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Pablo Groisman

Publications and source records attributed to Pablo Groisman.

At least 19 recordsLinked to original sources
arXiv2026

The tentacles landscape: geometric properties of high-dimensional basins of attraction

Basins of attraction in multistable high dimensional dynamical systems are expected to have universal features but very little has been proved rigorously. We consider phase oscillators coupled according to a cycle graph, $\dot\theta_i = f(\theta_{i+1}-\theta_i) + f(\theta_{i-1}-\theta_i)$, with coupling $f$ that is $C^1$, odd, $2\pi$-periodic, and strictly increasing on $(-\pi,\pi)$. We prove the full ``octopus'' picture of the basins of attraction observed numerically by Zhang and Strogatz [Phys.\ Rev.\ Lett.\ 127 (2021) 194101] and, beyond this model, across a wide family of high-dimensional multistable systems. In our case, we have a family of stable equilibria that can be indexed by their winding number $q \in \mathbb Z\cap(-n/2,n/2)$. Basin volumes obey a Gaussian law $\mu(\mathcal K_q)=\sqrt{6/(\pi n)}\,e^{-6q^2/n}(1+o(1))$ in the winding number. The distance from a uniform sample to its attractor, when divided by $\sqrt n$, concentrates at $\sqrt{\pi^2/3}\approx 1.814$. Along almost every straight line through any twisted state, the ray enters every other basin infinitely many times, with frequencies given by the basin volumes. The inscribed ball at a twisted state has radius $({\pi}/{\sqrt2})(1-{2|q|}/{n})$ for every $n$, while as $n\to\infty$, a typical ray travels distance $(\pi/2)\sqrt{n/\log n}$ before first leaving the basin: the head of the octopus is sharply anisotropic.

math-ph↗
arXiv2026

$K-$means with learned metrics

We study the Fr\'echet $k-$means of a metric measure space when both the measure and the distance are unknown and have to be estimated. We prove a general result that states that the $k-$means are continuous with respect to the measured Gromov-Hausdorff topology. In this situation, we also prove a stability result for the Voronoi clusters they determine. We do not assume uniqueness of the set of $k-$means, but when it is unique, the results are stronger. This framework provides a unified approach to proving consistency for a wide range of metric learning procedures. As concrete applications, we obtain new consistency results for several important estimators that were previously unestablished, even when $k=1$. These include $k-$means based on: (i) Isomap and Fermat geodesic distances on manifolds, (ii) difussion distances, (iii) Wasserstein distances computed with respect to learned ground metrics. Finally, we consider applications beyond the statistical inference paradigm like (iv) first passage percolation and (v) discrete approximations of length spaces.

math.ST↗
arXiv2026

A universality property for large deviations of RWRE close to the axis

We establish a general version of the strong KPZ universality conjecture near the axis for random walks in a random environment (RWRE) on $\mathbb{Z}^2$. For an i.i.d. elliptic random environment, we consider the quenched large deviations probabilities for trajectories starting at the origin and arriving at time $n+[n^a]$ to the position $(n,[n^a])$ and show that, if the logarithm of the right-jump probability has a finite moment of order $p>2$, then for $a < \frac{3}{7}(1-\frac{2}{p})$ the fluctuations of these propabilities are asymptotically governed by the GUE Tracy-Widom distribution. Our results are based on a comparison between RWRE and a last passage percolation model, whose asymptotic fluctuations near the axis were previously established independently by Bodineau-Martin and Baik-Suidan. Furthermore, we obtain also the full convergence to the directed landscape in this regime based on the extension of the aforementioned results to this setting by McKeown and Zhang.

math.PR↗
arXiv2025

Invariant Feature Extraction Through Conditional Independence and the Optimal Transport Barycenter Problem: the Gaussian case

A methodology is developed to extract $d$ invariant features $W=f(X)$ that predict a response variable $Y$ without being confounded by variables $Z$ that may influence both $X$ and $Y$. The methodology's main ingredient is the penalization of any statistical dependence between $W$ and $Z$ conditioned on $Y$, replaced by the more readily implementable plain independence between $W$ and the random variable $Z_Y = T(Z,Y)$ that solves the [Monge] Optimal Transport Barycenter Problem for $Z\mid Y$. In the Gaussian case considered in this article, the two statements are equivalent. When the true confounders $Z$ are unknown, other measurable contextual variables $S$ can be used as surrogates, a replacement that involves no relaxation in the Gaussian case if the covariance matrix $\Sigma_{ZS}$ has full range. The resulting linear feature extractor adopts a closed form in terms of the first $d$ eigenvectors of a known matrix. The procedure extends with little change to more general, non-Gaussian / non-linear cases.

math.ST↗
arXiv2025

The size of the sync basin resolved

Sparsely coupled Kuramoto oscillators offer a fertile playground for exploring high-dimensional basins of attraction due to their simple yet multistable dynamics. For $n$ identical Kuramoto oscillators on cycle graphs, it is well known that the only attractors are twisted states, whose phases wind around the circle with a constant gap between neighboring oscillators ($\theta_j = 2\pi q j/n$). It was conjectured in 2006 that basin sizes of these twisted states scale as $e^{-kq^2}$ to the winding number $q$. Here, we provide new numerical and analytical evidence supporting the conjecture and uncover the dynamical mechanism behind the Gaussian scaling. The key idea is that, when starting with a random initial condition, the winding number of the solution stabilizes rapidly at $t \propto \log n$, before long-range correlation can develop among oscillators. This timescale separation allows us to calculate the winding number as a sum of weakly-dependent random variables, leading to a Central Limit Theorem derivation of the basin scaling.

math-ph↗
arXiv2025

GUE Fluctuations Near the Axis in One-Sided Ballistic Deposition

We introduce a variation of the classic ballistic deposition model in which vertically falling blocks can only stick to the top or the upper right corner of growing columns. We establish that fluctuations of the height function at points near the $t$-axis are given by the GUE ensemble and its corresponding Tracy-Widom limiting distribution. The proof is based on a graphical construction of the process in terms of a directed Last Passage Percolation model. Using this graphical construction, we define the notion of geodesics for the height function and show that the wandering exponent governing the transversal fluctuations of these geodesics is $2/3$.

math.PR↗
arXiv2025

Phase Synchronization in Random Geometric Graphs on the 2D Sphere

The Kuramoto model is a classical nonlinear ODE system designed to study synchronization phenomena. Each equation represents the phase of an oscillator and the coupling between them is determined by a graph. There is an increasing interest in understanding the relation between the graph topology and the spontaneous synchronization of the oscillators. Abdalla, Bandeira and Invernizzi considered random geometric graphs on the $d$-dimensional sphere and proved that the system synchronizes with high probability as long as the mean number of neighbors and the dimension $d$ go to infinity. They posed the question about the behavior when $d$ is small. In this paper, we prove that synchronization holds for random geometric graphs on the two-dimensional sphere, with high probability as the number of nodes goes to infinity, as long as the initial conditions converge to a smooth function. We conjecture a similar behavior for more general simply-connected closed Riemannian manifolds but we expect global synchronization to fail if the manifold is not simply-connected, as was shown in [11] and suggested in [9].

math.PR↗
arXiv2025

On the uniqueness of quasi-stationary distributions for population models with spatial structure

Subcritical population processes are attracted to extinction and do not have non-trivial stationary distributions, which prompts the study of quasi-stationary distributions (QSDs) instead. In contrast to what generally happens for stationary distributions, QSDs may not be unique, even under irreducibility conditions. The general conditions for uniqueness of QSDs are not always easy to check. For the branching process, besides the quasi-limiting distribution there are many other QSDs. In this paper, we investigate whether adding little extra information to the continuous-time branching process is enough to obtain uniqueness. We consider the branching process with genealogy and branching random walks, and show that they have a unique QSD.

math.PR↗
arXiv2024

Siamese networks for Poincar\'e embeddings and the reconstruction of evolutionary trees

We present a method for reconstructing evolutionary trees from high-dimensional data, with a specific application to bird song spectrograms. We address the challenge of inferring phylogenetic relationships from phenotypic traits, like vocalizations, without predefined acoustic properties. Our approach combines two main components: Poincar\'e embeddings for dimensionality reduction and distance computation, and the neighbor joining algorithm for tree reconstruction. Unlike previous work, we employ Siamese networks to learn embeddings from only leaf node samples of the latent tree. We demonstrate our method's effectiveness on both synthetic data and spectrograms from six species of finches.

q-bio.PE↗
arXiv2024

Reconstructing Attractors with Autoencoders

We show how to train an autoencoder to reconstruct an attractor from recorded footage, preserving the topology of the underlying phase space. This is explicitly demonstrated for the classic finite-amplitude Lorenz atmospheric convection problem.

nlin.AO↗
arXiv2024

Scaling Limit of the Kuramoto Model on Random Geometric Graphs

We consider the Kuramoto model on a graph with nodes given by $n$ i.i.d. points uniformly distributed on the $d$ dimensional torus. Two nodes are declared neighbors if they are at distance less than $\epsilon$. We prove a scaling limit for this model in compact time intervals as $n\to\infty$ and $\epsilon \to 0$ such that $\epsilon^{d+2}n/\log n \to \infty$. The limiting object is given by the heat equation. On the one hand this shows that the nonlinearity given by the sine function disappears under this scaling and on the other hand, provides evidence that stable equilibria of the Kuramoto model on these graphs are, as $n\to\infty$, in correspondence with those of the heat equation, which are explicit and given by twisted states. In view of this, we conjecture the existence of twisted stable equilibria with high probability as $n\to \infty$.

math.PR↗
arXiv2024

The energy landscape of the Kuramoto model in random geometric graphs in a circle

We study the energy function of the Kuramoto model in random geometric graphs defined in the unit circle as the number of nodes diverges. We prove the existence of at least one local minimum for each winding number $q \in \mathbb{Z}$ with high probability. Hence providing a large family of graphs that support patterns that are generic. These states are in correspondence with the explicit twisted states found in WSG and other highly symmetric networks, but in our situation there is no explicit formula due to the lack of symmetry. The method of proof is simple and robust. It allows other types of graphs like $k-$nn graphs or the boolean model and holds also for graphs defined in any simple closed curve or even a small neighborhood of the curve and for weighted graphs. It seems plausible that the method can be extended also to higher dimensions, but a more careful analysis is required.

math.PR↗
arXiv2023

Choosing the parameter of the Fermat distance: navigating geometry and noise

The Fermat distance has been recently established as a useful tool for machine learning tasks when a natural distance is not directly available to the practitioner or to improve the results given by Euclidean distances by exploding the geometrical and statistical properties of the dataset. This distance depends on a parameter $\alpha$ that greatly impacts the performance of subsequent tasks. Ideally, the value of $\alpha$ should be large enough to navigate the geometric intricacies inherent to the problem. At the same, it should remain restrained enough to sidestep any deleterious ramifications stemming from noise during the process of distance estimation. We study both theoretically and through simulations how to select this parameter.

stat.ML↗
arXiv2022

The Kuramoto model on dynamic random graphs

We propose a Kuramoto model of coupled oscillators on a time-varying graph, whose dynamics is dictated by a Markov process in the space of graphs. The simplest representative is considering a base graph and then the subgraph determined by $N$ independent random walks on the underlying graph. We prove a synchronization result for solutions starting from a phase-cohesive set independent of the speed of the random walkers, an averaging principle and a global synchronization result with high probability for sufficiently fast processes. We also consider Kuramoto oscillators in a dynamical version of the Random Conductance Model.

math.PR↗
arXiv2020

Intrinsic persistent homology via density-based metric learning

We address the problem of estimating topological features from data in high dimensional Euclidean spaces under the manifold assumption. Our approach is based on the computation of persistent homology of the space of data points endowed with a sample metric known as Fermat distance. We prove that such metric space converges almost surely to the manifold itself endowed with an intrinsic metric that accounts for both the geometry of the manifold and the density that produces the sample. This fact implies the convergence of the associated persistence diagrams. The use of this intrinsic distance when computing persistent homology presents advantageous properties such as robustness to the presence of outliers in the input data and less sensitiveness to the particular embedding of the underlying manifold in the ambient space. We use these ideas to propose and implement a method for pattern recognition and anomaly detection in time series, which is evaluated in applications to real data.

stat.ML↗
arXiv2019

The quasi-stationary distribution of the subcritical contact process

We show that the quasi-stationary distribution of the subcritical contact process on $\mathbb{Z}^d$ is unique. This is in contrast with other processes which also do not come down from infinity, like stable queues and Galton-Watson, and it seems to be the first such example.

math.PR↗
arXiv2019

F-KPP Scaling limit and selection principle for a Brunet-Derrida type particle system

We study a particle system with the following diffusion-branching-selection mechanism. Particles perform independent one dimensional Brownian motions and on top of that, at a constant rate, a pair of particles is chosen uniformly at random and both particles adopt the position of the rightmost one among them. We show that the cumulative distribution function of the empirical measure converges to a solution of the Fisher-Kolmogorov-Petrovskii-Piskunov (F-KPP) equation and use this fact to prove that the system selects the minimal macroscopic speed as the number of particles goes to infinity.

math.PR↗