arXiv subjects

Koichi Tojo

Publications and source records attributed to Koichi Tojo.

8 recordsLinked to original sources

arXiv2026

Hurwitz-Radon numbers and proper actions of semisimple Lie groups

We study proper isometric actions of non-compact semisimple Lie groups on pseudo-Riemannian symmetric spaces. Motivated by Okuda's classification of semisimple symmetric spaces admitting proper $SL(2,\mathbb{R})$-actions [J. Differential Geom., 2013], we focus on symmetric spaces lying on the boundary of the existence of proper $SL(2,\mathbb{R})$-actions. As a rigidity result, we show that any connected non-compact semisimple Lie group acting properly on these symmetric spaces must be globally isomorphic to $Spin(n,1)$ up to compact factors. Moreover, the Hurwitz-Radon number arises as the largest value of $n$ for the existence of $Spin(n,1)$-proper actions. Our symmetric spaces include the pseudo-Riemannian hyperbolic space $\mathbf{H}_{+}^{N,N-1}$ of signature $(N,N-1)$.

math.DG↗

arXiv2023

Zariski dense discontinuous surface groups for reductive symmetric spaces

Let $G/H$ be a homogeneous space of reductive type with non-compact $H$. The study of deformations of discontinuous groups for $G/H$ was initiated by T.~Kobayashi. In this paper, we show that a standard discontinuous group $\Gamma$ admits a non-standard small deformation as a discontinuous group for $G/H$ if $\Gamma$ is isomorphic to a surface group of high genus and its Zariski closure is locally isomorphic to $SL(2,\mathbb{R})$. Furthermore, we also prove that if $G/H$ is a symmetric space and admits some non virtually abelian discontinuous groups, then $G$ contains a Zariski-dense discrete surface subgroup of high genus acting properly discontinuously on $G/H$. As a key part of our proofs, we show that for a discrete surface subgroup $\Gamma$ of high genus contained in a reductive group $G$, if the Zariski closure of $\Gamma$ is locally isomorphic to $SL(2,\mathbb{R})$, then $\Gamma$ admits a small deformation in $G$ whose Zariski closure is a reductive subgroup of the same real rank as $G$.

math.DG↗

arXiv2022

Universal approximation property of invertible neural networks

Invertible neural networks (INNs) are neural network architectures with invertibility by design. Thanks to their invertibility and the tractability of Jacobian, INNs have various machine learning applications such as probabilistic modeling, generative modeling, and representation learning. However, their attractive properties often come at the cost of restricting the layer designs, which poses a question on their representation power: can we use these models to approximate sufficiently diverse functions? To answer this question, we have developed a general theoretical framework to investigate the representation power of INNs, building on a structure theorem of differential geometry. The framework simplifies the approximation problem of diffeomorphisms, which enables us to show the universal approximation properties of INNs. We apply the framework to two representative classes of INNs, namely Coupling-Flow-based INNs (CF-INNs) and Neural Ordinary Differential Equations (NODEs), and elucidate their high representation power despite the restrictions on their architectures.

cs.LG↗

arXiv2021

Obstructions to the existence of compact Clifford-Klein forms for tangential symmetric spaces

For a homogeneous space $G/H$ of reductive type, we consider the tangential homogeneous space $G_\theta/H_\theta$. In this paper, we give obstructions to the existence of compact Clifford-Klein forms for such tangential symmetric spaces and obtain new tangential symmetric spaces which do not admit compact Clifford-Klein forms. As a result, in the class of irreducible classical semisimple symmetric spaces, we have only two types of symmetric spaces which are not proved not to admit compact Clifford-Klein forms. The existence problem of compact Clifford-Klein forms for homogeneous spaces of reductive type, which was initiated by T. Kobayashi in 1980s, has been studied by various methods but is not completely solved yet. On the other hand, the one for tangential homogeneous spaces has been studied since 2000s and an analogous criterion was proved by T. Kobayashi and T. Yoshino. In concrete examples, further works are needed to verify Kobayashi-Yoshino's condition by direct calculations. In this paper, some easy-to-check necessary conditions(=obstructions) for the existence of compact quotients in the tangential setting are given, and they are applied to the case of symmetric spaces. The conditions are related to various fields of mathematics such as associated pair of symmetric space, Calabi-Markus phenomenon, trivializability of vector bundle (parallelizability, Pontrjagin class), Hurwitz-Radon number and Pfister's theorem (the existence problem of common zero points of polynomials of odd degree).

math.RT↗

arXiv2020

Universal Approximation Property of Neural Ordinary Differential Equations

Neural ordinary differential equations (NODEs) is an invertible neural network architecture promising for its free-form Jacobian and the availability of a tractable Jacobian determinant estimator. Recently, the representation power of NODEs has been partly uncovered: they form an $L^p$-universal approximator for continuous maps under certain conditions. However, the $L^p$-universality may fail to guarantee an approximation for the entire input domain as it may still hold even if the approximator largely differs from the target function on a small region of the input space. To further uncover the potential of NODEs, we show their stronger approximation property, namely the $\sup$-universality for approximating a large class of diffeomorphisms. It is shown by leveraging a structure theorem of the diffeomorphism group, and the result complements the existing literature by establishing a fairly large set of mappings that NODEs can approximate with a stronger guarantee.

cs.LG↗

arXiv2020

Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators

Invertible neural networks based on coupling flows (CF-INNs) have various machine learning applications such as image synthesis and representation learning. However, their desirable characteristics such as analytic invertibility come at the cost of restricting the functional forms. This poses a question on their representation power: are CF-INNs universal approximators for invertible functions? Without a universality, there could be a well-behaved invertible transformation that the CF-INN can never approximate, hence it would render the model class unreliable. We answer this question by showing a convenient criterion: a CF-INN is universal if its layers contain affine coupling and invertible linear functions as special cases. As its corollary, we can affirmatively resolve a previously unsolved problem: whether normalizing flow models based on affine coupling can be universal distributional approximators. In the course of proving the universality, we prove a general theorem to show the equivalence of the universality for certain diffeomorphism classes, a theoretical insight that is of interest by itself.

cs.LG↗

arXiv2019

On a method to construct exponential families by representation theory

Exponential family plays an important role in information geometry. In arXiv:1811.01394, we introduced a method to construct an exponential family $\mathcal{P}=\{p_θ\}_{θ\inΘ}$ on a homogeneous space $G/H$ from a pair $(V,v_0)$. Here $V$ is a representation of $G$ and $v_0$ is an $H$-fixed vector in $V$. Then the following questions naturally arise: (Q1) when is the correspondence $θ\mapsto p_θ$ injective? (Q2) when do distinct pairs $(V,v_0)$ and $(V',v_0')$ generate the same family? In this paper, we answer these two questions (Theorems 1 and 2). Moreover, in Section 3, we consider the case $(G,H)=(\mathbb{R}_{>0}, \{1\})$ with a certain representation on $\mathbb{R}^2$. Then we see the family obtained by our method is essentially generalized inverse Gaussian distribution (GIG).

math.RT↗

arXiv2018

A method to construct exponential families by representation theory

In this paper, we give a method to construct "good" exponential families systematically by representation theory. More precisely, we consider a homogeneous space $G/H$ as a sample space and construct an exponential family invariant under the transformation group $G$ by using a representation of $G$. The method generates widely used exponential families such as normal, gamma, Bernoulli, categorical, Wishart, von Mises, Fisher-Bingham and hyperboloid distributions.

math.ST↗