arXiv subjects

Maria Fonoberova

Publications and source records attributed to Maria Fonoberova.

13 recordsLinked to original sources

arXiv2024

Koopman-driven grip force prediction through EMG sensing

Loss of hand function due to conditions like stroke or multiple sclerosis significantly impacts daily activities. Robotic rehabilitation provides tools to restore hand function, while novel methods based on surface electromyography (sEMG) enable the adaptation of the device's force output according to the user's condition, thereby improving rehabilitation outcomes. This study aims to achieve accurate force estimations during medium wrap grasps using a single sEMG sensor pair, thereby addressing the challenge of escalating sensor requirements for precise predictions. We conducted sEMG measurements on 13 subjects at two forearm positions, validating results with a hand dynamometer. We established flexible signal-processing steps, yielding high peak cross-correlations between the processed sEMG signal (representing meaningful muscle activity) and grip force. Influential parameters were subsequently identified through sensitivity analysis. Leveraging a novel data-driven Koopman operator theory-based approach and problem-specific data lifting techniques, we devised a methodology for the estimation and short-term prediction of grip force from processed sEMG signals. A weighted mean absolute percentage error (wMAPE) of approx. 5.5% was achieved for the estimated grip force, whereas predictions with a 0.5-second prediction horizon resulted in a wMAPE of approx. 17.9%. The methodology proved robust regarding precise electrode positioning, as the effect of sensing position on error metrics was non-significant. The algorithm executes exceptionally fast, processing, estimating, and predicting a 0.5-second sEMG signal batch in just approx. 30 ms, facilitating real-time implementation.

cs.RO↗

arXiv2023

Koopman Learning with Episodic Memory

Koopman operator theory has found significant success in learning models of complex, real-world dynamical systems, enabling prediction and control. The greater interpretability and lower computational costs of these models, compared to traditional machine learning methodologies, make Koopman learning an especially appealing approach. Despite this, little work has been performed on endowing Koopman learning with the ability to leverage its own failures. To address this, we equip Koopman methods -- developed for predicting non-autonomous time-series -- with an episodic memory mechanism, enabling global recall of (or attention to) periods in time where similar dynamics previously occurred. We find that a basic implementation of Koopman learning with episodic memory leads to significant improvements in prediction on synthetic and real-world data. Our framework has considerable potential for expansion, allowing for future advances, and opens exciting new directions for Koopman learning.

math.DS↗

arXiv2023

A Koopman Operator-Based Prediction Algorithm and its Application to COVID-19 Pandemic

The problem of prediction of behavior of dynamical systems has undergone a paradigm shift in the second half of the 20th century with the discovery of the possibility of chaotic dynamics in simple, physical, dynamical systems for which the laws of evolution do not change in time. The essence of the paradigm is the long term exponential divergence of trajectories. However, that paradigm does not account for another type of unpredictability: the ``Black Swan" event. It also does not account for the fact that short-term prediction is often possible even in systems with exponential divergence. In our framework, the Black Swan type dynamics occurs when an underlying dynamical system suddenly shifts between dynamics of different types. A learning and prediction system should be capable of recognizing the shift in behavior, exemplified by ``confidence loss". In this paradigm, the predictive power is assessed dynamically and confidence level is used to switch between long term prediction and local-in-time prediction. Here we explore the problem of prediction in systems that exhibit such behavior. The mathematical underpinnings of our theory and algorithms are based on an operator-theoretic approach in which the dynamics of the system are embedded into an infinite-dimensional space. We apply the algorithm to a number of case studies including prediction of influenza cases and the COVID-19 pandemic. The results show that the predictive algorithm is robust to perturbations of the available data, induced for example by delays in reporting or sudden increase in cases due to increase in testing capability. This is achieved in an entirely data-driven fashion, with no underlying mathematical model of the disease.

math.DS↗

arXiv2023

Identifying Equivalent Training Dynamics

Study of the nonlinear evolution deep neural network (DNN) parameters undergo during training has uncovered regimes of distinct dynamical behavior. While a detailed understanding of these phenomena has the potential to advance improvements in training efficiency and robustness, the lack of methods for identifying when DNN models have equivalent dynamics limits the insight that can be gained from prior work. Topological conjugacy, a notion from dynamical systems theory, provides a precise definition of dynamical equivalence, offering a possible route to address this need. However, topological conjugacies have historically been challenging to compute. By leveraging advances in Koopman operator theory, we develop a framework for identifying conjugate and non-conjugate training dynamics. To validate our approach, we demonstrate that comparing Koopman eigenvalues can correctly identify a known equivalence between online mirror descent and online gradient descent. We then utilize our approach to: (a) identify non-conjugate training dynamics between shallow and wide fully connected neural networks; (b) characterize the early phase of training dynamics in convolutional neural networks; (c) uncover non-conjugate training dynamics in Transformers that do and do not undergo grokking. Our results, across a range of DNN architectures, illustrate the flexibility of our framework and highlight its potential for shedding new light on training dynamics.

cs.LG↗

arXiv2022

Koopman Reduced Order Modeling with Confidence Bounds

This paper introduces a reduced order modeling technique based on Koopman operator theory that gives confidence bounds on the model's predictions. It is based on a data-driven spectral decomposition of the Koopman operator. The reduced order model is constructed using a finite number of Koopman eigenvalues and modes, while the rest of spectrum is treated as a noise process. This noise process is used to extract the confidence bounds. Additionally, we propose a heuristic algorithm to choose the number of deterministic modes to keep in the model.

math.DS↗

arXiv2022

Algorithmic (Semi-)Conjugacy via Koopman Operator Theory

Iterative algorithms are of utmost importance in decision and control. With an ever growing number of algorithms being developed, distributed, and proprietarized, there is a similarly growing need for methods that can provide classification and comparison. By viewing iterative algorithms as discrete-time dynamical systems, we leverage Koopman operator theory to identify (semi-)conjugacies between algorithms using their spectral properties. This provides a general framework with which to classify and compare algorithms.

cs.DS↗

arXiv2022

An Operator Theoretic View on Pruning Deep Neural Networks

The discovery of sparse subnetworks that are able to perform as well as full models has found broad applied and theoretical interest. While many pruning methods have been developed to this end, the naïve approach of removing parameters based on their magnitude has been found to be as robust as more complex, state-of-the-art algorithms. The lack of theory behind magnitude pruning's success, especially pre-convergence, and its relation to other pruning methods, such as gradient based pruning, are outstanding open questions in the field that are in need of being addressed. We make use of recent advances in dynamical systems theory, namely Koopman operator theory, to define a new class of theoretically motivated pruning algorithms. We show that these algorithms can be equivalent to magnitude and gradient based pruning, unifying these seemingly disparate methods, and find that they can be used to shed light on magnitude pruning's performance during the early part of training.

cs.LG↗

arXiv2020

Predicting the Critical Number of Layers for Hierarchical Support Vector Regression

Hierarchical support vector regression (HSVR) models a function from data as a linear combination of SVR models at a range of scales, starting at a coarse scale and moving to finer scales as the hierarchy continues. In the original formulation of HSVR, there were no rules for choosing the depth of the model. In this paper, we observe in a number of models a phase transition in the training error -- the error remains relatively constant as layers are added, until a critical scale is passed, at which point the training error drops close to zero and remains nearly constant for added layers. We introduce a method to predict this critical scale a priori with the prediction based on the support of either a Fourier transform of the data or the Dynamic Mode Decomposition (DMD) spectrum. This allows us to determine the required number of layers prior to training any models.

cs.LG↗

arXiv2020

Applications of Koopman Mode Analysis to Neural Networks

We consider the training process of a neural network as a dynamical system acting on the high-dimensional weight space. Each epoch is an application of the map induced by the optimization algorithm and the loss function. Using this induced map, we can apply observables on the weight space and measure their evolution. The evolution of the observables are given by the Koopman operator associated with the induced dynamical system. We use the spectrum and modes of the Koopman operator to realize the above objectives. Our methods can help to, a priori, determine the network depth; determine if we have a bad initialization of the network weights, allowing a restart before training too long; speeding up the training time. Additionally, our methods help enable noise rejection and improve robustness. We show how the Koopman spectrum can be used to determine the number of layers required for the architecture. Additionally, we show how we can elucidate the convergence versus non-convergence of the training process by monitoring the spectrum, in particular, how the existence of eigenvalues clustering around 1 determines when to terminate the learning process. We also show how using Koopman modes we can selectively prune the network to speed up the training procedure. Finally, we show that incorporating loss functions based on negative Sobolev norms can allow for the reconstruction of a multi-scale signal polluted by very large amounts of noise.

cs.LG↗

arXiv2020

Extended Dynamic Mode Decomposition with Learned Koopman Eigenfunctions for Prediction and Control

This paper presents a novel learning framework to construct Koopman eigenfunctions for unknown, nonlinear dynamics using data gathered from experiments. The learning framework can extract spectral information from the full nonlinear dynamics by learning the eigenvalues and eigenfunctions of the associated Koopman operator. We then exploit the learned Koopman eigenfunctions to learn a lifted linear state-space model. To the best of our knowledge, our method is the first to utilize Koopman eigenfunctions as lifting functions for EDMD-based methods. We demonstrate the performance of the framework in state prediction and closed loop trajectory tracking of a simulated cart pole system. Our method is able to significantly improve the controller performance while relying on linear control methods to do nonlinear control.

eess.SY↗

arXiv2019

Exponentially decaying modes and long-term prediction of sea ice concentration using Koopman Mode Decomposition

Sea ice cover in the Arctic and Antarctic is an important indicator of changes in the climate, with important environmental, economic and security consequences. The complexity of the spatio-temporal dynamics of sea ice makes it difficult to assess the temporal nature of the changes - e.g. linear or exponential - and their precise geographical loci. In this study, Koopman Mode Decomposition (KMD) was applied to satellite data of sea ice concentration for the northern and southern hemispheres to gain insight into the temporal and spatial dynamics of the sea ice behavior and to predict future sea ice behavior. We discover exponentially decaying spatial modes in both hemispheres and discuss their precise spatial extent, and also perform precise geographic predictions of sea ice concentration up to four years in the future. This data-driven decomposition technique gives insight in spatial and temporal dynamics not apparent in traditional linear approaches.

math.DS↗

arXiv2018

Spectral Complexity of Directed Graphs and Application to Structural Decomposition

We introduce a new measure of complexity (called spectral complexity) for directed graphs. We start with splitting of the directed graph into its recurrent and non-recurrent parts. We define the spectral complexity metric in terms of the spectrum of the recurrence matrix (associated with the reccurent part of the graph) and the Wasserstein distance. We show that the total complexity of the graph can then be defined in terms of the spectral complexity, complexities of individual components and edge weights. The essential property of the spectral complexity metric is that it accounts for directed cycles in the graph. In engineered and software systems, such cycles give rise to sub-system interdependencies and increase risk for unintended consequences through positive feedback loops, instabilities, and infinite execution loops in software. In addition, we present a structural decomposition technique that identifies such cycles using a spectral technique. We show that this decomposition complements the well-known spectral decomposition analysis based on the Fiedler vector. We provide several examples of computation of spectral and total complexities, including the demonstration that the complexity increases monotonically with the average degree of a random graph. We also provide an example of spectral complexity computation for the architecture of a realistic fixed wing aircraft system.

math.SP↗

arXiv2016

Quasi-Periodic Intermittency in Oscillating Cylinder Flow

Fluid dynamics induced by periodically forced flow around a cylinder is analyzed computationally for the case when the forcing frequency is much lower than the von K{á}rm{á}n vortex shedding frequency corresponding to the constant flow velocity condition. By using the Koopman Mode Decomposition approach, we find a new normal form equation that extends the classical Hopf bifurcation normal form by a time-dependent term for Reynolds numbers close to the Hopf bifurcation value. The normal form describes the dynamics of an observable and features a forcing (control) term that multiplies the state, and is thus a parametric - i.e. not an additive - forcing effect. We find that the dynamics of the flow in this regime are characterized by alternating instances of quiescent and strong oscillatory behavior, and that this pattern persists indefinitely. Furthermore, the spectrum of the associated Koopman operator is shown to possess quasi-periodic features. We establish the theoretical underpinnings of this phenomenon -- that we name Quasi-Periodic Intermittency -- using the new normal form model and show that the dynamics are caused by the tendency of the flow to oscillate between the unstable fixed point and the stable limit cycle of the unforced flow. The quasi-periodic intermittency phenomena is also characterized by positive Finite-Time Lyapunov Exponents that, over a long period of time, asymptotically approach zero.

physics.flu-dyn↗