arXiv subjects

Seonghyeon Jeong

Publications and source records attributed to Seonghyeon Jeong.

6 recordsLinked to original sources

arXiv2025

Synthetic MTW conditions and their equivalence under mild regularity assumption on the cost function

Loeper's condition in \cite{Loe09} and the quantitatively quasi-convex condition (QQconv) from \cite{GK15} are synthetic expressions of the analytic MTW condition from \cite{TW} since they only require $C^2$ differentiability of the cost function $c$. When the cost function $c$ is $C^4$, it is known that the two synthetic MTW conditions are equivalent to the analytic MTW condition. However, when the cost function has regularity weaker than $C^4$, it is not known that if the two synthetic MTW conditions are equivalent. In this paper, we show the equivalence of the synthetic MTW conditions when the cost function has only $C^2$ regularity.

math.AP↗

arXiv2025

An alternative definition for c-convex functions and another synthetic statement of MTW condition

The main theorem of this paper states that the c-convexity and the alternative c-convexity are equivalent if and only if the cost function c satisfies MTW condition. The alternative c-convex function is an analogy of the definition of the convex function that is using the inequality phi(t x1 + (1 - t) x0) <= t phi(x1) + (1 - t) phi(x0). We study properties of the alternative c-convex functions and MTW condition, then prove the main theorem.

math.OC↗

arXiv2025

Conditions for existence of single valued optimal transport maps on convex boundaries with nontwisted cost

We prove that if $Ω\subset \mathbb{R}^{n+1}$ is a (not necessarily strictly) convex, $C^1$ domain, and $μ$ and $\barμ$ are probability measures absolutely continuous with respect to surface measure on $\partial Ω$, with densities bounded away from zero and infinity, whose $2$-Monge-Kantorovich distance is sufficiently small, then there exists a continuous Monge solution to the optimal transport problem with cost function given by the quadratic distance on the ambient space $\mathbb{R}^{n+1}$. This result is also shown to be sharp, via a counterexample when $Ω$ is uniformly convex but not $C^1$. Additionally, if $Ω$ is $C^{1, α}$ regular for some $α$, then the Monge solution is shown to be Hölder continuous.

math.AP↗

arXiv2025

Quadratic optimal transportation problem with a positive semi definite structure on the cost function

Optimal transportation problem seeks for a coupling $π$ of two probability measures $μ$ and $ν$ which minimize the total cost $\int c dπ$, which is linear in $π$. In this paper, we introduce a variation of optimal transportation problem which we call quadratic transportation problem that considers a total cost $\iint c dπdπ$ which is quadratic in $π$. We compare this problem with other variations of optimal transportation problem, and prove some properties of the solutions to the problem. Then, we introduce squared cost function, which let us consider the total cost $\iint c dπdπ$ as a positive semi-definite bilinear operator on probability measures, and show Kantorovich duality formula when we have a squared cost function.

math.OC↗

arXiv2024

Convergence analysis of t-SNE as a gradient flow for point cloud on a manifold

We present a theoretical foundation regarding the boundedness of the t-SNE algorithm. t-SNE employs gradient descent iteration with Kullback-Leibler (KL) divergence as the objective function, aiming to identify a set of points that closely resemble the original data points in a high-dimensional space, minimizing KL divergence. Investigating t-SNE properties such as perplexity and affinity under a weak convergence assumption on the sampled dataset, we examine the behavior of points generated by t-SNE under continuous gradient flow. Demonstrating that points generated by t-SNE remain bounded, we leverage this insight to establish the existence of a minimizer for KL divergence.

stat.ML↗

arXiv2020

Hölder regularity of solutions to generated Jacobian equations

Generated Jacobian equations are Monge-Ampère type equations which contain optimal transport as a special case. Therefore, optimal transport case has its own special structure which is not necessarily true for more general generated Jacobian equations. Hence the theory for optimal transport can not be directly transplanted to generated Jacobian equations. In this paper, we point out the difficulties that prevent applying the proof the local Hölder regularity of solutions of optimal transport problem from \cite{Loeper2009OnTR} directly to Generated Jacobian Equations, we then discuss how to handle these difficulties, and prove local Hölder regularity in the generated Jacobian equation case.

math.AP↗