arXiv subjects

Joon-Hyeok Yim

Publications and source records attributed to Joon-Hyeok Yim.

4 recordsLinked to original sources

arXiv2026

A quantitative tree-likeness bound from average hyperbolicity

Chatterjee and Sloman proved that a bounded measurable similarity function with sufficiently small average Gromov hyperbolicity admits a tree representation with small mean approximation error. Their argument uses a weighted version of Szemerédi's regularity lemma and does not yield useful quantitative bounds. Here, we establish an explicit relation between average hyperbolicity and mean tree approximation error. For a similarity function $s:S\times S\to[0,b]$, we prove that $$\operatorname{Tree}(s) \leq (63/e)^{1/3} \sqrt[3]{b^2 \operatorname{Hyp}(s)} \leq 2.8512 \sqrt[3]{b^2 \operatorname{Hyp}(s)}.$$ The proof uses a simple pivoting construction inspired by KwikCluster. We also discuss the optimal dependence on average hyperbolicity, including a square-root lower bound, and connections with ultrametric fitting.

math.PR↗

arXiv2026

Towards stratified sampling for redistricting plans

Rapid algorithmic developments have accelerated the sampling of redistricting ensembles (balanced graph partitions), yet evaluating rare events and sampling complex target measures remains a core challenge due to the high-dimensional and combinatorial nature of the phase space. We address a prerequisite for stratified sampling on this space: constructing and diagnosing candidate strata with suitable coverage and overlap. We build a grammar on observed plans by clustering districts into representative ``letters'' and using them to form plan-level ``words.'' A partition of unity over these words gives a soft assignment of plans to strata and allows us to estimate stratum masses and an overlap-induced flux matrix. We demonstrate this computational pipeline using real-world congressional redistricting data from Connecticut and examine how strata learned from one target distribution behave under related distributions. The resulting construction provides a foundation for future stratified sampling on spaces of redistricting plans or balanced graph partitions. We do not implement a complete stratified sampler here; evaluating whether the proposed strata improve sampling efficiency or reduce estimator variance is left for future work.

physics.soc-ph↗

arXiv2024

Hyperbolicity, slimness, and minsize, on average

A metric space $(X,d)$ is said to be $δ$-hyperbolic if $d(x,y)+d(z,w)$ is at most $\max(d(x,z)+d(y,w), d(x,w)+d(y,z))$ by $2 δ$. A geodesic space is $δ$-slim if every geodesic triangle $Δ(x,y,z)$ is $δ$-slim. It is well-established that the notions of $δ$-slimness, $δ$-hyperbolicity, $δ$-thinness and similar concepts are equivalent up to a constant factor. In this paper, we investigate these properties under an average-case framework and reveal a surprising discrepancy: while $\mathbb{E}δ$-slimness implies $\mathbb{E}δ$-hyperbolicity, the converse does not hold. Furthermore, similar asymmetries emerge for other definitions when comparing average-case and worst-case formulations of hyperbolicity. We exploit these differences to analyze the random Gaussian distribution in Euclidean space, random $d$-regular graph, and the random Erdős-Rényi graph model, illustrating the implications of these average-case deviations.

math.PR↗

arXiv2024

Fitting trees to $\ell_1$-hyperbolic distances

Building trees to represent or to fit distances is a critical component of phylogenetic analysis, metric embeddings, approximation algorithms, geometric graph neural nets, and the analysis of hierarchical data. Much of the previous algorithmic work, however, has focused on generic metric spaces (i.e., those with no a priori constraints). Leveraging several ideas from the mathematical analysis of hyperbolic geometry and geometric group theory, we study the tree fitting problem as finding the relation between the hyperbolicity (ultrametricity) vector and the error of tree (ultrametric) embedding. That is, we define a vector of hyperbolicity (ultrametric) values over all triples of points and compare the $\ell_p$ norms of this vector with the $\ell_q$ norm of the distortion of the best tree fit to the distances. This formulation allows us to define the average hyperbolicity (ultrametricity) in terms of a normalized $\ell_1$ norm of the hyperbolicity vector. Furthermore, we can interpret the classical tree fitting result of Gromov as a $p = q = \infty$ result. We present an algorithm HCCRootedTreeFit such that the $\ell_1$ error of the output embedding is analytically bounded in terms of the $\ell_1$ norm of the hyperbolicity vector (i.e., $p = q = 1$) and that this result is tight. Furthermore, this algorithm has significantly different theoretical and empirical performance as compared to Gromov's result and related algorithms. Finally, we show using HCCRootedTreeFit and related tree fitting algorithms, that supposedly standard data sets for hierarchical data analysis and geometric graph neural networks have radically different tree fits than those of synthetic, truly tree-like data sets, suggesting that a much more refined analysis of these standard data sets is called for.

cs.DS↗