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Mihaly Petreczky

Publications and source records attributed to Mihaly Petreczky.

At least 19 recordsLinked to original sources
arXiv2026

PAC-Bayesian Bounds for Learning Partially Observed Stochastic Linear Time-Invariant State-Space Systems with Inputs and Sub-Gaussian Noise

In this paper we derive a Probably Approximately Correct (PAC)-Bayesian error bound for partially observed linear time-invariant (LTI) stochastic dynamical systems in state-space form with inputs and sub-Gaussian noise. Such bounds are widespread in machine learning, and they are useful for characterizing the predictive power of models learned from finitely many data points. The bound derived in this paper relates the expectation of prediction errors with the prediction error generated by the model on the data used for learning. In addition, we show that it can also be used to derive bounds for the parameter estimation error. In turn, this allows us to provide finite-sample error bounds for the prediction error and parameter estimation error for a wide class of system identification algorithms. Furthermore, as LTI systems are a sub-class of recurrent neural networks (RNNs), these error bounds could be a first step towards PAC-Bayesian bounds for RNNs.

cs.LG↗
arXiv2025

A finite-sample bound for identifying partially observed linear switched systems from a single trajectory

We derive a finite-sample probabilistic bound on the parameter estimation error of a system identification algorithm for Linear Switched Systems. The algorithm estimates Markov parameters from a single trajectory and applies a variant of the Ho-Kalman algorithm to recover the system matrices. Our bound guarantees statistical consistency under the assumption that the true system exhibits quadratic stability. The proof leverages the theory of weakly dependent processes. To the best of our knowledge, this is the first finite-sample bound for this algorithm in the single-trajectory setting.

cs.LG↗
arXiv2024

Loewner functions for bilinear systems

This work brings together the moment matching approach based on Loewner functions and the classical Loewner framework based on the Loewner pencil in the case of bilinear systems. New Loewner functions are defined based on the bilinear Loewner framework, and a Loewner equivalent model is produced using these functions. This model is composed of infinite series that needs to be truncated in order to be implemented in practice. In this context, a new notion of approximate Loewner equivalence is introduced. In the end, it is shown that the moment matching procedure based on the proposed Loewner functions and the classical interpolatory bilinear Loewner framework both result in $κ$-Loewner equivalent models, the main difference being that the latter preserves bilinearity at the expense of a higher order.

math.CA↗
arXiv2024

A four-bodies motorcycle dynamic model for observer design

Motivated by the need to predict dangerous scenarios, this article introduces a non-linear dynamic model for motorcycles consisting of four rigid bodies. Using Jourdain's principle, the model incorporates both longitudinal and lateral dynamics, targeting a balance between numerical complexity and accuracy of representation. The paper further employs the model to design a Luenberger observer based on linear quadratic regulator theory, for estimating physical states based on sensor measurements. In turn, the state estimates are useful for predicting dangerous scenarios (lowside, highside, fall). The relevance of the approach is demonstrated through simulations of various rectilinear trajectories and a lane-changing scenario using BikeSim simulator.

eess.SY↗
arXiv2024

A finite-sample generalization bound for stable LPV systems

One of the main theoretical challenges in learning dynamical systems from data is providing upper bounds on the generalization error, that is, the difference between the expected prediction error and the empirical prediction error measured on some finite sample. In machine learning, a popular class of such bounds are the so-called Probably Approximately Correct (PAC) bounds. In this paper, we derive a PAC bound for stable continuous-time linear parameter-varying (LPV) systems. Our bound depends on the H2 norm of the chosen class of the LPV systems, but does not depend on the time interval for which the signals are considered.

cs.LG↗
arXiv2024

Minimal covariance realization and system identification algorithm for a class of stochastic linear switched systems with i.i.d. switching

In this paper, we consider stochastic realization theory of Linear Switched Systems (LSS) with i.i.d. switching. We characterize minimality of stochastic LSSs and show existence and uniqueness (up to isomorphism) of minimal LSSs in innovation form. We present a realization algorithm to compute a minimal LSS in innovation form from output and input covariances. Finally, based on this realization algorithm, by replacing true covariances with empirical ones, we propose a statistically consistent system identification algorithm.

math.OC↗
arXiv2024

Towards stochastic realization theory for Generalized Linear Switched Systems with inputs: decomposition into stochastic and deterministic components and existence and uniqueness of innovation form

In this paper, we study a class of stochastic Generalized Linear Switched System (GLSS), which includes subclasses of jump-Markov, piecewide-linear and Linear Parameter-Varying (LPV) systems. We prove that the output of such systems can be decomposed into deterministic and stochastic components. Using this decomposition, we show existence of state-space representation in innovation form, and we provide sufficient conditions for such representations to be minimal and unique up to isomorphism.

math.OC↗
arXiv2023

PAC-Bayes Generalisation Bounds for Dynamical Systems Including Stable RNNs

In this paper, we derive a PAC-Bayes bound on the generalisation gap, in a supervised time-series setting for a special class of discrete-time non-linear dynamical systems. This class includes stable recurrent neural networks (RNN), and the motivation for this work was its application to RNNs. In order to achieve the results, we impose some stability constraints, on the allowed models. Here, stability is understood in the sense of dynamical systems. For RNNs, these stability conditions can be expressed in terms of conditions on the weights. We assume the processes involved are essentially bounded and the loss functions are Lipschitz. The proposed bound on the generalisation gap depends on the mixing coefficient of the data distribution, and the essential supremum of the data. Furthermore, the bound converges to zero as the dataset size increases. In this paper, we 1) formalize the learning problem, 2) derive a PAC-Bayesian error bound for such systems, 3) discuss various consequences of this error bound, and 4) show an illustrative example, with discussions on computing the proposed bound. Unlike other available bounds the derived bound holds for non i.i.d. data (time-series) and it does not grow with the number of steps of the RNN.

cs.LG↗
arXiv2023

PAC-Bayesian bounds for learning LTI-ss systems with input from empirical loss

In this paper we derive a Probably Approxilmately Correct(PAC)-Bayesian error bound for linear time-invariant (LTI) stochastic dynamical systems with inputs. Such bounds are widespread in machine learning, and they are useful for characterizing the predictive power of models learned from finitely many data points. In particular, with the bound derived in this paper relates future average prediction errors with the prediction error generated by the model on the data used for learning. In turn, this allows us to provide finite-sample error bounds for a wide class of learning/system identification algorithms. Furthermore, as LTI systems are a sub-class of recurrent neural networks (RNNs), these error bounds could be a first step towards PAC-Bayesian bounds for RNNs.

stat.ML↗
arXiv2022

PAC-Bayesian-Like Error Bound for a Class of Linear Time-Invariant Stochastic State-Space Models

In this paper we derive a PAC-Bayesian-Like error bound for a class of stochastic dynamical systems with inputs, namely, for linear time-invariant stochastic state-space models (stochastic LTI systems for short). This class of systems is widely used in control engineering and econometrics, in particular, they represent a special case of recurrent neural networks. In this paper we 1) formalize the learning problem for stochastic LTI systems with inputs, 2) derive a PAC-Bayesian-Like error bound for such systems, 3) discuss various consequences of this error bound.

stat.ML↗
arXiv2022

Realization Theory Of Recurrent Neural ODEs Using Polynomial System Embeddings

In this paper we show that neural ODE analogs of recurrent (ODE-RNN) and Long Short-Term Memory (ODE-LSTM) networks can be algorithmically embeddeded into the class of polynomial systems. This embedding preserves input-output behavior and can suitably be extended to other neural DE architectures. We then use realization theory of polynomial systems to provide necessary conditions for an input-output map to be realizable by an ODE-LSTM and sufficient conditions for minimality of such systems. These results represent the first steps towards realization theory of recurrent neural ODE architectures, which is is expected be useful for model reduction and learning algorithm analysis of recurrent neural ODEs.

math.OC↗
arXiv2022

Noisy Learning for Neural ODEs Acts as a Robustness Locus Widening

We investigate the problems and challenges of evaluating the robustness of Differential Equation-based (DE) networks against synthetic distribution shifts. We propose a novel and simple accuracy metric which can be used to evaluate intrinsic robustness and to validate dataset corruption simulators. We also propose methodology recommendations, destined for evaluating the many faces of neural DEs' robustness and for comparing them with their discrete counterparts rigorously. We then use this criteria to evaluate a cheap data augmentation technique as a reliable way for demonstrating the natural robustness of neural ODEs against simulated image corruptions across multiple datasets.

cs.LG↗
arXiv2022

On minimal LPV state-space representations in innovation form: an algebraic characterization

In this paper we will propose a definition of the concept of minimal state-space representations in innovation form for LPV. We also present algebraic conditions for a stochastic LPV state-space representation to be minimal in forward innovation form and discuss an algorithm for transforming any stochastic LPV state-space representation to a minimal one in innovation form.

math.OC↗
arXiv2021

Application of robust control for CSR formalization and stakeholders interest

In this paper, we propose a new definition of sustainability that includes dynamics and equity. We propose a theoretical framework that allows finding a fair equilibrium and sustainable strategies for all stakeholders. The goal is to propose a strategy such that in the long run the attributes get close to an equilibrium point which is Pareto-optimal for the stakeholders. In order to calculate a strategy, we model stakeholders and actors as dynamical systems in state-space form. Furthermore, we use robust control and linear matrix inequalities (LMIs) to calculate the desired feedback strategy. We use several simulation scenarios to show the effectiveness of our proposed framework.

math.OC↗
arXiv2021

Reduced-order modeling of LPV systems in the Loewner framework

We propose a model reduction method for LPV systems. We consider LPV state-space representations with an affine dependence on the scheduling variables. The main idea behind the proposed method is to compute the reduced order model in such a manner that its frequency domain transfer function coincides with that of the original model for some frequencies. The proposed method uses Loewner-like matrices, which can be calculated from the frequency domain representation of the system. The contribution of the paper represents an extension of the well-established Loewner framework to LPV models.

eess.SY↗
arXiv2021

PAC-Bayesian theory for stochastic LTI systems

In this paper we derive a PAC-Bayesian error bound for autonomous stochastic LTI state-space models. The motivation for deriving such error bounds is that they will allow deriving similar error bounds for more general dynamical systems, including recurrent neural networks. In turn, PACBayesian error bounds are known to be useful for analyzing machine learning algorithms and for deriving new ones.

stat.ML↗
arXiv2020

Model reduction of linear hybrid systems

The paper proposes a model reduction algorithm for linear hybrid systems, i.e., hybrid systems with externally induced discrete events, with linear continuous subsystems, and linear reset maps. The model reduction algorithm is based on balanced truncation. Moreover, the paper also proves an analytical error bound for the difference between the input-output behaviors of the original and the reduced order model. This error bound is formulated in terms of singular values of the Gramians used for model reduction.

math.DS↗