arXiv subjects

Guillaume Wang

Publications and source records attributed to Guillaume Wang.

14 recordsLinked to original sources

arXiv2026

Effective dynamics of the Sinkhorn algorithm in the regime of low entropy regularization

The Sinkhorn algorithm is the de facto standard method for numerically solving entropy-regularized optimal transport problems over finite sets. In this work, we investigate a phenomenon arising when Sinkhorn is applied with a small regularization parameter $τ$: the evolution of the dual variables (the logarithm of the scaling factors) is approximately piecewise-linear, while the primal variables (the approximate transport plans) exhibit a saddle-to-saddle type behavior. We prove that as $τ\to 0$, the Sinkhorn iterates indeed converge to a continuous-time curve consistent with these observations, when time is rescaled as $t = τk$, and we characterize the limiting "cold Sinkhorn" dynamics explicitly. In particular, we show that it acts as a dual optimization dynamics for the unregularized problem with properties analogous to the simplex algorithm. Notably, this dynamics converges in finite time to an unregularized solution, implying a novel guarantee for the Sinkhorn algorithm itself: it achieves $\tilde{O}(τ)$ dual suboptimality in $k = O(τ^{-1})$ iterations, instead of $k = O(τ^{-2})$ as existing analyses would suggest.

math.OC↗

arXiv2026

Almost-sharp $O(k^{-1} \log k)$ convergence rate for the Sinkhorn algorithm in the asymptotically scalable case

We prove that the Sinkhorn algorithm converges at a rate of $O(k^{-1} \log k)$ in $\ell_1$-norm marginal error, in the asymptotically scalable case. This almost closes the gap between the lower bound $Ω(k^{-1})$ (Qu et al., 2025) and the previously best known upper bound $O(k^{-1/2})$ (Léger, 2021), and generalizes the analysis for the positive case by Dvurechensky et al. (2018).

math.OC↗

arXiv2026

Sharp $O(1/k)$ convergence rate for the Sinkhorn algorithm via a local analysis

We prove that the Sinkhorn algorithm converges at the rate of $O(1/k)$ in $\ell_1$-norm marginal error and in joint relative entropy, which is known to be sharp in the asymptotically scalable case. The proof is based on examining the bipartite graph associated to the entropy-regularized optimal transport problem, and treating differently the edges that are assigned a positive mass in the optimal transport plan vs. those that are not. This yields a local convergence bound with the sharp rate, which is bootstrapped into a global bound using the author's previous result in arXiv:2604.26265 where we showed an almost-sharp rate up to a logarithmic factor.

math.OC↗

arXiv2026

Local convergence of mean-field Langevin dynamics: from gradient flows to linearly monotone games

We study the local convergence of diffusive mean-field systems, including Wasserstein gradient flows, min-max dynamics, and multi-species games. We establish exponential local convergence in $χ^2$-divergence with sharp rates, under two main assumptions: (i) the stationary measures satisfy a Poincaré inequality, and (ii) the velocity field satisfies a monotonicity condition, which reduces to linear convexity of the objective in the gradient flow case. We do not assume any form of displacement convexity or displacement monotonicity. In the gradient flow case, global exponential convergence is already known under our linear convexity assumption, with an asymptotic rate governed by the log-Sobolev constant of the stationary measure. Our contribution in this setting is to identify the sharp rate near equilibrium governed instead by the Poincaré constant. This rate coincides with the one suggested by Otto calculus (i.e. by a tight positivity estimate of the Wasserstein Hessian), and refines some results of Tamura (1984), extending them beyond quadratic objectives. More importantly, our proof technique extends to certain non-gradient systems, such as linearly monotone two-player and multi-player games. In this case, we obtain explicit local exponential convergence rates in $χ^2$-divergence, thereby partially answering the open question raised by the authors at COLT 2024. While that question concerns global convergence (which remains open), even local convergence results were previously unavailable. At the heart of our analysis is the design of a Lyapunov functional that mixes the $χ^2$-divergence with weighted negative Sobolev norms of the density relative to equilibrium.

math.OC↗

arXiv2025

Metallurgy at the nanoscale: domain walls in nanoalloys

In binary alloys, domain walls play a central role not only on the phase transitions but also on their physical properties and were at the heart of the 70's metallurgy research. Whereas it can be predicted, with simple physics arguments, that such domain walls cannot exist at the nanometer scale due to the typical lengths of the statistical fluctuations of the order parameter, here we show, with both experimental and numerical approaches how orientational domain walls are formed in CuAu nanoparticles binary model systems. We demonstrate that the formation of domains in larger NPs is driven by elastic strain relaxation which is not needed in smaller NPs where surface effects dominate. Finally, we show how the multivariants NPs tend to form an isotropic material through a continuous model of elasticity.

cond-mat.mtrl-sci↗

arXiv2025

A higher-order Otto calculus approach to the Gaussian completely monotone conjecture

The Gaussian completely monotone (GCM) conjecture states that the $m$-th time-derivative of the entropy along the heat flow on $\mathbb{R}^d$ is positive for $m$ even and negative for $m$ odd. We prove the GCM conjecture for orders up to $m=5$, assuming that the initial measure is log-concave, in any dimension. Our proof differs significantly from previous approaches to the GCM conjecture: it is based on Otto calculus and on the interpretation of the heat flow as the Wasserstein gradient flow of the entropy. Crucial to our methodology is the observation that the convective derivative behaves as a flat connection over probability measures on $\mathbb{R}^d$. In particular we prove a form of the univariate Faa di Bruno's formula on the Wasserstein space (despite it being curved), and we compute the higher-order Wasserstein differentials of internal energy functionals (including the entropy), both of which are of independent interest.

cs.IT↗

arXiv2025

An Exponentially Converging Particle Method for the Mixed Nash Equilibrium of Continuous Games

We consider the problem of computing mixed Nash equilibria of two-player zero-sum games with continuous sets of pure strategies and with first-order access to the payoff function. This problem arises for example in game-theory-inspired machine learning applications, such as distributionally-robust learning. In those applications, the strategy sets are high-dimensional and thus methods based on discretisation cannot tractably return high-accuracy solutions. In this paper, we introduce and analyze a particle-based method that enjoys guaranteed local convergence for this problem. This method consists in parametrizing the mixed strategies as atomic measures and applying proximal point updates to both the atoms' weights and positions. It can be interpreted as a time-implicit discretization of the "interacting" Wasserstein-Fisher-Rao gradient flow. We prove that, under non-degeneracy assumptions, this method converges at an exponential rate to the exact mixed Nash equilibrium from any initialization satisfying a natural notion of closeness to optimality. We illustrate our results with numerical experiments and discuss applications to max-margin and distributionally-robust classification using two-layer neural networks, where our method has a natural interpretation as a simultaneous training of the network's weights and of the adversarial distribution.

math.OC↗

arXiv2025

Mean-Field Langevin Dynamics for Signed Measures via a Bilevel Approach

Mean-field Langevin dynamics (MLFD) is a class of interacting particle methods that tackle convex optimization over probability measures on a manifold, which are scalable, versatile, and enjoy computational guarantees. However, some important problems -- such as risk minimization for infinite width two-layer neural networks, or sparse deconvolution -- are originally defined over the set of signed, rather than probability, measures. In this paper, we investigate how to extend the MFLD framework to convex optimization problems over signed measures. Among two known reductions from signed to probability measures -- the lifting and the bilevel approaches -- we show that the bilevel reduction leads to stronger guarantees and faster rates (at the price of a higher per-iteration complexity). In particular, we investigate the convergence rate of MFLD applied to the bilevel reduction in the low-noise regime and obtain two results. First, this dynamics is amenable to an annealing schedule, adapted from Suzuki et al. (2023), that results in improved convergence rates to a fixed multiplicative accuracy. Second, we investigate the problem of learning a single neuron with the bilevel approach and obtain local exponential convergence rates that depend polynomially on the dimension and noise level (to compare with the exponential dependence that would result from prior analyses).

math.OC↗

arXiv2024

The entropy power conjecture implies the McKean conjecture

After reviewing the entropy power, the McKean, and the Gaussian completely monotone conjectures, we prove that the first implies the second, for each order of the time-derivative. The proof is elementary and is based on manipulating the Bell polynomials.

cs.IT↗

arXiv2023

Local Convergence of Gradient Methods for Min-Max Games: Partial Curvature Generically Suffices

We study the convergence to local Nash equilibria of gradient methods for two-player zero-sum differentiable games. It is well-known that such dynamics converge locally when $S \succ 0$ and may diverge when $S=0$, where $S\succeq 0$ is the symmetric part of the Jacobian at equilibrium that accounts for the "potential" component of the game. We show that these dynamics also converge as soon as $S$ is nonzero (partial curvature) and the eigenvectors of the antisymmetric part $A$ are in general position with respect to the kernel of $S$. We then study the convergence rates when $S \ll A$ and prove that they typically depend on the average of the eigenvalues of $S$, instead of the minimum as an analogy with minimization problems would suggest. To illustrate our results, we consider the problem of computing mixed Nash equilibria of continuous games. We show that, thanks to partial curvature, conic particle methods -- which optimize over both weights and supports of the mixed strategies -- generically converge faster than fixed-support methods. For min-max games, it is thus beneficial to add degrees of freedom "with curvature": this can be interpreted as yet another benefit of over-parameterization.

math.OC↗

arXiv2023

Atomic Scale Surface Segregation in Copper-Gold Nanoparticles

In this work, we combine electron microscopy measurements of the surface compositions in Cu-Au nanoparticles and atomistic simulations to investigate the effect of gold segregation. While this mechanism has been extensively investigated within Cu-Au in the bulk state, it was never studied at the atomic level in nanoparticles. By using energy dispersive X-ray analysis across the (100) and (111) facets of nanoparticles, we provide evidence of gold segregation in Cu$_{3}$Au and CuAu$_{3}$ nanoparticles in the 10 nm size range grown by epitaxy on a salt surface with high control of the nanoparticles morphology. To get atomic-scale insights into the segregation properties in Cu-Au nanoparticles on the whole composition range, we perform Monte Carlo calculations employing $N$-body interatomic potentials. These simulations confirm this effect by showing a complete segregation of Au in the (100) and (111) faces of a truncated octahedron for gold nominal composition of the alloy above 70\% and 60\% respectively. Furthermore, we show that there is no size effect on the segregation behaviour since we evidence the same oscillating concentration profile from surface to the nanoparticles core as in the bulk. These results can shed new lights in the interpretation of the enhanced reactivity, selectivity and stability of Cu-Au nanoparticles in various catalytic reactions.

cond-mat.mtrl-sci↗

arXiv2022

Tight bounds for minimum l1-norm interpolation of noisy data

We provide matching upper and lower bounds of order $σ^2/\log(d/n)$ for the prediction error of the minimum $\ell_1$-norm interpolator, a.k.a. basis pursuit. Our result is tight up to negligible terms when $d \gg n$, and is the first to imply asymptotic consistency of noisy minimum-norm interpolation for isotropic features and sparse ground truths. Our work complements the literature on "benign overfitting" for minimum $\ell_2$-norm interpolation, where asymptotic consistency can be achieved only when the features are effectively low-dimensional.

math.ST↗

arXiv2017

Structural Properties of Double-Walled Carbon Nanotubes driven by Mechanical Interlayer Coupling

Structural identification of double-walled carbon nanotubes (DWNT) is presented through a robust procedure based on the latest generation of transmission electron microscope, making possible a statistical analysis based on numerous nano-objects. This approach reveals that inner and outer tubes of DWNTs are not randomly oriented, suggesting the existence of a mechanical coupling between the two concentric walls. With the support of atomic scale modelisations, we attribute it to the presence of incommensurate domains whose structures depend on the diameters and helicities of both tubes, and where inner tubes try to achieve a local stacking orientation to reduce strain effects.

cond-mat.mtrl-sci↗

arXiv2015

Origin of optical losses in gallium arsenide disk whispering gallery resonators

Whispering gallery modes in GaAs disk resonators reach half a million of optical quality factor. These high Qs remain still well below the ultimate design limit set by bending losses. Here we investigate the origin of residual optical dissipation in these devices. A Transmission Electron Microscope analysis is combined with an improved Volume Current Method to precisely quantify optical scattering losses by roughness and waviness of the structures, and gauge their importance relative to intrinsic material and radiation losses. The analysis also provides a qualitative description of the surface reconstruction layer, whose optical absorption is then revealed by comparing spectroscopy experiments in air and in different liquids. Other linear and nonlinear optical loss channels in the disks are evaluated likewise. Routes are given to further improve the performances of these miniature GaAs cavities.

physics.optics↗