arXiv subjects

Forest Kobayashi

Publications and source records attributed to Forest Kobayashi.

5 recordsLinked to original sources

arXiv2025

Principal Curves In Metric Spaces And The Space Of Probability Measures

We introduce principal curves in Wasserstein space, and in general compact metric spaces. Our motivation for the Wasserstein case comes from optimal-transport-based trajectory inference, where a developing population of cells traces out a curve in Wasserstein space. Our framework enables new experimental procedures for collecting high-density time-courses of developing populations of cells: time-points can be processed in parallel (making it easier to collect more time-points). However, then the time of collection is unknown, and must be recovered by solving a seriation problem (or one-dimensional manifold learning problem). We propose an estimator based on Wasserstein principal curves, and prove it is consistent for recovering a curve of probability measures in Wasserstein space from empirical samples. This consistency theorem is obtained via a series of results regarding principal curves in compact metric spaces. In particular, we establish the validity of certain numerical discretization schemes for principal curves, which is a new result even in the Euclidean setting.

math.ST↗

arXiv2025

Structure of average distance minimizers in general dimensions

For a fixed, compactly supported probability measure $\mu$ on the $d$-dimensional space $\mathbb{R}^d$, we consider the problem of minimizing the $p^{\mathrm{th}}$-power average distance functional over all compact, connected $\Sigma \subseteq \mathbb{R}^d$ with Hausdorff 1-measure $\mathcal{H}^1(\Sigma) \leq l$. This problem, known as the average distance problem, was first studied by Buttazzo, Oudet, and Stepanov in 2002, and has undergone a considerable amount of research since. We will provide a novel approach to studying this problem by analyzing it using the so-called \textit{barycentre field} considered previously by Hayase and two of the authors. This allows us to provide a complete topological description of minimizers in arbitrary dimensions when $p = 2$ and $p > \frac{1}{2}(3 + \sqrt{5}) \approx 2.618$, the first such result that includes the case when $d > 2$.

math.OC↗

arXiv2024

Monge-Kantorovich Fitting With Sobolev Budgets

Given $m < n$, we consider the problem of ``best'' approximating an $n\text{-d}$ probability measure $\rho$ via an $m\text{-d}$ measure $\nu$ such that $\mathrm{supp}\ \nu$ has bounded total ``complexity.'' When $\rho$ is concentrated near an $m\text{-d}$ set we may interpret this as a manifold learning problem with noisy data. However, we do not restrict our analysis to this case, as the more general formulation has broader applications. We quantify $\nu$'s performance in approximating $\rho$ via the Monge-Kantorovich (also called Wasserstein) $p$-cost $\mathbb{W}_p^p(\rho, \nu)$, and constrain the complexity by requiring $\mathrm{supp}\ \nu$ to be coverable by an $f : \mathbb{R}^{m} \to \mathbb{R}^{n}$ whose $W^{k,q}$ Sobolev norm is bounded by $\ell \geq 0$. This allows us to reformulate the problem as minimizing a functional $\mathscr J_p(f)$ under the Sobolev ``budget'' $\ell$. This problem is closely related to (but distinct from) principal curves with length constraints when $m=1, k = 1$ and an unsupervised analogue of smoothing splines when $k > 1$. New challenges arise from the higher-order differentiability condition. We study the ``gradient'' of $\mathscr J_p$, which is given by a certain vector field that we call the barycenter field, and use it to prove a nontrivial (almost) strict monotonicity result. We also provide a natural discretization scheme and establish its consistency. We use this scheme as a toy model for a generative learning task, and by analogy, propose novel interpretations for the role regularization plays in improving training.

cs.LG↗

arXiv2021

Uniform Convergence and Knot Equivalence

Given a uniformly convergent sequence of ambient isotopies $(H_n)_{n\in\mathbb{N}}$, bijectivity of the limit function $H_\infty$ together with a minor compactness condition guarantees that $H_\infty$ is also an ambient isotopy. By offloading the uniform convergence hypothesis to a more diagrammatic condition, we obtain sufficient conditions for performing countably-many Reidemeister moves. We use this to construct examples of tame knots with countably-many crossings and discuss what distinguishes these from similar-looking wild curves.

math.GT↗

arXiv2020

Kaestner Brackets

We introduce \textit{Kaestner brackets}, a generalization of biquandle brackets to the case of parity biquandles. This infinite set of quantum enhancements of the biquandle counting invariant for oriented virtual knots and links includes the classical quantum invariants, the quandle and biquandle $2$-cocycle invariants and the classical biquandle brackets as special cases, coinciding with them for oriented classical knots and links but defining generally stronger invariants for oriented virtual knots and links. We provide examples to illustrate the computation of the new invariant and to show that it is stronger than the classical biquandle bracket invariant for virtual knots.

math.GT↗