arXiv subjects

Hansol Park

Publications and source records attributed to Hansol Park.

At least 19 recordsLinked to original sources

arXiv2026

Stability of equilibria of an aggregation-diffusion energy on sphere

We consider an aggregation-diffusion energy on the sphere and investigate the stability of its equilibria. The energy consists of a porous-medium type nonlinear entropy $\frac{1}{m-1}\int ρ(x)^m\mathrm{d}S(x)$ with $m>1$, together with an interaction energy modeled by a quadratic interaction potential. The energy generalizes the Onsager free energy with dipolar potential, which models polymer orientation. Our study complements the authors' previous work [Nonlinearity {\bf 39} (2026), 055011], where the ground states of the energy functional were investigated. In the current paper we extend the previous results on existence of equilibria for $m>2$ from $d=2$ to arbitrary dimension $d \geq 2$. This allows us to present the bifurcation structure (with respect to the interaction strength) of all equilibria in any dimension $d \geq 2$, for all $m>1$. Furthermore, we provide a complete classification of the stability of all equilibria of the energy, by deriving a criterion for stability that can be checked explicitly. In particular, for $m>2$ we identify a saddle-node bifurcation of strictly supported equilibria, and a subcritical pitchfork bifurcation where the uniform distribution loses stability.

math.AP↗

arXiv2026

Mask2Flow-TSE: Two-Stage Target Speaker Extraction with Masking and Flow Matching

Target speaker extraction (TSE) extracts the target speaker's voice from overlapping speech given a reference utterance. Existing masking-based approaches are lightweight and effective but suffer from an inability to synthesize missing content, leading to degraded perceptual quality. On the other hand, recent generative TSE models typically synthesize high-quality speech with diffusion, but require numerous iterative steps resulting in high computational costs and latency. We propose Mask2Flow-TSE, a two-stage framework combining the strengths of both paradigms. We introduce the deletion/insertion (D/I) proportion, an analytical tool that reveals early flow steps predominantly remove signal components rather than synthesize them. Based on this finding, we decouple deletion from insertion: a masking-based module handles the deletion-dominant early steps, while a single flow-matching step performs the remaining insertion for high-quality reconstruction. Specifically, the first stage uses lightweight convolution for the masking module, while the second stage employs a Diffusion Transformer (DiT) adapted for TSE with speaker conditioning. Unlike prior approaches that start from Gaussian noise, our method starts from the masked spectrogram, enabling high-quality reconstruction in a single inference step. Experiments show that Mask2Flow-TSE produces high-quality extractions with only 85M parameters and one-step inference, while preserving clean single-speaker inputs with minimal degradation.

cs.SD↗

arXiv2026

Finite-dimensional reduction of a Wasserstein gradient flow and sharp decay rates

We study the Wasserstein gradient flow generated by a family of extended generalized variance functionals, defined as the expected squared $n$-dimensional volume of a simplex, which includes the classical variance-type interaction and generalized variance as special cases. The key structural observation is that this functional depends only on the covariance matrix. Consequently, the Wasserstein gradient flow reduces to a finite-dimensional system for the eigenvalues of the covariance matrix, and the full measure-valued solution can be recovered through an explicit linear pushforward representation. Using this representation, we establish global well-posedness for arbitrary initial data in $\mathcal P_2(\mathbb R^d)$ without assuming compact support. We also study the long-time behavior of the flow. For every initial datum in $\mathcal P_2(\mathbb R^d)$, the solution converges to a limiting equilibrium measure whose covariance has rank strictly less than $n$. Moreover, we obtain sharp convergence rates in all spectral regimes: exponential in the non-degenerate case and algebraic with the optimal exponent in the degenerate case.

math.AP↗

arXiv2026

Collective Optimization on Riemannian Manifolds with Bounded Curvature

In this paper, we develop an intrinsic consensus-based optimization framework on Riemannian manifolds with bounded sectional curvature. In contrast to extrinsic approaches based on an ambient Euclidean embedding, our model is formulated directly in terms of the Riemannian structure, using logarithmic and exponential maps induced by the intrinsic geodesic distance. We prove the global well-posedness of the proposed particle system and its associated McKean--Vlasov dynamics. We also establish the global convergence of the mean-field equation toward a global minimizer of the objective function under suitable conditions. Numerical experiments on the sphere, hyperbolic space, and the special orthogonal group demonstrate the effectiveness of the intrinsic CBO dynamics for nonconvex optimization problems on manifolds.

math.OC↗

arXiv2026

Evaluating Hallucinations in Audio-Visual Multimodal LLMs with Spoken Queries under Diverse Acoustic Conditions

Hallucinations in multimodal models have been extensively studied using benchmarks that probe reliability in image-text query settings. However, the effect of spoken queries on multimodal hallucinations remains largely unexplored, despite the growing role of voice interfaces. In this paper, we introduce a systematic pipeline that converts existing multimodal hallucination benchmarks into spoken-query versions while preserving the original tasks and labels. We instantiate this pipeline on RePOPE and release RePOPE-Spk, where all queries are provided as spoken audio under diverse input conditions. Experimental results show that hallucinations escalate when queries are spoken rather than written: error rates increase by 3-6% with clean speech and by up to 30% under environmental noise. Furthermore, many-shot prompting and chain-of-thought reasoning provide only partial mitigation. Our findings motivate new directions for building reliable voice interface systems and evaluations.

cs.SD↗

arXiv2025

Ground states and phase transitions for an aggregation model with fast diffusion on sphere

We consider a free energy on the sphere that contains an entropy associated to nonlinear fast diffusion, and a nonlocal interaction energy. The two components of the free energy compete with each other, as one favours spreading and the other promotes concentration, respectively. The model is a generalization of the Onsager free energy with dipolar potential, used to study polymer orientation. We study the global energy minimizers of the energy functional, and in particular the various phase transitions that occur with respect to the strength of the nonlocal attractive interactions. In the considered regime, diffusion reduces as the density increases, for which reason the global energy minimizers can contain Dirac mass concentrations. We identify various ranges of the fast diffusion exponent and of the interaction strength, which give qualitatively different equilibria and ground states. The theoretical results are supported by numerical illustrations.

math.AP↗

arXiv2025

Bifurcation of global energy minimizers for a diffusion-aggregation model on sphere

We consider a free energy functional defined on probability densities on the unit sphere $\mathbb{S}^d$, and investigate its global minimizers. The energy consists of two components: an entropy and a nonlocal interaction energy, which favour spreading and aggregation behaviour, respectively. We find a threshold value for the size of the attractive interactions, and establish the global energy minimizers in each case. The bifurcation at this threshold value is investigated. We also generalize the results to spaces consisting of an arbitrary number of spheres (e.g., the flat torus $\mathbb{S}^1 \times \mathbb{S}^1$).

math.AP↗

arXiv2025

Global energy minimizers for a diffusion-aggregation model on sphere

We investigate the ground states of a free energy functional on sphere. The energy consists of an entropy and a nonlocal interaction term that are in competition with each other, as they favour spreading and aggregation, respectively. Specifically, the entropy corresponds to slow nonlinear diffusion and the interaction term is modeled by a quadratic interaction potential. We investigate the transitions that occur in the equilibria and the global minimizers of the energy, in terms of the strength of the nonlocal attractive interactions. We consider separately various ranges of the diffusion exponent, which give qualitatively different behaviours of equilibria and ground states. In terms of applications, we note that the energy we consider here is a generalization to nonlinear diffusion of the Onsager free energy with dipolar potential, used to study phase transitions in polymer orientation.

math.AP↗

arXiv2025

Global minimizers for fast diffusion versus nonlocal interactions on negatively curved manifolds

We investigate the existence of ground states for a free energy functional on Cartan-Hadamard manifolds. The energy, which consists of an entropy and an interaction term, is associated to a macroscopic aggregation model that includes nonlinear diffusion and nonlocal interactions. We consider specifically the regime of fast diffusion, and establish necessary and sufficient conditions on the behaviour of the interaction potential for global energy minimizers to exist. We first consider the case of manifolds with constant bounds of sectional curvatures, then extend the results to manifolds with general curvature bounds. To establish our results we derive several new Carlson-Levin type inequalities for Cartan-Hadamard manifolds.

math.AP↗

arXiv2024

Existence of ground states for free energies on the hyperbolic space

We investigate a free energy functional that arises in aggregation-diffusion phenomena modelled by nonlocal interactions and local repulsion on the hyperbolic space $\bbh^\dm$. The free energy consists of two competing terms: an entropy, corresponding to slow nonlinear diffusion, that favours spreading, and an attractive interaction potential energy that favours aggregation. We establish necessary and sufficient conditions on the interaction potential for ground states to exist on the hyperbolic space $\bbh^\dm$. To prove our results we derived several Hardy-Littlewood-Sobolev (HLS)-type inequalities on general Cartan-Hadamard manifolds of bounded curvature, which have an interest in their own.

math.AP↗

arXiv2023

Aggregation-diffusion energies on Cartan-Hadamard manifolds of unbounded curvature

We consider an aggregation-diffusion energy on Cartan-Hadamard manifolds with sectional curvatures that can grow unbounded at infinity. The energy corresponds to a macroscopic aggregation model that involves nonlocal interactions and linear diffusion. We establish necessary and sufficient conditions on the growth at infinity of the attractive interaction potential for ground states to exist. Specifically, we derive explicit conditions on the attractive potential in terms of the bounds on the sectional curvatures at infinity. To prove our results we establish a new logarithmic Hardy-Littlewood inequality for Cartan-Hadamard manifolds of unbounded curvature.

math.AP↗

arXiv2023

Ground states for aggregation-diffusion models on Cartan-Hadamard manifolds

We consider a free energy functional on Cartan-Hadamard manifolds, and investigate the existence of its global minimizers. The energy functional consists of two components: an entropy (or internal energy) and an interaction energy modelled by an attractive potential. The two components have competing effects, as they favour spreading by linear diffusion and blow-up by nonlocal attractive interactions, respectively. We find necessary and sufficient conditions for existence of ground states for manifolds with sectional curvatures bounded above and below, respectively. In particular, for general Cartan-Hadamard manifolds, superlinear growth at infinity of the attractive potential prevents the spreading. The behaviour can be relaxed for homogeneous manifolds, for which only linear growth of the potential is sufficient for this purpose.

math.AP↗

arXiv2022

Higher-order interaction model from geometric measurements

We introduce a higher simplicial generalization of the linear consensus model which shares several common features. The well-known linear consensus model is a gradient flow with a sum of squares of distances between each pair of points. Our newly suggested model is also represented as a gradient flow equipped with total $n$-dimensional volume functional consisting of $n+1$ points as a potential. In this manner, the linear consensus model coincides with the case of $n=1$ where distance is understood as the 1-dimensional volume. From a simple mathematical analysis, one can easily show that the linear consensus model (a gradient flow with 1-dimensional volume functional) collapses to one single point, which can be considered as a 0-complex. By extending this result, we show that a solution to our model converges to an $(n-1)$-dimensional affine subspace. We also perform several numerical simulations with an efficient algorithm that reduces the computational cost.

math.OC↗

arXiv2022

Long-time behaviour of interaction models on Riemannian manifolds with bounded curvature

We investigate the long-time behaviour of solutions to a nonlocal partial differential equation on smooth Riemannian manifolds of bounded sectional curvature. The equation models self-collective behaviour with intrinsic interactions that are modelled by an interaction potential. We consider attractive interaction potentials and establish sufficient conditions for a consensus state to form asymptotically. In addition, we quantify the approach to consensus, by deriving a convergence rate for the diameter of the solution's support. The analytical results are supported by numerical simulations for the equation set up on the rotation group.

math.AP↗

arXiv2022

Optimal consensus control models on the sphere

In this paper, we investigate the consensus models on the sphere with control signals, where both the first and second order systems are considered. We provide the existence of the optimal control-trajectory pair and derive the first order optimality condition taking the form of the Pontryagin Minimum Principle. Numeric simulations are also presented to show that the obtained optimal control can help to accelerate the process of reaching a consensus.

math.DS↗

arXiv2022

Asymptotic convergence of heterogeneous first-order aggregation models: from the sphere to the unitary group

We provide the detailed asymptotic behavior for first-order aggregation models of heterogeneous oscillators. Due to the dissimilarity of natural frequencies, one could expect that all relative distances converge to definite positive value and furthermore that each oscillator converges to a possibly different stationary point. In order to establish the desired results, we introduce a novel method, called dimension reduction method that can be applied to a specific situation when the degree of freedom of the natural frequency is one. In this way, we would say that although a small perturbation is allowed, convergence toward an equilibrium of the gradient flow is still guaranteed. Several first-order aggregation models are provided as concrete examples by using the dimension reduction method to study the structure of the equilibrium, and numerical simulations are conducted to support theoretical results.

math.DS↗

arXiv2022

Equilibria and energy minimizers for an interaction model on the hyperbolic space

We study an intrinsic model for collective behaviour on the hyperbolic space $\bbh^\dm$. We investigate the equilibria of the aggregation equation (or equivalently, the critical points of the associated interaction energy) for interaction potentials that include Newtonian repulsion. By using the method of moving planes, we establish the radial symmetry and the monotonicity of equilibria supported on geodesic balls of $\bbh^\dm$. We find several explicit forms of equilibria and show that one such equilibrium is a global energy minimizer. We also consider more general potentials and utilize a technique used for $\bbr^\dm$ to establish the existence of compactly supported global minimizers. Numerical simulations are presented, suggesting that some of the equilibria studied here are global attractors. The key tool in our investigations is a family of isometries of $\bbh^\dm$ that we have developed for this purpose.

math.AP↗

arXiv2022

Mean field Kuramoto models on graphs

One of a classical synchronization model is the Kuramoto model. We propose both first and second order Kuramoto dynamical models on graphs using discrete optimal transport dynamics. We analyze the synchronization behaviors for some examples of Kuramoto models on graphs. We also provide a generalized Hopf-Cole transformation for discrete optimal transport systems. Focus on the two points graph, we derive analytical formulas of the Kuramoto dynamics with various potential induced from entropy functionals. Several numerical examples for the Kuramoto model on general graphs are presented.

math.DS↗