arXiv subjects

Vladimir Kostic

Publications and source records attributed to Vladimir Kostic.

11 recordsLinked to original sources

arXiv2026

kooplearn: A Scikit-Learn Compatible Library of Algorithms for Evolution Operator Learning

kooplearn is a machine-learning library that implements linear, kernel, and deep-learning estimators of dynamical operators and their spectral decompositions. kooplearn can model both discrete-time evolution operators (Koopman/Transfer) and continuous-time infinitesimal generators. By learning these operators, users can analyze dynamical systems via spectral methods, derive data-driven reduced-order models, and forecast future states and observables. kooplearn's interface is compliant with the scikit-learn API, facilitating its integration into existing machine learning and data science workflows. Additionally, kooplearn includes curated benchmark datasets to support experimentation, reproducibility, and the fair comparison of learning algorithms. The software is available at https://github.com/Machine-Learning-Dynamical-Systems/kooplearn.

cs.LG↗

arXiv2025

The seeds of the future are in the present: A blind exploration of metastable states

In this work, we present a novel type of molecular dynamics simulation that aims at discovering, in a blind way, new metastable states. Using only data coming from an initial unbiased simulation, and with the help of an appropriately defined loss function, we compute a bias that favors sampling yet unexplored configurational space regions, encouraging the system to leave the initial basin. In our work, we take advantage of what is normally thought to be a defect, namely the difficulty of neural networks to generalize. Contrary to most other enhanced sampling methods, which need previous knowledge of the reactive process, we are able to discover in a blind way new metastable states, overcoming otherwise insuperable kinetic bottlenecks. We illustrate the workings of the method with a number of instructive examples.

physics.chem-ph↗

arXiv2025

Morphological Symmetries in Robotics

We present a comprehensive framework for studying and leveraging morphological symmetries in robotic systems. These are intrinsic properties of the robot's morphology, frequently observed in animal biology and robotics, which stem from the replication of kinematic structures and the symmetrical distribution of mass. We illustrate how these symmetries extend to the robot's state space and both proprioceptive and exteroceptive sensor measurements, resulting in the equivariance of the robot's equations of motion and optimal control policies. Thus, we recognize morphological symmetries as a relevant and previously unexplored physics-informed geometric prior, with significant implications for both data-driven and analytical methods used in modeling, control, estimation and design in robotics. For data-driven methods, we demonstrate that morphological symmetries can enhance the sample efficiency and generalization of machine learning models through data augmentation, or by applying equivariant/invariant constraints on the model's architecture. In the context of analytical methods, we employ abstract harmonic analysis to decompose the robot's dynamics into a superposition of lower-dimensional, independent dynamics. We substantiate our claims with both synthetic and real-world experiments conducted on bipedal and quadrupedal robots. Lastly, we introduce the repository MorphoSymm to facilitate the practical use of the theory and applications outlined in this work.

cs.RO↗

arXiv2024

From Biased to Unbiased Dynamics: An Infinitesimal Generator Approach

We investigate learning the eigenfunctions of evolution operators for time-reversal invariant stochastic processes, a prime example being the Langevin equation used in molecular dynamics. Many physical or chemical processes described by this equation involve transitions between metastable states separated by high potential barriers that can hardly be crossed during a simulation. To overcome this bottleneck, data are collected via biased simulations that explore the state space more rapidly. We propose a framework for learning from biased simulations rooted in the infinitesimal generator of the process and the associated resolvent operator. We contrast our approach to more common ones based on the transfer operator, showing that it can provably learn the spectral properties of the unbiased system from biased data. In experiments, we highlight the advantages of our method over transfer operator approaches and recent developments based on generator learning, demonstrating its effectiveness in estimating eigenfunctions and eigenvalues. Importantly, we show that even with datasets containing only a few relevant transitions due to sub-optimal biasing, our approach recovers relevant information about the transition mechanism.

cs.LG↗

arXiv2024

Dynamics Harmonic Analysis of Robotic Systems: Application in Data-Driven Koopman Modelling

We introduce the use of harmonic analysis to decompose the state space of symmetric robotic systems into orthogonal isotypic subspaces. These are lower-dimensional spaces that capture distinct, symmetric, and synergistic motions. For linear dynamics, we characterize how this decomposition leads to a subdivision of the dynamics into independent linear systems on each subspace, a property we term dynamics harmonic analysis (DHA). To exploit this property, we use Koopman operator theory to propose an equivariant deep-learning architecture that leverages the properties of DHA to learn a global linear model of the system dynamics. Our architecture, validated on synthetic systems and the dynamics of locomotion of a quadrupedal robot, exhibits enhanced generalization, sample efficiency, and interpretability, with fewer trainable parameters and computational costs.

cs.RO↗

arXiv2023

A randomized algorithm to solve reduced rank operator regression

We present and analyze an algorithm designed for addressing vector-valued regression problems involving possibly infinite-dimensional input and output spaces. The algorithm is a randomized adaptation of reduced rank regression, a technique to optimally learn a low-rank vector-valued function (i.e. an operator) between sampled data via regularized empirical risk minimization with rank constraints. We propose Gaussian sketching techniques both for the primal and dual optimization objectives, yielding Randomized Reduced Rank Regression (R4) estimators that are efficient and accurate. For each of our R4 algorithms we prove that the resulting regularized empirical risk is, in expectation w.r.t. randomness of a sketch, arbitrarily close to the optimal value when hyper-parameteres are properly tuned. Numerical expreriments illustrate the tightness of our bounds and show advantages in two distinct scenarios: (i) solving a vector-valued regression problem using synthetic and large-scale neuroscience datasets, and (ii) regressing the Koopman operator of a nonlinear stochastic dynamical system.

cs.LG↗

arXiv2023

Consistent Long-Term Forecasting of Ergodic Dynamical Systems

We study the evolution of distributions under the action of an ergodic dynamical system, which may be stochastic in nature. By employing tools from Koopman and transfer operator theory one can evolve any initial distribution of the state forward in time, and we investigate how estimators of these operators perform on long-term forecasting. Motivated by the observation that standard estimators may fail at this task, we introduce a learning paradigm that neatly combines classical techniques of eigenvalue deflation from operator theory and feature centering from statistics. This paradigm applies to any operator estimator based on empirical risk minimization, making them satisfy learning bounds which hold uniformly on the entire trajectory of future distributions, and abide to the conservation of mass for each of the forecasted distributions. Numerical experiments illustrates the advantages of our approach in practice.

stat.ML↗

arXiv2023

Sharp Spectral Rates for Koopman Operator Learning

Nonlinear dynamical systems can be handily described by the associated Koopman operator, whose action evolves every observable of the system forward in time. Learning the Koopman operator and its spectral decomposition from data is enabled by a number of algorithms. In this work we present for the first time non-asymptotic learning bounds for the Koopman eigenvalues and eigenfunctions. We focus on time-reversal-invariant stochastic dynamical systems, including the important example of Langevin dynamics. We analyze two popular estimators: Extended Dynamic Mode Decomposition (EDMD) and Reduced Rank Regression (RRR). Our results critically hinge on novel {minimax} estimation bounds for the operator norm error, that may be of independent interest. Our spectral learning bounds are driven by the simultaneous control of the operator norm error and a novel metric distortion functional of the estimated eigenfunctions. The bounds indicates that both EDMD and RRR have similar variance, but EDMD suffers from a larger bias which might be detrimental to its learning rate. Our results shed new light on the emergence of spurious eigenvalues, an issue which is well known empirically. Numerical experiments illustrate the implications of the bounds in practice.

cs.LG↗

arXiv2022

Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces

We study a class of dynamical systems modelled as Markov chains that admit an invariant distribution via the corresponding transfer, or Koopman, operator. While data-driven algorithms to reconstruct such operators are well known, their relationship with statistical learning is largely unexplored. We formalize a framework to learn the Koopman operator from finite data trajectories of the dynamical system. We consider the restriction of this operator to a reproducing kernel Hilbert space and introduce a notion of risk, from which different estimators naturally arise. We link the risk with the estimation of the spectral decomposition of the Koopman operator. These observations motivate a reduced-rank operator regression (RRR) estimator. We derive learning bounds for the proposed estimator, holding both in i.i.d. and non i.i.d. settings, the latter in terms of mixing coefficients. Our results suggest RRR might be beneficial over other widely used estimators as confirmed in numerical experiments both for forecasting and mode decomposition.

cs.LG↗

arXiv2021

Convergence of Batch Greenkhorn for Regularized Multimarginal Optimal Transport

In this work we propose a batch version of the Greenkhorn algorithm for multimarginal regularized optimal transport problems. Our framework is general enough to cover, as particular cases, some existing algorithms like Sinkhorn and Greenkhorn algorithm for the bi-marginal setting, and (greedy) MultiSinkhorn for multimarginal optimal transport. We provide a complete convergence analysis, which is based on the properties of the iterative Bregman projections (IBP) method with greedy control. Global linear rate of convergence and explicit bound on the iteration complexity are obtained. When specialized to above mentioned algorithms, our results give new insights and/or improve existing ones.

stat.ML↗

arXiv2021

The method of Bregman projections in deterministic and stochastic convex feasibility problems

In this work we study the method of Bregman projections for deterministic and stochastic convex feasibility problems with three types of control sequences for the selection of sets during the algorithmic procedure: greedy, random, and adaptive random. We analyze in depth the case of affine feasibility problems showing that the iterates generated by the proposed methods converge Q-linearly and providing also explicit global and local rates of convergence. This work generalizes from one hand recent developments in randomized methods for the solution of linear systems based on orthogonal projection methods. On the other hand, our results yield global and local Q-linear rates of convergence for the Sinkhorn and Greenhorn algorithms in discrete entropic-regularized optimal transport, for the first time, even in the multimarginal setting.

math.OC↗