arXiv subjects

Pakawut Jiradilok

Publications and source records attributed to Pakawut Jiradilok.

16 recordsLinked to original sources

arXiv2026

Denoising Distances in Metric Measure Spaces

Recent work studied the problem of finding clusters and denoising pairwise distances from noisy distances of points sampled on a manifold. We study the same problems in more general metric measure spaces under a lower mass condition. We give an algorithm that extracts large localized clusters around every sampled point, which can be used to denoise distances, with near-linear running time in the dense regime for fixed target distance error $r$. When the target distance error $r$ is allowed to vanish as $n\to\infty$, we identify the sharp information-theoretic scale for achieving distance error $r$, suggesting a statistical-computational gap for high-accuracy denoising beyond the Riemannian setting.

cs.CG↗

arXiv2026

Reconstructing the Geometry of Random Geometric Graphs

Random geometric graphs are random graph models defined on metric spaces. Such a model is defined by first sampling points from a metric space and then connecting each pair of sampled points with probability that depends on their distance, independently among pairs. In this work, we show how to efficiently reconstruct the geometry of the underlying space from the sampled graph under the manifold assumption, i.e., assuming that the underlying space is a low dimensional manifold and that the connection probability is a strictly decreasing function of the Euclidean distance between the points in a given embedding of the manifold in $\mathbb{R}^N$. Our work complements a large body of work on manifold learning, where the goal is to recover a manifold from sampled points sampled in the manifold along with their (approximate) distances.

cs.LG↗

arXiv2026

Denoising distances beyond the volumetric barrier

We study the problem of reconstructing the latent geometry of a $d$-dimensional Riemannian manifold from a random geometric graph. While recent works have made significant progress in manifold recovery from random geometric graphs, and more generally from noisy distances, the precision of pairwise distance estimation has been fundamentally constrained by the volumetric barrier, namely the natural sample-spacing scale $n^{-1/d}$ coming from the fact that a generic point of the manifold typically lies at distance of order $n^{-1/d}$ from the nearest sampled point. In this paper, we introduce a novel approach, Orthogonal Ring Distance Estimation Routine (ORDER), which achieves a pointwise distance estimation precision of order $n^{-2/(d+5)}$ up to polylogarithmic factors in $n$ in polynomial time. This strictly beats the volumetric barrier for dimensions $d > 5$. As a consequence of obtaining pointwise precision better than $n^{-1/d}$, we prove that the Gromov--Wasserstein distance between the reconstructed metric measure space and the true latent manifold is of order $n^{-1/d}$. This matches the Wasserstein convergence rate of empirical measures, demonstrating that our reconstructed graph metric is asymptotically as good as having access to the full pairwise distance matrix of the sampled points. Our results are proven in a very general setting which includes general models of noisy pairwise distances, sparse random geometric graphs, and unknown connection probability functions.

stat.ML↗

arXiv2025

Reconstructing Riemannian Metrics From Random Geometric Graphs

Random geometric graphs are random graph models defined on metric measure spaces. A random geometric graph is generated by first sampling points from a metric space and then connecting each pair of sampled points independently with a probability that depends on their distance. In recent work of Huang, Jiradilok, and Mossel~\cite{HJM24}, the authors study the problem of reconstructing an embedded manifold form a random geometric graph sampled from the manifold, where edge probabilities depend monotonically on the Euclidean distance between the embedded points. They show that, under mild regularity assumptions on the manifold, the sampling measure, and the connection probability function, it is possible to recover the pairwise Euclidean distances of the embedded sampled points up to a vanishing error as the number of vertices grows. In this work we consider a similar and arguably more natural problem where the metric is the Riemannian metric on the manifold. Again points are sampled from the manifold and a random graph is generated where the connection probability is monotone in the Riemannian distance. Perhaps surprisingly we obtain stronger results in this setup. Unlike the previous work that only considered dense graph we provide reconstruction algorithms from sparse graphs with average degree $n^{1/2}{\rm polylog}(n)$, where $n$ denotes the number of vertices. Our algorithm is also a more efficient algorithm for distance reconstruction with improved error bounds. The running times of the algorithm is $O(n^2\,{\rm polylog}(n))$ which up to polylog factor matches the size of the input graph. Our distance error also nearly matches the volumetric lower bounds for distance estimation.

math.PR↗

arXiv2025

Random Subwords and Billiard Walks in Affine Weyl Groups

Let $W$ be an irreducible affine Weyl group, and let $\mathsf{b}$ be a finite word over the alphabet of simple reflections of $W$. Fix a probability $p\in(0,1)$. For each integer $K\geq 0$, let $\mathsf{sub}_p(\mathsf{b}^K)$ be the random subword of $\mathsf{b}^K$ obtained by deleting each letter independently with probability $1-p$. Let $v_p(\mathsf{b}^K)$ be the element of $W$ represented by $\mathsf{sub}_p(\mathsf{b}^K)$. One can view $v_p(\mathsf{b}^K)$ geometrically as a random alcove; in many cases, this alcove can be seen as the location after a certain amount of time of a random billiard trajectory that, upon hitting a hyperplane in the Coxeter arrangement of $W$, reflects off of the hyperplane with probability $1-p$. We show that the asymptotic distribution of $v_p(\mathsf{b}^K)$ is a central spherical multivariate normal distribution with some variance $σ_{\mathsf{b}}^2$ depending on $\mathsf{b}$ and $p$. We provide a formula to compute $σ_{\mathsf{b}}^2$ that is remarkably simple when $\mathsf{b}$ contains only one occurrence of the simple reflection that is not in the associated finite Weyl group. As a corollary, we provide an asymptotic formula for $\mathbb{E}[\ell(v_p(\mathsf{b}^K))]$, the expected Coxeter length of $v_p(\mathsf{b}^K)$. For example, when $W=\widetilde A_{r}$ and $\mathsf{b}$ contains each simple reflection exactly once, we find that \[\lim_{K\to\infty}\frac{1}{\sqrt{K}}\mathbb{E}[\ell(v_p(\mathsf{b}^K))]=\sqrt{\frac{2}πr(r+1)\frac{p}{1-p}}.\]

math.PR↗

arXiv2024

Repeatable patterns and the maximum multiplicity of a generator in a reduced word

We study the maximum multiplicity $\mathcal{M}(k,n)$ of a simple transposition $s_k=(k \: k+1)$ in a reduced word for the longest permutation $w_0=n \: n-1 \: \cdots \: 2 \: 1$, a problem closely related to much previous work on sorting networks and on the "$k$-set" problem. After reinterpreting the problem in terms of monotone weakly separated paths, we show that, for fixed $k$ and sufficiently large $n$, the optimal density is realized by paths which are periodic in a precise sense, so that \[ \mathcal{M}(k,n)=c_k n + p_k(n) \] for a periodic function $p_k$ and constant $c_k$. In fact we show that $c_k$ is always rational, and compute several bounds and exact values for this quantity with "repeatable patterns", which we introduce.

math.CO↗

arXiv2024

Some Combinatorial Formulas Related to Diagonal Ramsey Numbers

We derive some combinatorial formulas related to the diagonal Ramsey numbers $R(k)$. Each formula is a statement of the form "$F(n,k) = 0$ if and only if $n \ge R(k)$," where $F(n,k)$ is a combinatorial expression which depends on $n$ and $k$. Our work is closely related to a recent work by De Loera and Wesley.

math.CO↗

arXiv2024

Gaussian Broadcast on Grids

Motivated by the classical work on finite noisy automata (Gray 1982, Gács 2001, Gray 2001) and by the recent work on broadcasting on grids (Makur, Mossel, and Polyanskiy 2022), we introduce Gaussian variants of these models. These models are defined on graded posets. At time $0$, all nodes begin with $X_0$. At time $k\ge 1$, each node on layer $k$ computes a combination of its inputs at layer $k-1$ with independent Gaussian noise added. When is it possible to recover $X_0$ with non-vanishing correlation? We consider different notions of recovery including recovery from a single node, recovery from a bounded window, and recovery from an unbounded window. Our main interest is in two models defined on grids: In the infinite model, layer $k$ is the vertices of $\mathbb{Z}^{d+1}$ whose sum of entries is $k$ and for a vertex $v$ at layer $k \ge 1$, $X_v=α\sum (X_u + W_{u,v})$, summed over all $u$ on layer $k-1$ that differ from $v$ exactly in one coordinate, and $W_{u,v}$ are i.i.d. $\mathcal{N}(0,1)$. We show that when $α<1/(d+1)$, the correlation between $X_v$ and $X_0$ decays exponentially, and when $α>1/(d+1)$, the correlation is bounded away from $0$. The critical case when $α=1/(d+1)$ exhibits a phase transition in dimension, where $X_v$ has non-vanishing correlation with $X_0$ if and only if $d\ge 3$. The same results hold for any bounded window. In the finite model, layer $k$ is the vertices of $\mathbb{Z}^{d+1}$ with nonnegative entries with sum $k$. We identify the sub-critical and the super-critical regimes. In the sub-critical regime, the correlation decays to $0$ for unbounded windows. In the super-critical regime, there exists for every $t$ a convex combination of $X_u$ on layer $t$ whose correlation is bounded away from $0$. We find that for the critical parameters, the correlation is vanishing in all dimensions and for unbounded window sizes.

cs.IT↗

arXiv2023

Triangular-Grid Billiards and Plabic Graphs

Given a polygon $P$ in the triangular grid, we obtain a permutation $π_P$ via a natural billiards system in which beams of light bounce around inside of $P$. The different cycles in $π_P$ correspond to the different trajectories of light beams. We prove that \[\text{area}(P)\geq 6\text{cyc}(P)-6\quad\text{and}\quad\text{perim}(P)\geq\frac{7}{2}\text{cyc}(P)-\frac{3}{2},\] where $\text{area}(P)$ and $\text{perim}(P)$ are the (appropriately normalized) area and perimeter of $P$, respectively, and $\text{cyc}(P)$ is the number of cycles in $π_P$. The inequality concerning $\text{area}(P)$ is tight, and we characterize the polygons $P$ satisfying $\text{area}(P)=6\text{cyc}(P)-6$. These results can be reformulated in the language of Postnikov's plabic graphs as follows. Let $G$ be a connected reduced plabic graph with essential dimension $2$. Suppose $G$ has $n$ marked boundary points and $v$ (internal) vertices, and let $c$ be the number of cycles in the trip permutation of $G$. Then we have \[v\geq 6c-6\quad\text{and}\quad n\geq\frac{7}{2}c-\frac{3}{2}.\]

math.CO↗

arXiv2022

Large-scale Rook Placements

For each certain "nice" piecewise linear function $f:[0,1] \to [0,1]$, we consider a family of growing Young diagrams $\{λ(f,N)\}_{N=1}^{\infty}$ by enlarging the region under the graph of $f$. We compute asymptotic formulas for the number of rook placements of the shape $λ(f,N)$. We prove that the normalized cumulative X-ray of a uniformly random permutation, as the size of the permutation grows, exhibits a limit shape phenomenon.

math.CO↗

arXiv2021

Roots of descent polynomials and an algebraic inequality on hook lengths

We prove a conjecture by Diaz-Lopez et al. that bounds the roots of descent polynomials. To do so, we prove an algebraic inequality, which we refer to as the "Slice and Push Inequality." This inequality compares expressions that come from Naruse's hook-length formula for the number of standard Young tableaux of a skew shape.

math.CO↗

arXiv2021

Transportation Distance between Probability Measures on the Infinite Regular Tree

In the infinite regular tree $\mathbb{T}_{q+1}$ with $q \in \mathbb{Z}_{\ge 2}$, we consider families $\{μ_u^n\}$, indexed by vertices $u$ and nonnegative integers ("discrete time steps") $n$, of probability measures such that $μ_u^n(v) = μ_{u'}^n(v')$ if the distances $\operatorname{dist}(u,v)$ and $\operatorname{dist}(u',v')$ are equal. Let $d$ be a positive integer, and let $X$ and $Y$ be two vertices in the tree which are at distance $d$ apart. We compute a formula for the transportation distance $W_1\!\left( μ_X^n, μ_Y^n \right)$ in terms of generating functions. In the special case where $μ_u^n = \mathfrak{m}_u^n$ are measures from simple random walks after $n$ time steps, we establish the linear asymptotic formula $W_1\!\left( \mathfrak{m}_X^n, \mathfrak{m}_Y^n \right) = An + B + o(1)$, as $n \to \infty$, and give the formulas for the coefficients $A$ and $B$ in closed forms. We also obtain linear asymptotic formulas in the cases of spheres and uniform balls as the radii tend to infinity. We show that these six coefficients (two from simple random walks, two from spheres, and two from uniform balls) are related by inequalities.

math.CO↗

arXiv2021

Double Rim Hook Cluster Algebras

We describe an infinite family of non-Plücker cluster variables inside the double Bruhat cell cluster algebra defined by Berenstein, Fomin, and Zelevinsky. These cluster variables occur in a family of subalgebras of the double Bruhat cell cluster algebra which we call Double Rim Hook (DRH) cluster algebras. We discover that all of the cluster variables are determinants of matrices of special form. We conjecture that all the cluster variables of the double Bruhat-cell cluster algebra have similar determinant form. We notice the resemblance between our staircase diagram and Auslander-Reiten quivers.

math.CO↗

arXiv2018

Zonotopes whose cellular strings are all coherent

A cellular string of a polytope is a sequence of faces stacked on top of each other in a given direction. The poset of cellular strings, ordered by refinement, is known to be homotopy equivalent to a sphere. The subposet of coherent cellular strings is the face lattice of the fiber polytope, hence is homeomorphic to a sphere. In some special cases, every cellular string is coherent. Such polytopes are said to be all-coherent. We give a complete classification of zonotopes with the all-coherence property in terms of their oriented matroid structure. Although the face lattice of the fiber polytope in this case is not an oriented matroid invariant, we prove that the all-coherence property is invariant.

math.CO↗

arXiv2017

Theta characteristics of tropical $K_4$-curves

A $K_4$-curve is a smooth, proper curve X of genus 3 over a nonarchimedean field whose Berkovich skeleton $Γ$ is a complete graph on 4 vertices. The curve X has 28 effective theta characteristics, i.e. the 28 bitangents to a canonical embedding, while $Γ$ has exactly seven tropical theta characteristics, as shown by Zharkov. We prove that the 28 effective theta characteristics of a $K_4$-curve specialize to the theta characteristics of its minimal skeleton in seven groups of four.

math.AG↗

arXiv2016

Reconstructing Partitions from their Multisets of $k$-Minors

For non-negative integers $n$ and $k$ with $n \ge k$, a {\em $k$-minor} of a partition $λ= [λ_1, λ_2, \dots]$ of $n$ is a partition $μ= [μ_1, μ_2, \dots]$ of $n-k$ such that $μ_i \le λ_i$ for all $i$. The multiset $\widehat{M}_k(λ)$ of $k$-minors of $λ$ is defined as the multiset of $k$-minors $μ$ with multiplicity of $μ$ equal to the number of standard Young tableaux of skew shape $λ/ μ$. We show that there exists a function $G(n)$ such that the partitions of $n$ can be reconstructed from their multisets of $k$-minors if and only if $k \le G(n)$. Furthermore, we prove that $\lim_{n \rightarrow \infty} G(n)/n = 1$ with $n-G(n) = O(n/\log n)$. As a direct consequence of this result, the irreducible representations of the symmetric group $S_n$ can be reconstructed from their restrictions to $S_{n-k}$ if and only if $k \le G(n)$ for the same function $G(n)$. For a minor $μ$ of the partition $λ$, we study the excitation factor $E_μ(λ)$, which appears as a crucial part in Naruse's Skew-Shape Hook Length Formula. We observe that certain excitation factors of $λ$ can be expressed as a $\mathbb{Q}[k]$-linear combination of the elementary symmetric polynomials of the hook lengths in the first row of $λ$ where $k = λ_1$ is the number of cells in the first row of $λ$.

math.CO↗