arXiv subjects

Isao Ishikawa

Publications and source records attributed to Isao Ishikawa.

43 records · Page 3Linked to original sources

arXiv2020

Kernel Mean Embeddings of Von Neumann-Algebra-Valued Measures

Kernel mean embedding (KME) is a powerful tool to analyze probability measures for data, where the measures are conventionally embedded into a reproducing kernel Hilbert space (RKHS). In this paper, we generalize KME to that of von Neumann-algebra-valued measures into reproducing kernel Hilbert modules (RKHMs), which provides an inner product and distance between von Neumann-algebra-valued measures. Von Neumann-algebra-valued measures can, for example, encode relations between arbitrary pairs of variables in a multivariate distribution or positive operator-valued measures for quantum mechanics. Thus, this allows us to perform probabilistic analyses explicitly reflected with higher-order interactions among variables, and provides a way of applying machine learning frameworks to problems in quantum mechanics. We also show that the injectivity of the existing KME and the universality of RKHS are generalized to RKHM, which confirms many useful features of the existing KME remain in our generalized KME. And, we investigate the empirical performance of our methods using synthetic and real-world data.

stat.ML↗

arXiv2020

Analysis via Orthonormal Systems in Reproducing Kernel Hilbert $C^*$-Modules and Applications

Kernel methods have been among the most popular techniques in machine learning, where learning tasks are solved using the property of reproducing kernel Hilbert space (RKHS). In this paper, we propose a novel data analysis framework with reproducing kernel Hilbert $C^*$-module (RKHM), which is another generalization of RKHS than vector-valued RKHS (vv-RKHS). Analysis with RKHMs enables us to deal with structures among variables more explicitly than vv-RKHS. We show the theoretical validity for the construction of orthonormal systems in Hilbert $C^*$-modules, and derive concrete procedures for orthonormalization in RKHMs with those theoretical properties in numerical computations. Moreover, we apply those to generalize with RKHM kernel principal component analysis and the analysis of dynamical systems with Perron-Frobenius operators. The empirical performance of our methods is also investigated by using synthetic and real-world data.

stat.ML↗

arXiv2019

Metric on random dynamical systems with vector-valued reproducing kernel Hilbert spaces

Development of metrics for structural data-generating mechanisms is fundamental in machine learning and the related fields. In this paper, we give a general framework to construct metrics on random nonlinear dynamical systems, defined with the Perron-Frobenius operators in vector-valued reproducing kernel Hilbert spaces (vvRKHSs). We employ vvRKHSs to design mathematically manageable metrics and also to introduce operator-valued kernels, which enables us to handle randomness in systems. Our metric provides an extension of the existing metrics for deterministic systems, and gives a specification of the kernel maximal mean discrepancy of random processes. Moreover, by considering the time-wise independence of random processes, we clarify a connection between our metric and the independence criteria with kernels such as Hilbert-Schmidt independence criteria. We empirically illustrate our metric with synthetic data, and evaluate it in the context of the independence test for random processes. We also evaluate the performance with real time seris datas via clusering tasks.

stat.ML↗

arXiv2019

The global optimum of shallow neural network is attained by ridgelet transform

We prove that the global minimum of the backpropagation (BP) training problem of neural networks with an arbitrary nonlinear activation is given by the ridgelet transform. A series of computational experiments show that there exists an interesting similarity between the scatter plot of hidden parameters in a shallow neural network after the BP training and the spectrum of the ridgelet transform. By introducing a continuous model of neural networks, we reduce the training problem to a convex optimization in an infinite dimensional Hilbert space, and obtain the explicit expression of the global optimizer via the ridgelet transform.

stat.ML↗

arXiv2018

Metric on Nonlinear Dynamical Systems with Perron-Frobenius Operators

The development of a metric for structural data is a long-term problem in pattern recognition and machine learning. In this paper, we develop a general metric for comparing nonlinear dynamical systems that is defined with Perron-Frobenius operators in reproducing kernel Hilbert spaces. Our metric includes the existing fundamental metrics for dynamical systems, which are basically defined with principal angles between some appropriately-chosen subspaces, as its special cases. We also describe the estimation of our metric from finite data. We empirically illustrate our metric with an example of rotation dynamics in a unit disk in a complex plane, and evaluate the performance with real-world time-series data.

stat.ML↗

arXiv2018

Gamma factors for Asai representations of ${\rm GL}_2$

Let $E$ be a quadratic semisimple extension of a local field $F$ of characteristic zero. We determine explicit relation between gamma factors for Asai representations of $R_{E/F}{\rm GL}_{2/E}$ defined by the Weil-Deligne representations and local zeta integrals. When $E = F\times F$, the results were due to Henniart and Jacquet. We completed the theory in this article based on explicit calculation.

math.NT↗

arXiv2016

Integrals on $p$-adic upper half planes and Hida families over totally real fields

Bertolini-Darmon and Mok proved a formula of the second derivative of the two-variable $p$-adic $L$-function of a modular elliptic curve over a totally real field along the Hida family in terms of the image of a global point by some $p$-adic logarithm map. The theory of $p$-adic indefinite integrals and $p$-adic multiplicative integrals on $p$-adic upper half planes plays an important role in their work. In this paper, we generalize these integrals for $p$-adic measures which are not necessarily $\mathbf{Z}$-valued, and prove a formula of the second derivative of the two-variable $p$-adic $L$-function of an abelian variety of ${\rm GL}(2)$-type associated to a Hilbert modular form of weight 2.

math.NT↗