Yusuke Mizota | Searcharxiv

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.

math.OC↗

arXiv2014

Lowerable vector fields for a finitely L-determined multigerm

We show that the module of lowerable vector fields for a finitely L-determined multigerm is finitely generated in a constructive way.

math.DG↗

arXiv2011

Multicusps

For a given multicusp $f=c_{(θ_0,..., θ_i)}$ $(1\le i)$, we present a direct sum decomposition theorem of the source space of ${}_i\barωf$, where ${}_i\barωf$ is a higher version of the reduced Kodaira-Spencer-Mather map $\barωf$. As a corollary of our direct sum decomposition theorem, we show that for any $i\in \mathbb{N}$ and any $f=c_{(θ_0,..., θ_i)}$, ${}_i\barωf$ is bijective. The corollary is an affirmative answer to the question raised by M. A. S. Ruas during the 11th International Workshop on Real and Complex Singularities at the University of S${\tilde {\rm a}}$o Paulo in S${\tilde {\rm a}}$o Carlos (2010).

math.DG↗