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Shunsuke Ichiki

Publications and source records attributed to Shunsuke Ichiki.

18 recordsLinked to original sources
arXiv2026

A transversality theorem for multiple-point crossings under generic linear perturbations with Hausdorff measure estimates

We establish a transversality theorem for multiple-point crossings under generic linear perturbations with explicit Hausdorff measure estimates for the exceptional parameter set, and hence explicit upper bounds on its Hausdorff dimension. This strengthens our earlier result, which showed only that the exceptional parameter set has Lebesgue measure zero. As applications, we obtain results on normal crossings, injectivity, and embeddings under generic linear perturbations. The embedding result yields a refinement of Mather's stability theorem for generic projections when the target dimension is more than twice the source dimension, with an explicit upper bound on the Hausdorff dimension of the exceptional set.

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arXiv2023

Non-density of $C^0$-stable mappings on non-compact manifolds

The problem of density of $C^0$-stable mappings is a classical and venerable subject in singularity theory. In 1973, Mather showed that the set of proper $C^0$-stable mappings is dense in the set of all proper mappings, which implies that the set of $C^0$-stable mappings is dense in the set of all mappings if the source manifold is compact. The aim of this paper is to complement Mather's result and to provide new information to the subject. Namely, we show that the set of $C^0$-stable mappings is never dense in the set of all mappings if the source manifold is non-compact. As a corollary of this result and Mather's result, we can obtain a characterization of density of $C^0$-stable mappings, i.e., the set of $C^0$-stable mappings is dense in the set of all mappings if and only if the source manifold is compact. To prove the non-density result, we provide a more essential result by using the notion of topologically critical points.

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arXiv2022

Non-density of stable mappings on non-compact manifolds

Around 1970, Mather established a significant theory on the stability of $C^\infty$ mappings and gave a characterization of the density of proper stable mappings in the set of all proper mappings. The result yields a characterization of the density of stable mappings in the set of all mappings in the case where the source manifold is compact. The aim of this paper is to complement Mather's result. Namely, we show that the set of stable mappings in the set of all mappings is never dense if the source manifold is non-compact. Moreover, as a corollary of Mather's result and the main theorem of this paper, we give a characterization of the density of stable mappings in the set of all mappings in the case where the source manifold is not necessarily compact.

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arXiv2021

All unconstrained strongly convex problems are weakly simplicial

A multi-objective optimization problem is $C^r$ weakly simplicial if there exists a $C^r$ surjection from a simplex onto the Pareto set/front such that the image of each subsimplex is the Pareto set/front of a subproblem, where $0\leq r\leq \infty$. This property is helpful to compute a parametric-surface approximation of the entire Pareto set and Pareto front. It is known that all unconstrained strongly convex $C^r$ problems are $C^{r-1}$ weakly simplicial for $1\leq r \leq \infty$. In this paper, we show that all unconstrained strongly convex problems are $C^0$ weakly simplicial. The usefulness of this theorem is demonstrated in a sparse modeling application: we reformulate the elastic net as a non-differentiable multi-objective strongly convex problem and approximate its Pareto set (the set of all trained models with different hyper-parameters) and Pareto front (the set of performance metrics of the trained models) by using a Bézier simplex fitting method, which accelerates hyper-parameter search.

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arXiv2021

Characterization of the equality of weak efficiency and efficiency on convex free disposal hulls

In solving a multi-objective optimization problem by scalarization techniques, solutions to a scalarized problem are, in general, weakly efficient rather than efficient to the original problem. Thus, it is crucial to understand what problem ensures that all weakly efficient solutions are efficient. In this paper, we give a characterization of the equality of the weakly efficient set and the efficient set, provided that the free disposal hull of the domain is convex. By using this characterization, we obtain various mathematical applications. As a practical application, we show that all weakly efficient solutions to a multi-objective LASSO with mild modification are efficient.

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arXiv2020

Simpliciality of strongly convex problems

A multiobjective optimization problem is $C^r$ simplicial if the Pareto set and the Pareto front are $C^r$ diffeomorphic to a simplex and, under the $C^r$ diffeomorphisms, each face of the simplex corresponds to the Pareto set and the Pareto front of a subproblem, where $0\leq r\leq \infty$. In the paper titled "Topology of Pareto sets of strongly convex problems," it has been shown that a strongly convex $C^r$ problem is $C^{r-1}$ simplicial under a mild assumption on the ranks of the differentials of the mapping for $2\leq r \leq \infty$. On the other hand, in this paper, we show that a strongly convex $C^1$ problem is $C^0$ simplicial under the same assumption. Moreover, we establish a specialized transversality theorem on generic linear perturbations of a strongly convex $C^r$ mapping $(r\geq 2)$. By the transversality theorem, we also give an application of singularity theory to a strongly convex $C^r$ problem for $2\leq r \leq \infty$.

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arXiv2019

Characterization of generic transversality

In this paper, the notion of generic transversality and its characterization are given. The characterization is also a further improvement of the basic transversality result and its strengthening which was given by John Mather.

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arXiv2019

Topology of Pareto sets of strongly convex problems

A multiobjective optimization problem is simplicial if the Pareto set and front are homeomorphic to a simplex and, under the homeomorphisms, each face of the simplex corresponds to the Pareto set and front of a subproblem. In this paper, we show that strongly convex problems are simplicial under a mild assumption on the ranks of the differentials of the objective mappings. We further prove that one can make any strongly convex problem satisfy the assumption by a generic linear perturbation, provided that the dimension of the source is sufficiently larger than that of the target. We demonstrate that the location problems, a biological modeling, and the ridge regression can be reduced to multiobjective strongly convex problems via appropriate transformations preserving the Pareto ordering and the topology.

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arXiv2018

Transversality theorems on generic linearly perturbed mappings

In his celebrated paper "Generic projections", John Mather has given a striking transversality theorem and its applications on generic projections. On the other hand, in this paper, two transversality theorems on generic linearly perturbed $C^r$ mappings are shown $(r\geq 1)$. Moreover, some applications of the two theorems are also given.

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arXiv2018

Generic linear perturbations

In his celebrated paper "Generic projections", John Mather has shown that almost all linear projections from a submanifold of a vector space into a subspace are transverse with respect to a given modular submanifold. In this paper, an improvement of Mather's result is stated. Namely, we show that almost all linear perturbations of a smooth mapping from a submanifold of $\mathbb{R}^m$ into $\mathbb{R}^\ell$ yield a transverse mapping with respect to a given modular submanifold. Moreover, applications of this result are given.

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arXiv2018

Generic distance-squared mappings on plane curves

A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. Moreover, distance-squared mappings are naturally extended mappings of distance-squared functions, wherein each component is a distance-squared function. In this paper, compositions of a given plane curve and generic distance-squared mappings on the plane into the plane are investigated from the viewpoint of stability.

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arXiv2017

Composing generic linearly perturbed mappings and immersions/injections

Let $N$ (resp., $U$) be a manifold (resp., an open subset of $\mathbb{R}^m$). Let $f:N\to U$ and $F:U\to \mathbb{R}^\ell$ be an immersion and a $C^{\infty}$ mapping, respectively. Generally, the composition $F\circ f$ does not necessarily yield a mapping transverse to a given subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$. Nevertheless, in this paper, for any $\mathcal{A}^1$-invariant fiber, we show that composing generic linearly perturbed mappings of $F$ and the given immersion $f$ yields a mapping transverse to the subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$ with the given fiber. Moreover, we show a specialized transversality theorem on crossings of compositions of generic linearly perturbed mappings of a given mapping $F:U\to \mathbb{R}^\ell$ and a given injection $f:N\to U$. Furthermore, applications of the two main theorems are given.

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arXiv2017

Geometric interpretation of generalized distance-squared mappings of $\mathbb{R}^2$ into $\mathbb{R}^\ell$ $(\ell \geq 3)$

Generalized distance-squared mappings are quadratic mappings of $\mathbb{R}^m$ into $\mathbb{R}^\ell$ of special type. In the case that matrices $A$ constructed by coefficients of generalized distance-squared mappings of $\mathbb{R}^2$ into $\mathbb{R}^\ell$ ($\ell \geq3$) are full rank, the generalized distance-squared mappings having a generic central point have the following properties. In the case of $\ell=3$, they have only one singular point. On the other hand, in the case of $\ell>3$, they have no singular points. Hence, in this paper, the reason why in the case of $\ell=3$ (resp., in the case of $\ell>3$), they have only one singular point (resp., no singular points) is explained by giving a geometric interpretation to these phenomena.

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arXiv2016

Preservation of immersed or injective properties by composing generic generalized distance-squared mappings

Any generalized distance-squared mapping of equidimensional case has singularities, and their singularity types are wrapped into mystery in higher dimensional cases. Any generalized distance-squared mapping of equidimensional case is not injective. Nevertheless, in this paper, it is shown that the non-singular property or the injective property of a mapping is preserved by composing a generic generalized distance-squared mapping of equidimensional case.

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arXiv2013

Recognizable classification of Lorentzian distance-squared mappings

The Lorentzian length, which is one of the most significant functions in Lorentzian geometry, is a complex-valued function. Its square gives a real-valued non-degenerate quadratic function. In this paper, we define naturally extended mappings of Lorentzian distance-squared functions, wherein each component is a Lorentzian distance-squared function; and classify these mappings completely by the likeness of recognition subspaces.

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arXiv2013

Distance-squared mappings

A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. In this paper, we define naturally extended mappings of distance-squared functions, wherein each component is a distance-squared function. We investigate the properties of these mappings from the viewpoint of differential topology.

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