arXiv subjects

Vladimir Jaćimović

Publications and source records attributed to Vladimir Jaćimović.

10 recordsLinked to original sources

arXiv2026

Busemann energy-based attention for emotion analysis in Poincaré discs

We present EmBolic - a novel fully hyperbolic deep learning architecture for fine-grained emotion analysis from textual messages. The underlying idea is that hyperbolic geometry efficiently captures hierarchies between both words and emotions. In our context, these hierarchical relationships arise from semantic ambiguities. EmBolic aims to infer the curvature on the continuous space of emotions, rather than treating them as a categorical set without any metric structure. In the heart of our architecture is the attention mechanism in the hyperbolic disc. The model is trained to generate queries (points in the hyperbolic disc) from textual messages, while keys (points at the boundary) emerge automatically from the generated queries. Predictions are based on the Busemann energy between queries and keys, evaluating how well a certain textual message aligns with the class directions representing emotions. Our experiments demonstrate strong generalization properties and reasonably good prediction accuracy even for small dimensions of the representation space. Overall, this study supports our claim that affective computing is one of the application domains where hyperbolic representations are particularly advantageous.

cs.LG↗

arXiv2025

Clustering in hyperbolic balls

The idea of representations of the data in negatively curved manifolds recently attracted a lot of attention and gave a rise to the new research direction named {\it hyperbolic machine learning} (ML). In order to unveil the full potential of this new paradigm, efficient techniques for data analysis and statistical modeling in hyperbolic spaces are necessary. In the present paper rigorous mathematical framework for clustering in hyperbolic spaces is established. First, we introduce the $k$-means clustering in hyperbolic balls, based on the novel definition of barycenter. Second, we present the expectation-maximization (EM) algorithm for learning mixtures of novel probability distributions in hyperbolic balls. In such a way we lay the foundation of unsupervised learning in hyperbolic spaces.

cs.LG↗

arXiv2025

A group-theoretic framework for machine learning in hyperbolic spaces

Embedding the data in hyperbolic spaces can preserve complex relationships in very few dimensions, thus enabling compact models and improving efficiency of machine learning (ML) algorithms. The underlying idea is that hyperbolic representations can prevent the loss of important structural information for certain ubiquitous types of data. However, further advances in hyperbolic ML require more principled mathematical approaches and adequate geometric methods. The present study aims at enhancing mathematical foundations of hyperbolic ML by combining group-theoretic and conformal-geometric arguments with optimization and statistical techniques. Precisely, we introduce the notion of the mean (barycenter) and the novel family of probability distributions on hyperbolic balls. We further propose efficient optimization algorithms for computation of the barycenter and for maximum likelihood estimation. One can build upon basic concepts presented here in order to design more demanding algorithms and implement hyperbolic deep learning pipelines.

cs.LG↗

arXiv2024

Reinforcement Learning in Hyperbolic Spaces: Models and Experiments

We examine five setups where an agent (or two agents) seeks to explore unknown environment without any prior information. Although seemingly very different, all of them can be formalized as Reinforcement Learning (RL) problems in hyperbolic spaces. More precisely, it is natural to endow the action spaces with the hyperbolic metric. We introduce statistical and dynamical models necessary for addressing problems of this kind and implement algorithms based on this framework. Throughout the paper we view RL through the lens of the black-box optimization.

cs.LG↗

arXiv2024

Phase holonomy underlies puzzling temporal patterns in Kuramoto models with two sub-populations

We present a geometric investigation of curious dynamical behaviors previously reported in Kuramoto models with two sub-populations. Our study demonstrates that chimeras and traveling waves in such models are associated with the birth of geometric phase. Although manifestations of geometric phase are frequent in various fields of Physics, this is the first time (to our best knowledge) that such a phenomenon is exposed in ensembles of Kuramoto oscillators or, more broadly, in complex systems.

nlin.AO↗

arXiv2022

Natural Gradient Ascent in Evolutionary Games

We study evolutionary games with a continuous trait space in which replicator dynamics are restricted to the manifold of multidimensional Gaussian distributions. We demonstrate that the replicator equations are natural gradient flow for maximization of the mean fitness. Our findings extend previous results on information-geometric aspects of evolutionary games with a finite strategy set. Throughout the paper we exploit the information-geometric approach and the relation between evolutionary dynamics and Natural Evolution Strategies, the concept that has been developed within the framework of black-box optimization. This relation sheds a new light on the replicator dynamics as a compromise between maximization of the mean fitness and preservation of diversity in the population.

cs.GT↗

arXiv2021

A new dynamical model for solving rotation averaging problem

The paper analyzes the rotation averaging problem as a minimization problem for a potential function of the corresponding gradient system. This dynamical system is one generalization of the famous Kuramoto model on special orthogonal group SO(3), which is known as the non-Abelian Kuramoto model. We have proposed a novel method for finding weighted and unweighted rotation average. In order to verify the correctness of our algorithms, we have compared the simulation results with geometric and projected average using real and random data sets. In particular, we have discovered that our method gives approximately the same results as geometric average.

cs.CV↗

arXiv2021

On synchronization in Kuramoto models on spheres

We analyze two classes of Kuramoto models on spheres that have been introduced in previous studies. Our analysis is restricted to ensembles of identical oscillators with the global coupling. In such a setup, with an additional assumption that the initial distribution of oscillators is uniform on the sphere, one can derive equations for order parameters in closed form. The rate of synchronization in a real Kuramoto model depends on the dimension of the sphere. Specifically, synchronization is faster on higher-dimensional spheres. On the other side, real order parameter in complex Kuramoto models always satisfies the same ODE, regardless of the dimension. The derivation of equations for real order parameters in Kuramoto models on spheres is based on recently unveiled connections of these models with geometries of unit balls. Simulations of the system with several hundreds of oscillators yield perfect fits with the theoretical predictions, that are obtained by solving equations for the order parameter.

nlin.AO↗

arXiv2021

Möbius group actions in the solvable chimera model

We study actions of Möbius group on two sub-populations in the solvable chimera model proposed by Abrams et al. Dynamics of global variables are given by two coupled Watanabe-Strogatz systems, one for each sub-population. At the first glance, asymptotic dynamics in the model seem to be very simple. For instance, in the stable chimera state distributions of oscillators perform a simple rotations after a certain (sufficiently large) moment. However, a closer look unveils that dynamics are subtler that what can be observed from evolution of densities of oscillators' phases. In order to gain the full picture, one needs to investigate dynamics on the transformation group that acts on these densities. Such an approach emphasizes impact of the "hidden" variable that is not visible on macroscopic level.

nlin.AO↗

arXiv2020

Collective dynamics of phase-repulsive oscillators solves graph coloring problem

We show how to couple phase-oscillators on a graph so that collective dynamics "searches" for the coloring of that graph as it relaxes toward the dynamical equilibrium. This translates a combinatorial optimization problem (graph coloring) into a functional optimization problem (finding and evaluating the global minimum of dynamical non-equilibrium potential, done by the natural system's evolution). Using a sample of graphs, we show that our method can serve as a viable alternative to the traditional combinatorial algorithms. Moreover, we show that, with the same computational cost, our method efficiently solves the harder problem of improper coloring of weighed graphs.

nlin.CD↗