Showing 1–50 of 93 results for author: Lim, S

The Gromov-Hausdorff Distance Between Consecutive Spheres

Authors: Donghan Kim, Sunhyuk Lim, Facundo Memoli

Abstract: We determine the Gromov-Hausdorff distance between consecutive unit round spheres equipped with their geodesic metrics. Put $ζ_n:=\arccos(-\tfrac{1}{n+1}),$ the common geodesic distance between distinct vertices of a regular simplex with $n+2$ vertices inscribed in $\mathbb{S}^n$. We prove that $$ d_{\mathrm{GH}}(\mathbb{S}^n,\mathbb{S}^{n+1})=\frac{ζ_n}{2} \qquad(n\geq1), $$ resolving a conject… ▽ More We determine the Gromov-Hausdorff distance between consecutive unit round spheres equipped with their geodesic metrics. Put $ζ_n:=\arccos(-\tfrac{1}{n+1}),$ the common geodesic distance between distinct vertices of a regular simplex with $n+2$ vertices inscribed in $\mathbb{S}^n$. We prove that $$ d_{\mathrm{GH}}(\mathbb{S}^n,\mathbb{S}^{n+1})=\frac{ζ_n}{2} \qquad(n\geq1), $$ resolving a conjecture of Lim, Mémoli, and Smith. All cases $n\geq4$ were previously open. This equality is established by explicitly constructing a family of correspondences $\mathcal R_n\subseteq \mathbb{S}^{n+1}\times \mathbb{S}^n$, whose distortion matches the known quantitative Borsuk-Ulam lower bound $ζ_n$. We also introduce synchronized spherical joins and suspensions of correspondences and prove that the distortion of a join is exactly the maximum of the distortions of its factors. In particular, suspension preserves distortion. Applying these join and suspension operations to the optimal correspondences $\mathcal R_n$ yields new bounds for spheres of nonconsecutive dimensions, including $$ \lim_{m\to\infty} d_{\mathrm{GH}}\bigl(\mathbb{S}^m,\mathbb{S}^{m+d(m)}\bigr) = \fracπ{4} \qquad\text{whenever } d(m)\geq1,\ \text{and }d(m)=o(m).$$ △ Less

Submitted 13 August, 2026; originally announced August 2026.

MSC Class: 53C23 (Primary) 51F30; 55M20 (Secondary)

Optimal Couplings of Levy Processes in the Class of Immersion Couplings

Authors: Tau Shean Lim, Ray Shua Ooi

Abstract: We study the optimal coupling problem for Levy processes on R^d with respect to the quadratic cost. For any two such processes with finite second moments, we prove that the optimal Levy coupling constructed in Kang and Lim (2025), which was previously shown to be optimal among Feller couplings, is in fact optimal among the larger class of immersion couplings. The proof makes use of a characterizat… ▽ More We study the optimal coupling problem for Levy processes on R^d with respect to the quadratic cost. For any two such processes with finite second moments, we prove that the optimal Levy coupling constructed in Kang and Lim (2025), which was previously shown to be optimal among Feller couplings, is in fact optimal among the larger class of immersion couplings. The proof makes use of a characterization of immersion couplings, which is equivalent to the classical martingale preservation definition but more convenient for our purposes. The construction is based on two fundamental ingredients: the existence of an optimal coupling within the class of Levy couplings, and a dual formulation of the associated optimization problem. While both results were previously established in Kang and Lim (2025), we provide here simpler and more transparent proofs relying only on optimal transport between infinitely divisible measures and a generalized minimax principle. These arguments are self-contained and may be of independent interest. △ Less

Submitted 23 June, 2026; originally announced June 2026.

Comments: 48

MSC Class: 60G51; 60J25; 49Q22

Extreme value theorem for geodesic flow on the quotient of the theta group

Authors: Jaelin Kim, Seul Bee Lee, Seonhee Lim

Abstract: We establish an extreme value theorem for the geodesic flow on the hyperbolic surface $Θ\backslash\mathbb{H}^2$ associated with the theta group $Θ$. To capture excursions into both cusps of this surface, we introduce a generalized continued fraction algorithm obtained by splicing the even and odd-odd continued fraction maps into a single dynamical system. We prove that the natural extension of thi… ▽ More We establish an extreme value theorem for the geodesic flow on the hyperbolic surface $Θ\backslash\mathbb{H}^2$ associated with the theta group $Θ$. To capture excursions into both cusps of this surface, we introduce a generalized continued fraction algorithm obtained by splicing the even and odd-odd continued fraction maps into a single dynamical system. We prove that the natural extension of this map is isomorphic to the first return map of the geodesic flow on a suitable cross section. Using spectral properties of the associated transfer operator, we derive a Galambos-type extreme value law for the digits of the spliced continued fraction. This symbolic result is then translated into a geometric extreme value theorem describing maximal cusp excursions of geodesics on $Θ\backslash\mathbb{H}^2$. △ Less

Submitted 8 March, 2026; originally announced March 2026.

Comments: 34 pages, 7 figures

MSC Class: 37D40; 11J70; 11K50

Vector-Valued Period Polynomials and Zeta Values of Quadratic Fields

Authors: Yeong-Wook Kwon, Subong Lim, Wissam Raji

Abstract: Let $k\ge 2$ and $N\ge 1$ be integers. Let $D$ be a positive integer that is congruent to a square modulo $4N$, and fix $ρ$ with $ρ^2\equiv D\pmod{4N}$. In this paper, we consider two weight $2k$ cusp forms $f^{\pm}_{k,N,D,ρ}$ on $Γ_0(N)$ defined by sums over binary quadratic forms, and investigate the vector-valued period polynomial arising from these forms. Our first main result gives a closed f… ▽ More Let $k\ge 2$ and $N\ge 1$ be integers. Let $D$ be a positive integer that is congruent to a square modulo $4N$, and fix $ρ$ with $ρ^2\equiv D\pmod{4N}$. In this paper, we consider two weight $2k$ cusp forms $f^{\pm}_{k,N,D,ρ}$ on $Γ_0(N)$ defined by sums over binary quadratic forms, and investigate the vector-valued period polynomial arising from these forms. Our first main result gives a closed formula for this vector-valued period polynomial. The identity component of this formula is particularly explicit: it separates as the sum of a finite \textit{algebraic part} coming from some binary forms and a \textit{zeta part} involving the values at $s=k$ of certain zeta functions. Using this formula together with a symmetry of vector-valued period polynomials, we explicitly evaluate, for odd $k$, the difference between the zeta values corresponding to the two choices of square root of $D$ modulo $4N$, in terms of Bernoulli numbers and a finite quadratic-form sum. Finally, under a vanishing condition on Fricke-invariant cusp forms at lower levels, we obtain a finite divisor-sum formula for the Dedekind zeta values $ζ_{\mathbb{Q}(\sqrt{D})}(k)$ at even integers $k$. △ Less

Submitted 31 January, 2026; originally announced February 2026.

Rankin-Cohen Bracket for Vector-Valued Modular Forms

Authors: Youngmin Lee, Subong Lim, Wissam Raji

Abstract: In this paper, we explore the relationship between Rankin-Cohen brackets for vector-valued modular forms and Petersson's inner products, deriving an explicit description of the adjoint map for the bracket operator. The study extends to the cases of Jacobi forms and skew-holomorphic Jacobi forms, establishing connections between their respective Rankin-Cohen brackets and those defined for vector-va… ▽ More In this paper, we explore the relationship between Rankin-Cohen brackets for vector-valued modular forms and Petersson's inner products, deriving an explicit description of the adjoint map for the bracket operator. The study extends to the cases of Jacobi forms and skew-holomorphic Jacobi forms, establishing connections between their respective Rankin-Cohen brackets and those defined for vector-valued modular forms through an isomorphism. Adjoint maps for these extended bracket operators are also examined. △ Less

Submitted 19 January, 2026; originally announced January 2026.

Comments: 18 pages

MSC Class: 11F11; 11FXX

On The Hidden Biases of Flow Matching Samplers

Authors: Soon Hoe Lim

Abstract: Flow matching (FM) constructs continuous-time ODE samplers by prescribing probability paths between a base distribution and a target distribution. In this note, we study FM through the lens of finite-sample plug-in estimation. In addition to replacing population expectations by sample averages, one may replace the target distribution itself by a finite-sample surrogate, ranging from the empirical… ▽ More Flow matching (FM) constructs continuous-time ODE samplers by prescribing probability paths between a base distribution and a target distribution. In this note, we study FM through the lens of finite-sample plug-in estimation. In addition to replacing population expectations by sample averages, one may replace the target distribution itself by a finite-sample surrogate, ranging from the empirical measure to a smoothed estimator. This viewpoint yields a natural hierarchy of empirical FM models. For affine conditional flows, we derive the exact empirical minimizer and identify a smoothed plug-in regime in which the terminal law is exactly a kernel-mixture estimator. This plug-in perspective clarifies several coupled finite-sample biases of empirical FM. First, replacing the target law by a finite-sample surrogate changes the statistical target. Second, the empirical minimizer is generally not a gradient field, even when each conditional flow is. Third, a fixed empirical marginal path does not determine a unique particle dynamics: one may add extra vector fields whose probability flux has zero divergence without changing the marginal path. For Gaussian affine conditional paths, we give explicit families of such flux-null corrections. Finally, the source distribution provides a primary mechanism controlling upper tails of kinetic energy. In particular, Gaussian bases yield exponential upper-tail bounds for instantaneous and integrated kinetic energies, whereas polynomially tailed bases yield corresponding polynomial upper-tail bounds. △ Less

Submitted 14 May, 2026; v1 submitted 18 December, 2025; originally announced December 2025.

Comments: 41 pages

Entropic Chaos of Mixed Mean-Field Jump Processes

Authors: Tau Shean Lim, Shuoning Zhang

Abstract: This paper studies a class of mixed mean-field jump processes on an abstract state space $Π$, together with their associated $N$-particle systems. The dynamics consist of the superposition of an independent Markovian component and a bounded mean-field jump interaction; in particular, piecewise deterministic Markov processes (PDMPs) with mean-field interactions are covered by this framework. Under… ▽ More This paper studies a class of mixed mean-field jump processes on an abstract state space $Π$, together with their associated $N$-particle systems. The dynamics consist of the superposition of an independent Markovian component and a bounded mean-field jump interaction; in particular, piecewise deterministic Markov processes (PDMPs) with mean-field interactions are covered by this framework. Under a second-order bounded difference condition on the mean-field jump kernel, we establish entropic propagation of chaos as $N \to \infty$. In particular, we obtain an explicit qualitative bound on the relative entropy between the law of the $N$-particle system and the product measure induced by the mean-field limit. The proof relies on the second-order concentration inequality introduced in Götze and Sambale, 2020. △ Less

Submitted 28 November, 2025; originally announced November 2025.

Comments: 61 pages

MSC Class: 60K35; 82C22 ACM Class: G.3

Analytic and Stochastic Approach to Quantum Advantages in Ground State and Quantum State Preparation Problems

Authors: Taehee Ko, Sungbin Lim

Abstract: We study the problems of state preparation, ground state preparation and quantum state preparation. We propose an analytic approach to a stochastic quantum algorithm which prepares the ground state for $n$-qubit Hamiltonian that is represented by $\text{poly}(n)$ Pauli operators and has an inverse-polynomial gap, requiring only $\text{poly}(n)$ Pauli rotations, measurements, and classical time com… ▽ More We study the problems of state preparation, ground state preparation and quantum state preparation. We propose an analytic approach to a stochastic quantum algorithm which prepares the ground state for $n$-qubit Hamiltonian that is represented by $\text{poly}(n)$ Pauli operators and has an inverse-polynomial gap, requiring only $\text{poly}(n)$ Pauli rotations, measurements, and classical time complexity when $n$ exceeds a threshold, to inverse-polynomial precision given the initial overlap being lower bounded by $\frac{1}{2^n}$. Extending this result, we prove that any $n$-qubit quantum state can be prepared in two regimes: (1) with a constant number of Pauli rotations to constant precision, or (2) with a polynomial number of rotations to inverse-polynomial precision. Our results improve over previous approaches to quantum state preparation in terms of gate complexity, thereby yielding quantum space advantage. As an application, we identify a practical condition under which quadratic unconstrained binary optimization (QUBO) problems can be solved with exponential quantum speedups. △ Less

Submitted 29 November, 2025; v1 submitted 1 October, 2025; originally announced October 2025.

Comments: We found a mistake in the analysis. For example, eq (72) is not true in the appendix A.3

On Optimal Markovian Couplings of Levy Processes

Authors: Wei Yang Kang, Tau Shean Lim

Abstract: We study the optimal Markovian coupling problem for two Pi-valued Feller processes {X_t} and {Y_t}, which seeks a coupling process {(X_t, Y_t)} that minimizes the right derivative at t = 0 of the expected cost E^{(x,y)}[c(X_t, Y_t)], for all initial states (x,y) in Pi^2 and a given cost function c on Pi. This problem was first formulated and solved by Chen (1994) for drift-diffusion processes and… ▽ More We study the optimal Markovian coupling problem for two Pi-valued Feller processes {X_t} and {Y_t}, which seeks a coupling process {(X_t, Y_t)} that minimizes the right derivative at t = 0 of the expected cost E^{(x,y)}[c(X_t, Y_t)], for all initial states (x,y) in Pi^2 and a given cost function c on Pi. This problem was first formulated and solved by Chen (1994) for drift-diffusion processes and later extended by Zhang (2000) to Markov processes with bounded jumps. In this work, we resolve the case of Levy processes under the quadratic cost c(x,y) = 1/2 |x - y|^2 by introducing a new formulation of the "Levy optimal transport problem" between Levy measures. We show that the resulting optimal coupling process {(X_t*, Y_t*)}_{t >= 0} satisfies a minimal growth property: for each t >= 0 and x,y in R^d, the expectation E^{(x,y)}|X_t* - Y_t*|^2 is minimized among all Feller couplings. A key feature of our approach is the development of a dual problem, expressed as a variational principle over test functions of the generators. We prove strong duality for this formulation, thereby closing the optimality gap. As a byproduct, we obtain a Wasserstein-type metric on the space of Levy generators and Levy measures with finite second moment, and establish several of its fundamental properties. △ Less

Submitted 26 September, 2025; originally announced September 2025.

Comments: 82 pages

MSC Class: 60G51; 49Q22

Abstract Formulation of Mean-Field Models and Propagation of Chaos

Authors: Tau Shean Lim, Chao Dun Teoh

Abstract: In this work, we formulate an abstract framework to study mean-field systems. In contrast to most approaches in the available literature which primarily rely on the analysis of SDEs, ours is based on optimal transport and semigroup theory. This allows for the inclusion of a wider range of mean-field particle systems within a unified structure. This new approach involves: (1) constructing an abstra… ▽ More In this work, we formulate an abstract framework to study mean-field systems. In contrast to most approaches in the available literature which primarily rely on the analysis of SDEs, ours is based on optimal transport and semigroup theory. This allows for the inclusion of a wider range of mean-field particle systems within a unified structure. This new approach involves: (1) constructing an abstract framework using semigroups and generators; (2) formulating a corresponding mean-field evolution problem, and proving its well-posedness; (3) demonstrating the propagation of chaos for a class of N-particle systems associated with the mean-field model. Our results are readily applicable to various mean-field models. To demonstrate this, we apply our findings to obtain a new result for Levy-type mean-field systems, which encompass the McKean-Vlasov diffusion. △ Less

Submitted 4 August, 2025; originally announced August 2025.

Comments: 96 pages

MSC Class: 60K35; 60J25; 47D07 ACM Class: G.3

The G-Gromov-Hausdorff Distance and Equivariant Topology

Authors: Sunhyuk Lim, Facundo Memoli

Abstract: For each arbitrary finite group $G$, we consider a suitable notion of Gromov Hausdorff distance between compact $G$ metric spaces and derive lower bounds based on equivariant topology methods. As applications, we prove equivariant rigidity and finiteness theorems, and obtain sharp bounds on the Gromov Hausdorff distance between spheres. For each arbitrary finite group $G$, we consider a suitable notion of Gromov Hausdorff distance between compact $G$ metric spaces and derive lower bounds based on equivariant topology methods. As applications, we prove equivariant rigidity and finiteness theorems, and obtain sharp bounds on the Gromov Hausdorff distance between spheres. △ Less

Submitted 28 January, 2026; v1 submitted 18 June, 2025; originally announced June 2025.

Comments: We added [AFH+25] ('Quantifying Discontinuity') to the list of references and explained its relationship/connection to our paper. We also added Remark 6.9

MSC Class: 53C23 (Primary) 49Q22; 55N31 (Secondary)

L-Series for Vector-Valued Weakly Holomorphic Modular Forms and Converse Theorems

Authors: Subong Lim, Wissam Raji

Abstract: We introduce the $L$-series of weakly holomorphic modular forms using Laplace transforms and give their functional equations. We then determine converse theorems for vector-valued harmonic weak Maass forms, Jacobi forms, and elliptic modular forms of half-integer weight in Kohnen plus space. We introduce the $L$-series of weakly holomorphic modular forms using Laplace transforms and give their functional equations. We then determine converse theorems for vector-valued harmonic weak Maass forms, Jacobi forms, and elliptic modular forms of half-integer weight in Kohnen plus space. △ Less

Submitted 6 September, 2024; originally announced September 2024.

Stochastic Processes: From Classical to Quantum

Authors: Soon Hoe Lim

Abstract: The main goal of these notes is to give an introduction to the mathematics of quantum noise and some of its applications in non-equilibrium statistical mechanics. We start with some reminders from the theory of classical stochastic processes. We then provide a brief overview of quantum mechanics and quantum field theory, from the viewpoint of quantum probability and adopting the language of Hudson… ▽ More The main goal of these notes is to give an introduction to the mathematics of quantum noise and some of its applications in non-equilibrium statistical mechanics. We start with some reminders from the theory of classical stochastic processes. We then provide a brief overview of quantum mechanics and quantum field theory, from the viewpoint of quantum probability and adopting the language of Hudson and Parthasarathy. We introduce quantum stochastic processes on a boson Fock space and their calculus. Whenever possible, we make connections with the relevant concepts in classical probability theory. As an application of the theory, we introduce the theory of open quantum systems, with emphasis on the physics and modeling aspects of these systems. △ Less

Submitted 4 July, 2024; originally announced July 2024.

Comments: 52 pages

Singular systems of linear forms over global function fields

Authors: Gukyeong Bang, Taehyeong Kim, Seonhee Lim

Abstract: In this paper, we consider singular systems of linear forms over global function fields of class number one and give an upper bound for the Hausdorff dimension of the set of singular systems of linear forms by constructing an appropriate Margulis height function on the space of lattices over global function fields. In this paper, we consider singular systems of linear forms over global function fields of class number one and give an upper bound for the Hausdorff dimension of the set of singular systems of linear forms by constructing an appropriate Margulis height function on the space of lattices over global function fields. △ Less

Submitted 20 February, 2026; v1 submitted 11 April, 2024; originally announced April 2024.

Comments: 35 pages. Accepted for publication in Journal of Number Theory

The number of automorphic representations of $\mathrm{GL}_2$ with exceptional eigenvalues

Authors: Dohoon Choi, Min Lee, Youngmin Lee, Subong Lim

Abstract: We obtain an upper bound for the dimension of the cuspidal automorphic forms for $\mathrm{GL}_2$ over a number field, whose archimedean local representations are not tempered. More precisely, we prove the following result. Let $F$ be a number field and $\mathbb{A}_{F}$ be the ring of adeles of $F$. Let $\mathcal{O}_{F}$ be the ring of integers of $F$. Let $\mathfrak{X}_{F,\mathrm{ex}}$ be the se… ▽ More We obtain an upper bound for the dimension of the cuspidal automorphic forms for $\mathrm{GL}_2$ over a number field, whose archimedean local representations are not tempered. More precisely, we prove the following result. Let $F$ be a number field and $\mathbb{A}_{F}$ be the ring of adeles of $F$. Let $\mathcal{O}_{F}$ be the ring of integers of $F$. Let $\mathfrak{X}_{F,\mathrm{ex}}$ be the set of irreducible cuspidal automorphic representations $π$ of $\mathrm{GL}_2(\mathbb{A}_{F})$ with the trivial central character such that for each archimedean place $v$ of $F$, the local representation of $π$ at $v$ is an unramified principal series and is not tempered. For an ideal $J$ of $\mathcal{O}_{F}$, let $\mathrm{K}_{0}(J)$ be the subgroup of $\mathrm{GL}_2(\mathbb{A}_{F})$ corresponding to $Γ_0(J) \subset \mathrm{SL}_2(\mathcal{O}_F)$. Let $r_1$ be the number of real embeddings of $F$ and $r_2$ be the number of conjugate pairs of complex embeddings of $F$. Using the Arthur-Selberg trace formula, we have \begin{equation*} \sum_{π\in \mathfrak{X}_{F,\mathrm{ex}}} \dim π^{\mathrm{K}_0(J)} \ll_{F} \frac{[\mathrm{SL}_2(\mathcal{O}_{F}) : Γ_0(J)]}{(\log (N_{F/\mathbb{Q}}(J)))^{2r_1+3r_2}} \quad \text{ as } \quad |N_{F/\mathbb{Q}}(J)|\to \infty. \end{equation*} From this result, we obtain the result on an upper bound for the number of Hecke-Maass cusp forms of weight $0$ on $Γ_0(N)$ which do not satisfy the Selberg eigenvalue conjecture. △ Less

Submitted 18 February, 2024; originally announced February 2024.

MSC Class: 11F72 (Primary); 11F12 (Secondary)

Euclidean algorithms are Gaussian over imaginary quadratic fields

Authors: Dohyeong Kim, Jungwon Lee, Seonhee Lim

Abstract: The distributional analysis of Euclidean algorithms was carried out by Baladi and Vallée. They showed the asymptotic normality of the number of division steps and associated costs in the Euclidean algorithm as a random variable on the set of rational numbers with bounded denominator based on the transfer operator methods. We extend their result to the Euclidean algorithm over appropriate imaginary… ▽ More The distributional analysis of Euclidean algorithms was carried out by Baladi and Vallée. They showed the asymptotic normality of the number of division steps and associated costs in the Euclidean algorithm as a random variable on the set of rational numbers with bounded denominator based on the transfer operator methods. We extend their result to the Euclidean algorithm over appropriate imaginary quadratic fields by studying dynamics of the nearest integer complex continued fraction map, which is piecewise analytic and expanding but not a full branch map. By observing a finite Markov partition with a regular CW-structure, which enables us to associate the transfer operator acting on a direct sum of spaces of $C^1$-functions, we obtain the limit Gaussian distribution as well as residual equidistribution. △ Less

Submitted 24 October, 2025; v1 submitted 1 January, 2024; originally announced January 2024.

Comments: 40 pages

Report number: MPIM-Bonn-2025

$\operatorname{PGL}_{2}(\mathbb{Q}_{p})$-orbit closures on a $p$-adic homogeneous space of infinite volume

Authors: Jinho Jeoung, Seonhee Lim

Abstract: Let $\mathbb{K}$ be an unramified quadratic extension of $\mathbb{Q}_{p}$ for a fixed $p>2$. Projective general linear groups $G=\operatorname{PGL}_{2}(\mathbb{K})$ and $H=\operatorname{PGL}_{2}(\mathbb{Q}_{p})$ act transitively on Bruhat-Tits trees $T_G$ and $T_H$, respectively. We identify $G/H$ with the set of $H$-subtrees $G.T_{H}$. Let $Γ$ be a Schottky subgroup such that $Γ\backslash T_{G}$… ▽ More Let $\mathbb{K}$ be an unramified quadratic extension of $\mathbb{Q}_{p}$ for a fixed $p>2$. Projective general linear groups $G=\operatorname{PGL}_{2}(\mathbb{K})$ and $H=\operatorname{PGL}_{2}(\mathbb{Q}_{p})$ act transitively on Bruhat-Tits trees $T_G$ and $T_H$, respectively. We identify $G/H$ with the set of $H$-subtrees $G.T_{H}$. Let $Γ$ be a Schottky subgroup such that $Γ\backslash T_{G}$ is infinite volume and has an additional condition named high-branchedness, and let $Λ$ be its limit set. We classify $Γ$-orbits in $G/H$. Let $C=g_{C}H\in G/H$. As a generalization of Ratner's theorem, if $Γ\backslash g_{C}.T_{H}$ meets the convex core of $Γ\backslash T_{G}$, then the $Γ$-orbit of $C$ is either dense or closed in $ {\cal{C}}_Λ=\{g H: \partial(g.T_{H})\capΛ\neq\varnothing\}$. △ Less

Submitted 18 November, 2023; originally announced November 2023.

Comments: 35 pages, 5 figures

MSC Class: 37D40; 20G25

Threshold-aware Learning to Generate Feasible Solutions for Mixed Integer Programs

Authors: Taehyun Yoon, Jinwon Choi, Hyokun Yun, Sungbin Lim

Abstract: Finding a high-quality feasible solution to a combinatorial optimization (CO) problem in a limited time is challenging due to its discrete nature. Recently, there has been an increasing number of machine learning (ML) methods for addressing CO problems. Neural diving (ND) is one of the learning-based approaches to generating partial discrete variable assignments in Mixed Integer Programs (MIP), a… ▽ More Finding a high-quality feasible solution to a combinatorial optimization (CO) problem in a limited time is challenging due to its discrete nature. Recently, there has been an increasing number of machine learning (ML) methods for addressing CO problems. Neural diving (ND) is one of the learning-based approaches to generating partial discrete variable assignments in Mixed Integer Programs (MIP), a framework for modeling CO problems. However, a major drawback of ND is a large discrepancy between the ML and MIP objectives, i.e., variable value classification accuracy over primal bound. Our study investigates that a specific range of variable assignment rates (coverage) yields high-quality feasible solutions, where we suggest optimizing the coverage bridges the gap between the learning and MIP objectives. Consequently, we introduce a post-hoc method and a learning-based approach for optimizing the coverage. A key idea of our approach is to jointly learn to restrict the coverage search space and to predict the coverage in the learned search space. Experimental results demonstrate that learning a deep neural network to estimate the coverage for finding high-quality feasible solutions achieves state-of-the-art performance in NeurIPS ML4CO datasets. In particular, our method shows outstanding performance in the workload apportionment dataset, achieving the optimality gap of 0.45%, a ten-fold improvement over SCIP within the one-minute time limit. △ Less

Submitted 1 August, 2023; originally announced August 2023.

The Gromov-Wasserstein distance between spheres

Authors: Shreya Arya, Arnab Auddy, Ranthony Edmonds, Sunhyuk Lim, Facundo Memoli, Daniel Packer

Abstract: In this paper we consider a two-parameter family {dGWp,q}p,q of Gromov- Wasserstein distances between metric measure spaces. By exploiting a suitable interaction between specific values of the parameters p and q and the metric of the underlying spaces, we determine the exact value of the distance dGW4,2 between all pairs of unit spheres of different dimension endowed with their Euclidean distance… ▽ More In this paper we consider a two-parameter family {dGWp,q}p,q of Gromov- Wasserstein distances between metric measure spaces. By exploiting a suitable interaction between specific values of the parameters p and q and the metric of the underlying spaces, we determine the exact value of the distance dGW4,2 between all pairs of unit spheres of different dimension endowed with their Euclidean distance and their uniform measure. △ Less

Submitted 12 July, 2024; v1 submitted 18 June, 2023; originally announced June 2023.

Comments: 1. Added a Section 4, Section 5, and Appendix C; 2. Swapped Sections 2 and 3; 3. "Relagated Proofs" section (Section 4 in the old version) is now in appendix

Results on the Non-Vanishing of Derivatives of L-Functions of Vector-Valued Modular Forms

Authors: Subong Lim, Wissam Raji

Abstract: We show a non-vanishing result for the averages of the derivatives of $L$-functions associated with the orthogonal basis of the space of vector-valued cusp forms of weight $k\in \frac12 \mathbb{Z}$ on the full group in the critical strip. We also show the existence of at least one basis element whose $L$-function does not vanish under certain conditions. As an application, we generalize our result… ▽ More We show a non-vanishing result for the averages of the derivatives of $L$-functions associated with the orthogonal basis of the space of vector-valued cusp forms of weight $k\in \frac12 \mathbb{Z}$ on the full group in the critical strip. We also show the existence of at least one basis element whose $L$-function does not vanish under certain conditions. As an application, we generalize our result to Kohnen's plus space and prove an analogous result for Jacobi forms. △ Less

Submitted 19 May, 2023; originally announced May 2023.

Comments: arXiv admin note: substantial text overlap with arXiv:2302.12576

Journal ref: Proceedings of the Edinburgh Mathematical Society 68 (2025) 63-79

Non-Vanishing of L-Functions of Vector-Valued Modular Forms

Authors: Subong Lim, Wissam Raji

Abstract: We show a non-vanishing result for the averages of L-functions associated with the orthogonal basis of the space of cusp forms of vector-valued modular forms on the full group. We also show the existence of at least one basis element whose L-function does not vanish under certain conditions. We show a non-vanishing result for the averages of L-functions associated with the orthogonal basis of the space of cusp forms of vector-valued modular forms on the full group. We also show the existence of at least one basis element whose L-function does not vanish under certain conditions. △ Less

Submitted 11 May, 2023; v1 submitted 24 February, 2023; originally announced February 2023.

Reverse Bernstein Inequality on the Circle

Authors: Parvaneh Joharinad, Jürgen Jost, Sunhyuk Lim, Rostislav Matveev

Abstract: The more then hundred years old Bernstein inequality states that the supremum norm of the derivative of a trigonometric polynomial of fixed degree can be bounded from above by supremum norm of the polynomial itself. The reversed Bernstein inequality, that we prove in this note, says that the reverse inequality holds for functions in the orthogonal complement of the space of polynomials of fixed de… ▽ More The more then hundred years old Bernstein inequality states that the supremum norm of the derivative of a trigonometric polynomial of fixed degree can be bounded from above by supremum norm of the polynomial itself. The reversed Bernstein inequality, that we prove in this note, says that the reverse inequality holds for functions in the orthogonal complement of the space of polynomials of fixed degree. In fact, we derived a more general result for the lower bounds on higher derivatives. These bounds are better then those obtained by applying bound for the first derivative successively several times. △ Less

Submitted 8 March, 2023; v1 submitted 20 February, 2023; originally announced February 2023.

Comments: 8 pages, additional references in v2

MSC Class: 42A05

Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes

Authors: Henry Adams, Johnathan Bush, Nate Clause, Florian Frick, Mario Gómez, Michael Harrison, R. Amzi Jeffs, Evgeniya Lagoda, Sunhyuk Lim, Facundo Mémoli, Michael Moy, Nikola Sadovek, Matt Superdock, Daniel Vargas, Qingsong Wang, Ling Zhou

Abstract: We explore emerging relationships between the Gromov--Hausdorff distance, Borsuk--Ulam theorems, and Vietoris--Rips simplicial complexes. The Gromov--Hausdorff distance between two metric spaces $X$ and~$Y$ can be lower bounded by the distortion of (possibly discontinuous) functions between them. The more these functions must distort the metrics, the larger the Gromov--Hausdorff distance must be.… ▽ More We explore emerging relationships between the Gromov--Hausdorff distance, Borsuk--Ulam theorems, and Vietoris--Rips simplicial complexes. The Gromov--Hausdorff distance between two metric spaces $X$ and~$Y$ can be lower bounded by the distortion of (possibly discontinuous) functions between them. The more these functions must distort the metrics, the larger the Gromov--Hausdorff distance must be. Topology has few tools to obstruct the existence of discontinuous functions. However, an arbitrary function $f\colon X\to Y$ induces a continuous map between their Vietoris--Rips simplicial complexes, where the allowable choices of scale parameters depend on how much the function $f$ distorts distances. We can then use equivariant topology to obstruct the existence of certain continuous maps between Vietoris--Rips complexes. With these ideas we bound how discontinuous an odd map between spheres $S^k\to S^n$ with $k>n$ must be, generalizing a result by Dubins and Schwarz (1981), which is the case $k=n+1$. As an application, we recover or improve upon all of the lower bounds from Lim, M{é}moli, and Smith (2022) on the Gromov--Hausdorff distances between spheres of different dimensions. We also provide new upper bounds on the Gromov--Hausdorff distance between spheres of adjacent dimensions. △ Less

Submitted 5 July, 2025; v1 submitted 31 December, 2022; originally announced January 2023.

MSC Class: 51F30; 53C23; 55N31; 55P91

Option pricing under path-dependent stock models

Authors: Kiseop Lee, Seongje Lim, Hyungbin Park

Abstract: This paper studies how to price and hedge options under stock models given as a path-dependent SDE solution. When the path-dependent SDE coefficients have Fréchet derivatives, an option price is differentiable with respect to time and the path, and is given as a solution to the path-dependent PDE. This can be regarded as a path-dependent version of the Feynman-Kac formula. As a byproduct, we obtai… ▽ More This paper studies how to price and hedge options under stock models given as a path-dependent SDE solution. When the path-dependent SDE coefficients have Fréchet derivatives, an option price is differentiable with respect to time and the path, and is given as a solution to the path-dependent PDE. This can be regarded as a path-dependent version of the Feynman-Kac formula. As a byproduct, we obtain the differentiability of path-dependent SDE solutions and the SDE representation of their derivatives. In addition, we provide formulas for Greeks with path-dependent coefficient perturbations. A stock model having coefficients with time integration forms of paths is covered as an example. △ Less

Submitted 11 August, 2023; v1 submitted 20 November, 2022; originally announced November 2022.

Learning Symmetric Rules with SATNet

Authors: Sangho Lim, Eun-Gyeol Oh, Hongseok Yang

Abstract: SATNet is a differentiable constraint solver with a custom backpropagation algorithm, which can be used as a layer in a deep-learning system. It is a promising proposal for bridging deep learning and logical reasoning. In fact, SATNet has been successfully applied to learn, among others, the rules of a complex logical puzzle, such as Sudoku, just from input and output pairs where inputs are given… ▽ More SATNet is a differentiable constraint solver with a custom backpropagation algorithm, which can be used as a layer in a deep-learning system. It is a promising proposal for bridging deep learning and logical reasoning. In fact, SATNet has been successfully applied to learn, among others, the rules of a complex logical puzzle, such as Sudoku, just from input and output pairs where inputs are given as images. In this paper, we show how to improve the learning of SATNet by exploiting symmetries in the target rules of a given but unknown logical puzzle or more generally a logical formula. We present SymSATNet, a variant of SATNet that translates the given symmetries of the target rules to a condition on the parameters of SATNet and requires that the parameters should have a particular parametric form that guarantees the condition. The requirement dramatically reduces the number of parameters to learn for the rules with enough symmetries, and makes the parameter learning of SymSATNet much easier than that of SATNet. We also describe a technique for automatically discovering symmetries of the target rules from examples. Our experiments with Sudoku and Rubik's cube show the substantial improvement of SymSATNet over the baseline SATNet. △ Less

Submitted 25 November, 2022; v1 submitted 28 June, 2022; originally announced June 2022.

Comments: 27 pages, 10 figures, the first two authors contributed equally to this work, accepted at NeurIPS'22

Chaotic Regularization and Heavy-Tailed Limits for Deterministic Gradient Descent

Authors: Soon Hoe Lim, Yijun Wan, Umut Şimşekli

Abstract: Recent studies have shown that gradient descent (GD) can achieve improved generalization when its dynamics exhibits a chaotic behavior. However, to obtain the desired effect, the step-size should be chosen sufficiently large, a task which is problem dependent and can be difficult in practice. In this study, we incorporate a chaotic component to GD in a controlled manner, and introduce multiscale p… ▽ More Recent studies have shown that gradient descent (GD) can achieve improved generalization when its dynamics exhibits a chaotic behavior. However, to obtain the desired effect, the step-size should be chosen sufficiently large, a task which is problem dependent and can be difficult in practice. In this study, we incorporate a chaotic component to GD in a controlled manner, and introduce multiscale perturbed GD (MPGD), a novel optimization framework where the GD recursion is augmented with chaotic perturbations that evolve via an independent dynamical system. We analyze MPGD from three different angles: (i) By building up on recent advances in rough paths theory, we show that, under appropriate assumptions, as the step-size decreases, the MPGD recursion converges weakly to a stochastic differential equation (SDE) driven by a heavy-tailed Lévy-stable process. (ii) By making connections to recently developed generalization bounds for heavy-tailed processes, we derive a generalization bound for the limiting SDE and relate the worst-case generalization error over the trajectories of the process to the parameters of MPGD. (iii) We analyze the implicit regularization effect brought by the dynamical regularization and show that, in the weak perturbation regime, MPGD introduces terms that penalize the Hessian of the loss function. Empirical results are provided to demonstrate the advantages of MPGD. △ Less

Submitted 22 October, 2022; v1 submitted 23 May, 2022; originally announced May 2022.

Comments: 24 pages, accepted at NeurIPS 2022

Weisfeiler-Lehman meets Gromov-Wasserstein

Authors: Samantha Chen, Sunhyuk Lim, Facundo Mémoli, Zhengchao Wan, Yusu Wang

Abstract: The Weisfeiler-Lehman (WL) test is a classical procedure for graph isomorphism testing. The WL test has also been widely used both for designing graph kernels and for analyzing graph neural networks. In this paper, we propose the Weisfeiler-Lehman (WL) distance, a notion of distance between labeled measure Markov chains (LMMCs), of which labeled graphs are special cases. The WL distance is polynom… ▽ More The Weisfeiler-Lehman (WL) test is a classical procedure for graph isomorphism testing. The WL test has also been widely used both for designing graph kernels and for analyzing graph neural networks. In this paper, we propose the Weisfeiler-Lehman (WL) distance, a notion of distance between labeled measure Markov chains (LMMCs), of which labeled graphs are special cases. The WL distance is polynomial time computable and is also compatible with the WL test in the sense that the former is positive if and only if the WL test can distinguish the two involved graphs. The WL distance captures and compares subtle structures of the underlying LMMCs and, as a consequence of this, it is more discriminating than the distance between graphs used for defining the state-of-the-art Wasserstein Weisfeiler-Lehman graph kernel. Inspired by the structure of the WL distance we identify a neural network architecture on LMMCs which turns out to be universal w.r.t. continuous functions defined on the space of all LMMCs (which includes all graphs) endowed with the WL distance. Finally, the WL distance turns out to be stable w.r.t. a natural variant of the Gromov-Wasserstein (GW) distance for comparing metric Markov chains that we identify. Hence, the WL distance can also be construed as a polynomial time lower bound for the GW distance which is in general NP-hard to compute. △ Less

Submitted 5 February, 2022; originally announced February 2022.

Classical Multidimensional Scaling on Metric Measure Spaces

Authors: Sunhyuk Lim, Facundo Memoli

Abstract: We generalize the classical Multidimensional Scaling procedure to the setting of general metric measure spaces. We develop a related spectral theory for the generalized cMDS operator, which provides a more natural and rigorous mathematical background for cMDS. Also, we show that the sum of all negative eigenvalues of the cMDS operator is a new invariant measuring non-flatness of a metric measure s… ▽ More We generalize the classical Multidimensional Scaling procedure to the setting of general metric measure spaces. We develop a related spectral theory for the generalized cMDS operator, which provides a more natural and rigorous mathematical background for cMDS. Also, we show that the sum of all negative eigenvalues of the cMDS operator is a new invariant measuring non-flatness of a metric measure space. Furthermore, the cMDS output of several non-finite exemplar metric measures spaces, in particular the cMDS for spheres S^{d-1} and subsets of Euclidean space, are studied. Finally, we prove the stability of the generalized cMDS process with respect to the Gromov-Wasserstein distance. △ Less

Submitted 13 May, 2024; v1 submitted 23 January, 2022; originally announced January 2022.

Comments: Major changes are the following: (1) Fixed the proof of Proposition 3.25 (2) We wrote a new Section 7 for further discussion

Some results about the Tight Span of spheres

Authors: Sunhyuk Lim, Facundo Memoli, Zhengchao Wan, Qingsong Wang, Ling Zhou

Abstract: The smallest hyperconvex metric space containing a given metric space X is called the tight span of X. It is known that tight spans have many nice geometric and topological properties, and they are gradually becoming a target of research of both the metric geometry community and the topological/geometric data analysis community. In this paper, we study the tight span of n-spheres (with either geod… ▽ More The smallest hyperconvex metric space containing a given metric space X is called the tight span of X. It is known that tight spans have many nice geometric and topological properties, and they are gradually becoming a target of research of both the metric geometry community and the topological/geometric data analysis community. In this paper, we study the tight span of n-spheres (with either geodesic metric or l infinity-metric). △ Less

Submitted 23 December, 2021; originally announced December 2021.

A Deep Reinforcement Learning Approach for Solving the Traveling Salesman Problem with Drone

Authors: Aigerim Bogyrbayeva, Taehyun Yoon, Hanbum Ko, Sungbin Lim, Hyokun Yun, Changhyun Kwon

Abstract: Reinforcement learning has recently shown promise in learning quality solutions in many combinatorial optimization problems. In particular, the attention-based encoder-decoder models show high effectiveness on various routing problems, including the Traveling Salesman Problem (TSP). Unfortunately, they perform poorly for the TSP with Drone (TSP-D), requiring routing a heterogeneous fleet of vehicl… ▽ More Reinforcement learning has recently shown promise in learning quality solutions in many combinatorial optimization problems. In particular, the attention-based encoder-decoder models show high effectiveness on various routing problems, including the Traveling Salesman Problem (TSP). Unfortunately, they perform poorly for the TSP with Drone (TSP-D), requiring routing a heterogeneous fleet of vehicles in coordination -- a truck and a drone. In TSP-D, the two vehicles are moving in tandem and may need to wait at a node for the other vehicle to join. State-less attention-based decoder fails to make such coordination between vehicles. We propose a hybrid model that uses an attention encoder and a Long Short-Term Memory (LSTM) network decoder, in which the decoder's hidden state can represent the sequence of actions made. We empirically demonstrate that such a hybrid model improves upon a purely attention-based model for both solution quality and computational efficiency. Our experiments on the min-max Capacitated Vehicle Routing Problem (mmCVRP) also confirm that the hybrid model is more suitable for the coordinated routing of multiple vehicles than the attention-based model. The proposed model demonstrates comparable results as the operations research baseline methods. △ Less

Submitted 5 December, 2022; v1 submitted 21 December, 2021; originally announced December 2021.

On Hausdorff dimension in inhomogeneous Diophantine approximation over global function fields

Authors: Taehyeong Kim, Seonhee Lim, Frédéric Paulin

Abstract: In this paper, we study inhomogeneous Diophantine approximation over the completion $K_v$ of a global function field $K$ (over a finite field) for a discrete valuation $v$, with affine algebra $R_v$. We obtain an effective upper bound for the Hausdorff dimension of the set \[ \mathbf{Bad}_A(ε)=\left\{\boldsymbolθ\in K_v^{\,m} : \liminf_{(\mathbf{p},\mathbf{q})\in R_v^{\,m} \times R_v^{\,n}, \|\m… ▽ More In this paper, we study inhomogeneous Diophantine approximation over the completion $K_v$ of a global function field $K$ (over a finite field) for a discrete valuation $v$, with affine algebra $R_v$. We obtain an effective upper bound for the Hausdorff dimension of the set \[ \mathbf{Bad}_A(ε)=\left\{\boldsymbolθ\in K_v^{\,m} : \liminf_{(\mathbf{p},\mathbf{q})\in R_v^{\,m} \times R_v^{\,n}, \|\mathbf{q}\|\to \infty} \|\mathbf{q}\|^n \|A\mathbf{q}-\boldsymbolθ-\mathbf{p}\|^m \geq ε\right\}, \] of $ε$-badly approximable targets $\boldsymbolθ\in K_v^{\,m}$ for a fixed matrix $A\in\mathscr{M}_{m,n}(K_v)$, using an effective version of entropy rigidity in homogeneous dynamics for an appropriate diagonal action on the space of $R_v$-grids. We further characterize matrices $A$ for which $\mathbf{Bad}_A(ε)$ has full Hausdorff dimension for some $ε>0$ by a Diophantine condition of singularity on average. Our methods also work for the approximation using weighted ultrametric distances. △ Less

Submitted 25 April, 2023; v1 submitted 8 December, 2021; originally announced December 2021.

Comments: 54 pages

Dimension estimates for badly approximable affine forms

Authors: Taehyeong Kim, Wooyeon Kim, Seonhee Lim

Abstract: For given $ε>0$ and $b\in\mathbb{R}^m$, we say that a real $m\times n$ matrix $A$ is $ε$-badly approximable for the target $b$ if $$\liminf_{q\in\mathbb{Z}^n, \|q\|\to\infty} \|q\|^n \langle Aq-b \rangle^m \geq ε,$$ where $\langle \cdot \rangle$ denotes the distance from the nearest integral point. In this article, we obtain upper bounds for the Hausdorff dimensions of the set of $ε$-badly approxi… ▽ More For given $ε>0$ and $b\in\mathbb{R}^m$, we say that a real $m\times n$ matrix $A$ is $ε$-badly approximable for the target $b$ if $$\liminf_{q\in\mathbb{Z}^n, \|q\|\to\infty} \|q\|^n \langle Aq-b \rangle^m \geq ε,$$ where $\langle \cdot \rangle$ denotes the distance from the nearest integral point. In this article, we obtain upper bounds for the Hausdorff dimensions of the set of $ε$-badly approximable matrices for fixed target $b$ and the set of $ε$-badly approximable targets for fixed matrix $A$. Moreover, we give an equivalent Diophantine condition of $A$ for which the set of $ε$-badly approximable targets for fixed $A$ has full Hausdorff dimension for some $ε>0$. The upper bounds are established by effectivizing entropy rigidity in homogeneous dynamics, which is of independent interest. For the $A$-fixed case, our method also works for the weighted setting where the supremum norms are replaced by certain weighted quasinorms. △ Less

Submitted 15 September, 2022; v1 submitted 30 November, 2021; originally announced November 2021.

Comments: 50 pages, 1 figure. This paper supersedes the posting arXiv:1904.07476, making the latter obsolete. v1->v2: Appendix is removed, Section 2 is revised

Asymptotic distribution for pairs of linear and quadratic forms at integral vectors

Authors: Jiyoung Han, Seonhee Lim, Keivan Mallahi-Karai

Abstract: We study the joint distribution of values of a pair consisting of a quadratic form $q$ and a linear form $\mathbf l$ over the set of integral vectors, a problem initiated by Dani-Margulis (1989). In the spirit of the celebrated theorem of Eskin, Margulis and Mozes on the quantitative version of the Oppenheim conjecture, we show that if $n \ge 5$ then under the assumptions that for every… ▽ More We study the joint distribution of values of a pair consisting of a quadratic form $q$ and a linear form $\mathbf l$ over the set of integral vectors, a problem initiated by Dani-Margulis (1989). In the spirit of the celebrated theorem of Eskin, Margulis and Mozes on the quantitative version of the Oppenheim conjecture, we show that if $n \ge 5$ then under the assumptions that for every $(α, β) \in \mathbb R^2 \setminus \{ (0,0) \}$, the form $αq + β\mathbf l^2$ is irrational and that the signature of the restriction of $q$ to the kernel of $\mathbf l$ is $(p, n-1-p)$, where $3\le p \le n-2$, the number of vectors $v \in \mathbb Z^n$ for which $\|v\| < T$, $a < q(v) < b$ and $c< \mathbf l(v) < d$ is asymptotically $$ C(q, \mathbf l)(d-c)(b-a)T^{n-3} , $$ as $T \to \infty$, where $C(q, \mathbf l)$ only depends on $q$ and $\mathbf l$. The density of the set of joint values of $(q, \mathbf l)$ under the same assumptions is shown by Gorodnik (2004). △ Less

Submitted 21 February, 2024; v1 submitted 11 November, 2021; originally announced November 2021.

Comments: 23 pages

MSC Class: 60B15

Journal ref: Ergod. Th. Dynam. Sys. 45 (2025) 111-139

Equivariant Manifold Flows

Authors: Isay Katsman, Aaron Lou, Derek Lim, Qingxuan Jiang, Ser-Nam Lim, Christopher De Sa

Abstract: Tractably modelling distributions over manifolds has long been an important goal in the natural sciences. Recent work has focused on developing general machine learning models to learn such distributions. However, for many applications these distributions must respect manifold symmetries -- a trait which most previous models disregard. In this paper, we lay the theoretical foundations for learning… ▽ More Tractably modelling distributions over manifolds has long been an important goal in the natural sciences. Recent work has focused on developing general machine learning models to learn such distributions. However, for many applications these distributions must respect manifold symmetries -- a trait which most previous models disregard. In this paper, we lay the theoretical foundations for learning symmetry-invariant distributions on arbitrary manifolds via equivariant manifold flows. We demonstrate the utility of our approach by using it to learn gauge invariant densities over $SU(n)$ in the context of quantum field theory. △ Less

Submitted 27 January, 2022; v1 submitted 18 July, 2021; originally announced July 2021.

Comments: Published at NeurIPS 2021

The Gromov-Hausdorff distance between spheres

Authors: Sunhyuk Lim, Facundo Mémoli, Zane Smith

Abstract: We provide general upper and lower bounds for the Gromov-Hausdorff distance $d_{\mathrm{GH}}(\mathbb{S}^m,\mathbb{S}^n)$ between spheres $\mathbb{S}^m$ and $\mathbb{S}^n$ (endowed with the round metric) for $0\leq m< n\leq \infty$. Some of these lower bounds are based on certain topological ideas related to the Borsuk-Ulam theorem. Via explicit constructions of (optimal) correspondences we prove t… ▽ More We provide general upper and lower bounds for the Gromov-Hausdorff distance $d_{\mathrm{GH}}(\mathbb{S}^m,\mathbb{S}^n)$ between spheres $\mathbb{S}^m$ and $\mathbb{S}^n$ (endowed with the round metric) for $0\leq m< n\leq \infty$. Some of these lower bounds are based on certain topological ideas related to the Borsuk-Ulam theorem. Via explicit constructions of (optimal) correspondences we prove that our lower bounds are tight in the cases of $d_{\mathrm{GH}}(\mathbb{S}^0,\mathbb{S}^n)$, $d_{\mathrm{GH}}(\mathbb{S}^m,\mathbb{S}^\infty)$, $d_{\mathrm{GH}}(\mathbb{S}^1,\mathbb{S}^2)$, $d_{\mathrm{GH}}(\mathbb{S}^1,\mathbb{S}^3)$ and $d_{\mathrm{GH}}(\mathbb{S}^2,\mathbb{S}^3)$. We also formulate a number of open questions. △ Less

Submitted 18 October, 2022; v1 submitted 2 May, 2021; originally announced May 2021.

Comments: * We made some structural changes for better readability

Journal ref: Geom. Topol. 27 (2023) 3733-3800

Tempered Fractional Brownian Motion with Variable Index and Variable Tempering Parameter

Authors: S. C. Lim, Chai Hok Eab

Abstract: Generalizations of tempered fractional Brownian from single index to two indices and variable index or tempered multifractional Brownian motion are studied. Tempered fractional Brownian motion and tempered multifractional Brownian motion with variable tempering parameter are considered. Generalizations of tempered fractional Brownian from single index to two indices and variable index or tempered multifractional Brownian motion are studied. Tempered fractional Brownian motion and tempered multifractional Brownian motion with variable tempering parameter are considered. △ Less

Submitted 10 April, 2021; originally announced April 2021.

Comments: 20 page, 1 figure. arXiv admin note: text overlap with arXiv:1907.08974

Noisy Recurrent Neural Networks

Authors: Soon Hoe Lim, N. Benjamin Erichson, Liam Hodgkinson, Michael W. Mahoney

Abstract: We provide a general framework for studying recurrent neural networks (RNNs) trained by injecting noise into hidden states. Specifically, we consider RNNs that can be viewed as discretizations of stochastic differential equations driven by input data. This framework allows us to study the implicit regularization effect of general noise injection schemes by deriving an approximate explicit regulari… ▽ More We provide a general framework for studying recurrent neural networks (RNNs) trained by injecting noise into hidden states. Specifically, we consider RNNs that can be viewed as discretizations of stochastic differential equations driven by input data. This framework allows us to study the implicit regularization effect of general noise injection schemes by deriving an approximate explicit regularizer in the small noise regime. We find that, under reasonable assumptions, this implicit regularization promotes flatter minima; it biases towards models with more stable dynamics; and, in classification tasks, it favors models with larger classification margin. Sufficient conditions for global stability are obtained, highlighting the phenomenon of stochastic stabilization, where noise injection can improve stability during training. Our theory is supported by empirical results which demonstrate that the RNNs have improved robustness with respect to various input perturbations. △ Less

Submitted 1 December, 2021; v1 submitted 9 February, 2021; originally announced February 2021.

Comments: 38 pages

Journal ref: NeurIPS 2021 (https://proceedings.neurips.cc/paper/2021/hash/29301521774ff3cbd26652b2d5c95996-Abstract.html)

Robustness and Personalization in Federated Learning: A Unified Approach via Regularization

Authors: Achintya Kundu, Pengqian Yu, Laura Wynter, Shiau Hong Lim

Abstract: We present a class of methods for robust, personalized federated learning, called Fed+, that unifies many federated learning algorithms. The principal advantage of this class of methods is to better accommodate the real-world characteristics found in federated training, such as the lack of IID data across parties, the need for robustness to outliers or stragglers, and the requirement to perform we… ▽ More We present a class of methods for robust, personalized federated learning, called Fed+, that unifies many federated learning algorithms. The principal advantage of this class of methods is to better accommodate the real-world characteristics found in federated training, such as the lack of IID data across parties, the need for robustness to outliers or stragglers, and the requirement to perform well on party-specific datasets. We achieve this through a problem formulation that allows the central server to employ robust ways of aggregating the local models while keeping the structure of local computation intact. Without making any statistical assumption on the degree of heterogeneity of local data across parties, we provide convergence guarantees for Fed+ for convex and non-convex loss functions under different (robust) aggregation methods. The Fed+ theory is also equipped to handle heterogeneous computing environments including stragglers without additional assumptions; specifically, the convergence results cover the general setting where the number of local update steps across parties can vary. We demonstrate the benefits of Fed+ through extensive experiments across standard benchmark datasets. △ Less

Submitted 12 July, 2022; v1 submitted 14 September, 2020; originally announced September 2020.

Comments: Accepted by IEEE EDGE 2022 (16 pages, 4 figures, 2 tables)

Understanding Recurrent Neural Networks Using Nonequilibrium Response Theory

Authors: Soon Hoe Lim

Abstract: Recurrent neural networks (RNNs) are brain-inspired models widely used in machine learning for analyzing sequential data. The present work is a contribution towards a deeper understanding of how RNNs process input signals using the response theory from nonequilibrium statistical mechanics. For a class of continuous-time stochastic RNNs (SRNNs) driven by an input signal, we derive a Volterra type s… ▽ More Recurrent neural networks (RNNs) are brain-inspired models widely used in machine learning for analyzing sequential data. The present work is a contribution towards a deeper understanding of how RNNs process input signals using the response theory from nonequilibrium statistical mechanics. For a class of continuous-time stochastic RNNs (SRNNs) driven by an input signal, we derive a Volterra type series representation for their output. This representation is interpretable and disentangles the input signal from the SRNN architecture. The kernels of the series are certain recursively defined correlation functions with respect to the unperturbed dynamics that completely determine the output. Exploiting connections of this representation and its implications to rough paths theory, we identify a universal feature -- the response feature, which turns out to be the signature of tensor product of the input signal and a natural support basis. In particular, we show that SRNNs, with only the weights in the readout layer optimized and the weights in the hidden layer kept fixed and not optimized, can be viewed as kernel machines operating on a reproducing kernel Hilbert space associated with the response feature. △ Less

Submitted 18 January, 2021; v1 submitted 19 June, 2020; originally announced June 2020.

Comments: 48 pages

Journal ref: Journal of Machine Learning Research (2021)

Neural Manifold Ordinary Differential Equations

Authors: Aaron Lou, Derek Lim, Isay Katsman, Leo Huang, Qingxuan Jiang, Ser-Nam Lim, Christopher De Sa

Abstract: To better conform to data geometry, recent deep generative modelling techniques adapt Euclidean constructions to non-Euclidean spaces. In this paper, we study normalizing flows on manifolds. Previous work has developed flow models for specific cases; however, these advancements hand craft layers on a manifold-by-manifold basis, restricting generality and inducing cumbersome design constraints. We… ▽ More To better conform to data geometry, recent deep generative modelling techniques adapt Euclidean constructions to non-Euclidean spaces. In this paper, we study normalizing flows on manifolds. Previous work has developed flow models for specific cases; however, these advancements hand craft layers on a manifold-by-manifold basis, restricting generality and inducing cumbersome design constraints. We overcome these issues by introducing Neural Manifold Ordinary Differential Equations, a manifold generalization of Neural ODEs, which enables the construction of Manifold Continuous Normalizing Flows (MCNFs). MCNFs require only local geometry (therefore generalizing to arbitrary manifolds) and compute probabilities with continuous change of variables (allowing for a simple and expressive flow construction). We find that leveraging continuous manifold dynamics produces a marked improvement for both density estimation and downstream tasks. △ Less

Submitted 17 June, 2020; originally announced June 2020.

Comments: Submitted to NeurIPS 2020

Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius

Authors: Sunhyuk Lim, Facundo Memoli, Osman Berat Okutan

Abstract: In the applied algebraic topology community, the persistent homology induced by the Vietoris-Rips simplicial filtration is a standard method for capturing topological information from metric spaces. In this paper, we consider a different, more geometric way of generating persistent homology of metric spaces which arises by first embedding a given metric space into a larger space and then consideri… ▽ More In the applied algebraic topology community, the persistent homology induced by the Vietoris-Rips simplicial filtration is a standard method for capturing topological information from metric spaces. In this paper, we consider a different, more geometric way of generating persistent homology of metric spaces which arises by first embedding a given metric space into a larger space and then considering thickenings of the original space inside this ambient metric space. In the course of doing this, we construct an appropriate category for studying this notion of persistent homology and show that, in a category theoretic sense, the standard persistent homology of the Vietoris-Rips filtration is isomorphic to our geometric persistent homology provided that the ambient metric space satisfies a property called injectivity. As an application of this isomorphism result we are able to precisely characterize the type of intervals that appear in the persistence barcodes of the Vietoris-Rips filtration of any compact metric space and also to give succinct proofs of the characterization of the persistent homology of products and metric gluings of metric spaces. Our results also permit proving several bounds on the length of intervals in the Vietoris-Rips barcode by other metric invariants. Finally, as another application, we connect this geometric persistent homology to the notion of filling radius of manifolds introduced by Gromov \cite{G07} and show some consequences related to (1) the homotopy type of the Vietoris-Rips complexes of spheres which follow from work of M.~Katz and (2) characterization (rigidity) results for spheres in terms of their Vietoris-Rips persistence barcodes which follow from work of F.~Wilhelm. △ Less

Submitted 28 March, 2024; v1 submitted 21 January, 2020; originally announced January 2020.

Comments: We added Remark 9.8

Journal ref: Algebr. Geom. Topol. 24 (2024) 1019-1100

Anomalous Thermodynamics in Homogenized Generalized Langevin Systems

Authors: Soon Hoe Lim

Abstract: We study functionals, such as heat and work, along trajectories of a class of multi-dimensional generalized Langevin systems in various limiting situations that correspond to different level of homogenization. These are the situations where one or more of the inertial time scale(s), the memory time scale(s) and the noise correlation time scale(s) of the systems are taken to zero. We find that, unl… ▽ More We study functionals, such as heat and work, along trajectories of a class of multi-dimensional generalized Langevin systems in various limiting situations that correspond to different level of homogenization. These are the situations where one or more of the inertial time scale(s), the memory time scale(s) and the noise correlation time scale(s) of the systems are taken to zero. We find that, unless one restricts to special situations that do not break symmetry of the Onsager matrix associated with the fast dynamics, it is generally not possible to express the effective evolution of these functionals solely in terms of trajectory of the homogenized process describing the system dynamics via the widely adopted Stratonovich convention. In fact, an anomalous term is often needed for a complete description, implying that convergence of these functionals needs more information than simply the limit of the dynamical process. We trace the origin of such impossibility to area anomaly, thereby linking the symmetry breaking and area anomaly. This hold important consequences for many nonequilibrium systems that can be modeled by generalized Langevin equations. Our convergence results hold in a strong pathwise sense. △ Less

Submitted 24 February, 2021; v1 submitted 18 November, 2019; originally announced November 2019.

Comments: 41 pages, to appear in Journal of Physics A: Mathematical and Theoretical

Making an Invisibility Cloak: Real World Adversarial Attacks on Object Detectors

Authors: Zuxuan Wu, Ser-Nam Lim, Larry Davis, Tom Goldstein

Abstract: We present a systematic study of adversarial attacks on state-of-the-art object detection frameworks. Using standard detection datasets, we train patterns that suppress the objectness scores produced by a range of commonly used detectors, and ensembles of detectors. Through extensive experiments, we benchmark the effectiveness of adversarially trained patches under both white-box and black-box set… ▽ More We present a systematic study of adversarial attacks on state-of-the-art object detection frameworks. Using standard detection datasets, we train patterns that suppress the objectness scores produced by a range of commonly used detectors, and ensembles of detectors. Through extensive experiments, we benchmark the effectiveness of adversarially trained patches under both white-box and black-box settings, and quantify transferability of attacks between datasets, object classes, and detector models. Finally, we present a detailed study of physical world attacks using printed posters and wearable clothes, and rigorously quantify the performance of such attacks with different metrics. △ Less

Submitted 22 July, 2020; v1 submitted 31 October, 2019; originally announced October 2019.

Comments: ECCV 2020

The Milnor $K$-theory and the Shintani cocycle

Authors: Sung Hyun Lim, Jeehoon Park

Abstract: The goal of this article is to complete the unfinished construction (due to Glenn Stevens in an old preprint) of a certain Milnor $K$-group valued group cocycle for $GL_n(\mathbb{Q})$ where $n$ is a positive integer, which we call the Stevens cocycle. Moreover, we give a precise relationship between the Stevens cocycle and the Shintani cocycle, which encodes key informations on the zeta values of… ▽ More The goal of this article is to complete the unfinished construction (due to Glenn Stevens in an old preprint) of a certain Milnor $K$-group valued group cocycle for $GL_n(\mathbb{Q})$ where $n$ is a positive integer, which we call the Stevens cocycle. Moreover, we give a precise relationship between the Stevens cocycle and the Shintani cocycle, which encodes key informations on the zeta values of totally real fields of degree $n$, using the $\operatorname{dlog}$ map of $K$-theory and the Fourier transform of locally constant functions on $\mathbb{Q}^n$ with bounded support. Roughly speaking, the Stevens cocycle is a multiplicative version of the Shintani cocyle. △ Less

Submitted 8 September, 2019; originally announced September 2019.

Comments: 19 pages

MSC Class: 11F15; 11R70

Predicting Critical Transitions in Multiscale Dynamical Systems Using Reservoir Computing

Authors: Soon Hoe Lim, Ludovico Theo Giorgini, Woosok Moon, J. S. Wettlaufer

Abstract: We study the problem of predicting rare critical transition events for a class of slow-fast nonlinear dynamical systems. The state of the system of interest is described by a slow process, whereas a faster process drives its evolution and induces critical transitions. By taking advantage of recent advances in reservoir computing, we present a data-driven method to predict the future evolution of t… ▽ More We study the problem of predicting rare critical transition events for a class of slow-fast nonlinear dynamical systems. The state of the system of interest is described by a slow process, whereas a faster process drives its evolution and induces critical transitions. By taking advantage of recent advances in reservoir computing, we present a data-driven method to predict the future evolution of the state. We show that our method is capable of predicting a critical transition event at least several numerical time steps in advance. We demonstrate the success as well as the limitations of our method using numerical experiments on three examples of systems, ranging from low dimensional to high dimensional. We discuss the mathematical and broader implications of our results. △ Less

Submitted 5 December, 2020; v1 submitted 10 August, 2019; originally announced August 2019.

Comments: 21 pages

Journal ref: Chaos: An Interdisciplinary Journal of Nonlinear Science (2020)

Tempered Fractional Brownian Motion Revisited Via Fractional Ornstein-Uhlenbeck Processes

Authors: S. C. Lim, Chai Hok Eab

Abstract: Tempered fractional Brownian motion is revisited from the viewpoint of reduced fractional Ornstein-Uhlenbeck process. Many of the basic properties of the tempered fractional Brownian motion can be shown to be direct consequences or modifications of the properties of fractional Ornstein-Uhlenbeck process. Mixed tempered fractional Brownian motion is introduced and its properties are considered. Tem… ▽ More Tempered fractional Brownian motion is revisited from the viewpoint of reduced fractional Ornstein-Uhlenbeck process. Many of the basic properties of the tempered fractional Brownian motion can be shown to be direct consequences or modifications of the properties of fractional Ornstein-Uhlenbeck process. Mixed tempered fractional Brownian motion is introduced and its properties are considered. Tempered fractional Brownian motion is generalised from single index to two indices. Finally, tempered multifractional Brownian motion and its properties are studied. △ Less

Submitted 21 July, 2019; originally announced July 2019.

Comments: 24 page 1 figure

Quantitative Propagation of Chaos in the bimolecular chemical reaction-diffusion model

Authors: Tau Shean Lim, Yulong Lu, James Nolen

Abstract: We study a stochastic system of $N$ interacting particles which models bimolecular chemical reaction-diffusion. In this model, each particle $i$ carries two attributes: the spatial location $X_t^i\in \mathbb{T}^d$, and the type $Ξ_t^i\in \{1,\cdots,n\}$. While $X_t^i$ is a standard (independent) diffusion process, the evolution of the type $Ξ_t^i$ is described by pairwise interactions between diff… ▽ More We study a stochastic system of $N$ interacting particles which models bimolecular chemical reaction-diffusion. In this model, each particle $i$ carries two attributes: the spatial location $X_t^i\in \mathbb{T}^d$, and the type $Ξ_t^i\in \{1,\cdots,n\}$. While $X_t^i$ is a standard (independent) diffusion process, the evolution of the type $Ξ_t^i$ is described by pairwise interactions between different particles under a series of chemical reactions described by a chemical reaction network. We prove that in the large particle limit the stochastic dynamics converges to a mean field limit which is described by a nonlocal reaction-diffusion partial differential equation. In particular, we obtain a quantitative propagation of chaos result for the interacting particle system. Our proof is based on the relative entropy method used recently by Jabin and Wang \cite{JW18}. The key ingredient of the relative entropy method is a large deviation estimate for a special partition function, which was proved previously by technical combinatorial estimates. We give a simple probabilistic proof based on a novel martingale argument. △ Less

Submitted 22 January, 2020; v1 submitted 3 June, 2019; originally announced June 2019.

Comments: 38 pages

MSC Class: Analysis of PDEs

Martin boundary of Brownian motion on Gromov hyperbolic metric graphs

Authors: Soonki Hong, Seonhee Lim

Abstract: Let $\widetilde{X}$ be a locally finite complete Gromov hyperbolic metric graph with the geometric boundary consisting of infinitely many points. Suppose that there is a discrete subgroup of the isometry group $Iso(\widetilde{X})$ acting geometrically on $\widetilde{X}$. The $λ$-Martin boundary is the boundary of the image of an embedding from $\widetilde{X}$ to the space of $λ$-superharmonic func… ▽ More Let $\widetilde{X}$ be a locally finite complete Gromov hyperbolic metric graph with the geometric boundary consisting of infinitely many points. Suppose that there is a discrete subgroup of the isometry group $Iso(\widetilde{X})$ acting geometrically on $\widetilde{X}$. The $λ$-Martin boundary is the boundary of the image of an embedding from $\widetilde{X}$ to the space of $λ$-superharmonic functions. We show that the $λ$-Martin boundary coincides with the geometric boundary for any $λ\in [0, λ_0],$ in particular at the bottom of the spectrum $λ_0$. △ Less

Submitted 15 December, 2020; v1 submitted 23 May, 2019; originally announced May 2019.

Comments: 32 pages, 4 figures

MSC Class: 37H05; 31C25; 31C35

Dimension bound for doubly badly approximable affine forms

Authors: Wooyeon Kim, Seonhee Lim

Abstract: We prove that for all $b$, the Hausdorff dimension of the set of $m \times n$ matrices $ε$-badly approximable for the target $b$ is not full. The doubly metric case follows. It was known that for almost every matrix $A$, the Hausdorff dimension of the set $Bad_A(ε)$ of $ε$-badly approximable target $b$ is not full, and that for real numbers $α$, $\dim_H Bad_α(ε)=1$ if and only if $α$ is singular… ▽ More We prove that for all $b$, the Hausdorff dimension of the set of $m \times n$ matrices $ε$-badly approximable for the target $b$ is not full. The doubly metric case follows. It was known that for almost every matrix $A$, the Hausdorff dimension of the set $Bad_A(ε)$ of $ε$-badly approximable target $b$ is not full, and that for real numbers $α$, $\dim_H Bad_α(ε)=1$ if and only if $α$ is singular on average. We show that if $\dim_H Bad_A(ε)=m$, then $A$ is singular on average. △ Less

Submitted 30 August, 2019; v1 submitted 16 April, 2019; originally announced April 2019.

Comments: 20 pages. Some typos are corrected and a section on preliminaries (section 2.2) are added

MSC Class: 11K60; 28A80; 37E10

Homogenization for Generalized Langevin Equations with Applications to Anomalous Diffusion

Authors: Soon Hoe Lim, Jan Wehr, Maciej Lewenstein

Abstract: We study homogenization for a class of generalized Langevin equations (GLEs) with state-dependent coefficients and exhibiting multiple time scales. In addition to the small mass limit, we focus on homogenization limits, which involve taking to zero the inertial time scale and, possibly, some of the memory time scales and noise correlation time scales. The latter are meaningful limits for a class o… ▽ More We study homogenization for a class of generalized Langevin equations (GLEs) with state-dependent coefficients and exhibiting multiple time scales. In addition to the small mass limit, we focus on homogenization limits, which involve taking to zero the inertial time scale and, possibly, some of the memory time scales and noise correlation time scales. The latter are meaningful limits for a class of GLEs modeling anomalous diffusion. We find that, in general, the limiting stochastic differential equations (SDEs) for the slow degrees of freedom contain non-trivial drift correction terms and are driven by non-Markov noise processes. These results follow from a general homogenization theorem stated and proven here. We illustrate them using stochastic models of particle diffusion. △ Less

Submitted 1 November, 2019; v1 submitted 18 February, 2019; originally announced February 2019.

Comments: 66 pages

Journal ref: Annales Henri Poincaré (2020)