In this paper, we give a general framework for Koopman and Perron-Frobenius operators on reproducing kernel Banach spaces (RKBSs). More precisely, we extend basic known properties of these operators from RKHSs to RKBSs and state new results, including symmetry and sparsity concepts, on these operators on RKBS for discrete and continuous time systems.

math.DS↗
arXiv2023

Global existence and asymptotic behavior for semilinear damped wave equations on measure spaces

This paper is concerned with the semilinear damped wave equation on a measure space with a self-adjoint operator, instead of the standard Laplace operator. Under a certain decay estimate on the corresponding heat semigroup, we establish the linear estimates which generalize the so-called Matsumura estimates, and prove the small data global existence of solutions to the damped wave equation based on the linear estimates. Our approach is based on a direct spectral analysis analogous to the Fourier analysis. The self-adjoint operators treated in this paper include some important examples such as the Laplace operators on Euclidean spaces, the Dirichlet Laplacian on an arbitrary open set, the Robin Laplacian on an exterior domain, the Schrödinger operator, the elliptic operator, the Laplacian on Sierpinski gasket, and the fractional Laplacian.

math.AP↗