arXiv subjects

Sungwoo Jeong

Publications and source records attributed to Sungwoo Jeong.

9 recordsLinked to original sources

arXiv2026

Sampling Pfaffian point processes and the symplectic Arnoldi method

We present an exact sampling algorithm for Pfaffian point processes based on a skew-symmetric analogue of the Cholesky factorization. This algorithm enables efficient sampling of a wide range of statistics arising in random matrix theory and combinatorics. For instance, we can sample eigenvalues of the orthogonal and symplectic ensembles ($β= 1,4$). In addition, we introduce a symplectic Arnoldi method for computing skew-orthogonal polynomials associated with a general weight function. This method can be used to efficiently construct the $2 \times 2$ matrix valued skew-symmetric kernels that arise in $β= 1,4$ polynomial ensembles. We illustrate our approach with several numerical examples and experiments, including the symmetric corner growth model, the finite-$N$ Gaussian (Hermite) orthogonal and symplectic ensembles, and the $β= 1,4$ Airy point processes and Tracy-Widom distributions.

math.NA↗

arXiv2026

On the Limit of the Tridiagonal Model for $β$-Dyson Brownian Motion

In previous work, a description of the result of applying the Householder tridiagonalization algorithm to a G$β$E random matrix is provided by Edelman and Dumitriu. The resulting tridiagonal ensemble makes sense for all $β>0$, and has spectrum given by the $β$-ensemble for all $β>0$. Moreover, the tridiagonal model has useful stochastic operator limits which was introduced and analyzed in subsequent studies. In this work, we analogously study the result of applying the Householder tridiagonalization algorithm to a G$β$E process which has eigenvalues governed by $β$-Dyson Brownian motion. We propose an explicit limit of the upper left $k \times k$ minor of the $n \times n$ tridiagonal process as $n \to \infty$ and $k$ remains fixed. We prove the result for $β=1$, and also provide numerical evidence for $β=1,2,4$. This leads us to conjecture the form of a dynamical $β$-stochastic Airy operator with smallest $k$ eigenvalues evolving according to the $n \to \infty$ limit of the largest, centered and re-scaled, $k$ eigenvalues of $β$-Dyson Brownian motion.

math.PR↗

arXiv2025

The smallest eigenvalue of $β$-Laguerre and $β$-Jacobi ensembles and multivariate orthogonal polynomials

We study the smallest eigenvalue statistics of the $β$-Laguerre and $β$-Jacobi ensembles. Using Kaneko's integral formula, we show that the smallest eigenvalue marginal density and distribution functions of the two ensembles for any $β>0$ can be represented in terms of multivariate Laguerre and Jacobi polynomials evaluated at a multiple of the identity, provided that the exponent of $x$ in the Laguerre and Jacobi weights is an integer. These representations are readily computable in explicit form using existing symbolic algorithms for multivariate orthogonal polynomials. From these expressions, we derive new differentiation formulas for the multivariate Laguerre and Jacobi polynomials. Furthermore, we derive explicit solutions to the Painleve V and VI differential equations associated with the smallest eigenvalue of the LUE and JUE. We provide numerical experiments and examples.

math-ph↗

arXiv2025

Convergence of Pivoted Cholesky Algorithm for Lipschitz Kernels

We investigate the continuous analogue of the Cholesky factorization, namely the pivoted Cholesky algorithm. Our analysis establishes quantitative convergence guarantees for kernels of minimal smoothness. We prove that for a symmetric positive definite Lipschitz continuous kernel $K:Ω\times Ω\rightarrow \mathbb{R}$ on a compact domain $Ω\subset\mathbb{R}^d$, the residual of the Cholesky algorithm with any pivoting strategy is uniformly bounded above by a constant multiple of the fill distance of pivots. In particular, our result implies that under complete pivoting (where the maximum value of the diagonal of the residual is selected as the next pivot): \begin{equation*} \|R_n\|_{\infty} = O(n^{-1/d}), \end{equation*} where $R_n$ is the residual after $n$ Cholesky steps and $\|\cdot\|_\infty$ is the absolute maximum value of $R_n$. Moreover, if $K$ is differentiable in both variables with a Lipschitz derivative, our convergence rate improves to $O(n^{-2/d})$. Our result closes a gap between theory and practice as previous analyses required $C^2$-regularity of $K$ to establish convergence, whereas empirical evidence indicated robust performance even for non-differentiable kernels. We further detail how our convergence results propagate to downstream applications, including discrete analogues, Gaussian process regression, and the P-greedy interpolation method.

math.NA↗

arXiv2025

Extending Mercer's expansion to indefinite and asymmetric kernels

Mercer's expansion and Mercer's theorem are cornerstone results in kernel theory. While the classical Mercer's theorem only considers continuous symmetric positive definite kernels, analogous expansions are effective in practice for indefinite and asymmetric kernels. In this paper we extend Mercer's expansion to continuous kernels, providing a rigorous theoretical underpinning for indefinite and asymmetric kernels. We begin by demonstrating that Mercer's expansion may not be pointwise convergent for continuous indefinite kernels, before proving that the expansion of continuous kernels with bounded variation uniformly in each variable separably converges pointwise almost everywhere, almost uniformly, and unconditionally almost everywhere. We also describe an algorithm for computing Mercer's expansion for general kernels and give new decay bounds on its terms.

math.NA↗

arXiv2023

The conditional DPP approach to random matrix distributions

We present the conditional determinantal point process (DPP) approach to obtain new (mostly Fredholm determinantal) expressions for various eigenvalue statistics in random matrix theory. It is well-known that many (especially $β=2$) eigenvalue $n$-point correlation functions are given in terms of $n\times n$ determinants, i.e., they are continuous DPPs. We exploit a derived kernel of the conditional DPP which gives the $n$-point correlation function conditioned on the event of some eigenvalues already existing at fixed locations. Using such kernels we obtain new determinantal expressions for the joint densities of the $k$ largest eigenvalues, probability density functions of the $k^\text{th}$ largest eigenvalue, density of the first eigenvalue spacing, and more. Our formulae are highly amenable to numerical computations and we provide various numerical experiments. Several numerical values that required hours of computing time could now be computed in seconds with our expressions, which proves the effectiveness of our approach. We also demonstrate that our technique can be applied to an efficient sampling of DR paths of the Aztec diamond domino tiling. Further extending the conditional DPP sampling technique, we sample Airy processes from the extended Airy kernel. Additionally we propose a sampling method for non-Hermitian projection DPPs.

math-ph↗

arXiv2022

On the structure of the solutions to the matrix equation $G^*JG=J$

We study the mathematical structure of the solution set (and its tangent space) to the matrix equation $G^*JG=J$ for a given square matrix $J$. In the language of pure mathematics, this is a Lie group which is the isometry group for a bilinear (or a sesquilinear) form. Generally these groups are described as intersections of a few special groups. The tangent space to $\{G: G^*JG=J \}$ consists of solutions to the linear matrix equation $X^*J+JX=0$. For the complex case, the solution set of this linear equation was computed by De Ter{á}n and Dopico. We found that on its own, the equation $X^*J+JX=0$ is hard to solve. By throwing into the mix the complementary linear equation $X^*J-JX=0$, we find that rather than increasing the complexity, we reduce the complexity. Not only is it possible to now solve the original problem, but we can approach the broader algebraic and geometric structure. One implication is that the two equations form an $\mathfrak{h}$ and $\mathfrak{m}$ pair familiar in the study of pseudo-Riemannian symmetric spaces. We explicitly demonstrate the computation of the solutions to the equation $X^*J\pm XJ=0$ for real and complex matrices. However, any real, complex or quaternionic case with an arbitrary involution (e.g., transpose, conjugate transpose, and the various quaternion transposes) can be effectively solved with the same strategy. We provide numerical examples and visualizations.

math.NA↗

arXiv2022

On the Cartan Decomposition for Classical Random Matrix Ensembles

We complete Dyson's dream by cementing the links between symmetric spaces and classical random matrix ensembles. Previous work has focused on a one-to-one correspondence between symmetric spaces and many but not all of the classical random matrix ensembles. This work shows that we can completely capture all of the classical random matrix ensembles from Cartan's symmetric spaces through the use of alternative coordinate systems. In the end, we have to let go of the notion of a one-to-one correspondence. We emphasize that the KAK decomposition traditionally favored by mathematicians is merely one coordinate system on the symmetric space, albeit a beautiful one. However, other matrix factorizations, especially the generalized singular value decomposition from numerical linear algebra reveal themselves to be perfectly valid coordinate systems revealing that one symmetric space can lead to many classical random matrix theories. We establish the connection between this numerical linear algebra viewpoint and the theory of generalized Cartan decomposition. This in turn allows us to produce yet more random matrix theories from a single symmetric space. Yet again these random matrix theories arise from matrix factorizations, through ones that we are not aware have appeared in the literature.

math-ph↗

arXiv2022

Fifty Three Matrix Factorizations: A systematic approach

The success of matrix factorizations such as the singular value decomposition (SVD) has motivated the search for even more factorizations. We catalog 53 matrix factorizations, most of which we believe to be new. Our systematic approach, inspired by the generalized Cartan decomposition of Lie theory, also encompasses known factorizations such as the SVD, the symmetric eigendecomposition, the CS decomposition, the hyperbolic SVD, structured SVDs, the Takagi factorization, and others thereby covering familiar matrix factorizations as well as ones that were waiting to be discovered. We suggest that Lie theory has one way or another been lurking hidden in the foundations of the very successful field of matrix computations with applications routinely used in so many areas of computation. In this paper, we investigate consequences of the Cartan decomposition and the little known generalized Cartan decomposition for matrix factorizations. We believe that these factorizations once properly identified can lead to further work on algorithmic computations and applications.

math.NA↗