Sérgio Pequito

Jingqi Li, Ximing Chen, Sérgio Pequito et al.

In this paper, we consider the structural stabilizability problem of undirected networks. More specifically, we are tasked to infer the stabilizability of an undirected network from its underlying topology, where the undirected networks are modeled as continuous-time linear time-invariant (LTI) systems involving symmetric state matrices. Firstly, we derive a graph-theoretic necessary and sufficient condition for structural stabilizability of undirected networks. Then, we propose a method to infer the maximum dimension of stabilizable subspace solely based on the network structure. Based on these results, on one hand, we study the optimal actuator-disabling attack problem, i.e., removing a limited number of actuators to minimize the maximum dimension of stabilizable subspace. We show this problem is NP-hard. On the other hand, we study the optimal recovery problem with respect to the same kind of attacks, i.e., adding a limited number of new actuators such that the maximum dimension of stabilizable subspace is maximized. We prove the optimal recovery problem is also NP-hard, and we develop a (1-1/e) approximation algorithm to this problem.

Sarthak Chatterjee, Orlando Romero, Sérgio Pequito

Closed-loop neurotechnology requires the capability to predict the state evolution and its regulation under (possibly) partial measurements. There is evidence that neurophysiological dynamics can be modeled by fractional-order dynamical systems. Therefore, we propose to establish a separation principle for discrete-time fractional-order dynamical systems, which are inherently nonlinear and are able to capture spatiotemporal relations that exhibit non-Markovian properties. The separation principle states that the problems of controller and state estimator design can be done independently of each other while ensuring proper estimation and control in closed-loop setups. Lastly, we illustrate, as proof-of-concept, the application of the separation principle when designing controllers and estimators for these classes of systems in the context of neurophysiological data. In particular, we rely on real data to derive the models used to assess and regulate the evolution of closed-loop neurotechnologies based on electroencephalographic data.

Mohammadreza Doostmohammadian, Sérgio Pequito

Distributed optimization finds many applications in machine learning, signal processing, and control systems. In these real-world applications, the constraints of communication networks, particularly limited bandwidth, necessitate implementing quantization techniques. In this paper, we propose distributed optimization dynamics over multi-agent networks subject to logarithmically quantized data transmission. Under this condition, data exchange benefits from representing smaller values with more bits and larger values with fewer bits. As compared to uniform quantization, this allows for higher precision in representing near-optimal values and more accuracy of the distributed optimization algorithm. The proposed optimization dynamics comprise a primary state variable converging to the optimizer and an auxiliary variable tracking the objective function's gradient. Our setting accommodates dynamic network topologies, resulting in a hybrid system requiring convergence analysis using matrix perturbation theory and eigenspectrum analysis.

Guilherme Ramos, A. Pedro Aguiar, Soummya Kar et al.

Average consensus protocols emerge with a central role in distributed systems and decision-making such as distributed information fusion, distributed optimization, distributed estimation, and control. A key advantage of these protocols is that agents exchange and reveal their state information only to their neighbors. Yet, it can raise privacy concerns in situations where the agents' states contain sensitive information. In this paper, we propose a novel (noiseless) privacy preserving distributed algorithms for multi-agent systems to reach an average consensus. The main idea of the algorithms is that each agent runs a (small) network with a crafted structure and dynamics to form a network of networks (i.e., the connection between the newly created networks and their interconnections respecting the initial network connections). Together with a re-weighting of the dynamic parameters dictating the inter-agent dynamics and the initial states, we show that it is possible to ensure that the value of each node converges to the consensus value of the original network. Furthermore, we show that, under mild assumptions, it is possible to craft the dynamics such that the design can be achieved in a distributed fashion. Finally, we illustrate the proposed algorithm with examples.

Orlando Romero, Subhro Das, Pin-Yu Chen et al.

Out of the recent advances in systems and control (S\&C)-based analysis of optimization algorithms, not enough work has been specifically dedicated to machine learning (ML) algorithms and its applications. This paper addresses this gap by illustrating how (discrete-time) Lyapunov stability theory can serve as a powerful tool to aid, or even lead, in the analysis (and potential design) of optimization algorithms that are not necessarily gradient-based. The particular ML problem that this paper focuses on is that of parameter estimation in an incomplete-data Bayesian framework via the popular optimization algorithm known as maximum a posteriori expectation-maximization (MAP-EM). Following first principles from dynamical systems stability theory, conditions for convergence of MAP-EM are developed. Furthermore, if additional assumptions are met, we show that fast convergence (linear or quadratic) is achieved, which could have been difficult to unveil without our adopted S\&C approach. The convergence guarantees in this paper effectively expand the set of sufficient conditions for EM applications, thereby demonstrating the potential of similar S\&C-based convergence analysis of other ML algorithms.

Matilde Tristany Farinha, Sérgio Pequito, Pedro A. Santos et al.

Artificial neural networks, one of the most successful approaches to supervised learning, were originally inspired by their biological counterparts. However, the most successful learning algorithm for artificial neural networks, backpropagation, is considered biologically implausible. We contribute to the topic of biologically plausible neuronal learning by building upon and extending the equilibrium propagation learning framework. Specifically, we introduce: a new neuronal dynamics and learning rule for arbitrary network architectures; a sparsity-inducing method able to prune irrelevant connections; a dynamical-systems characterization of the models, using Lyapunov theory.

Sarthak Chatterjee, Orlando Romero, Sérgio Pequito

The Expectation-Maximization (EM) algorithm is one of the most popular methods used to solve the problem of parametric distribution-based clustering in unsupervised learning. In this paper, we propose to analyze a generalized EM (GEM) algorithm in the context of Gaussian mixture models, where the maximization step in the EM is replaced by an increasing step. We show that this GEM algorithm can be understood as a linear time-invariant (LTI) system with a feedback nonlinearity. Therefore, we explore some of its convergence properties by leveraging tools from robust control theory. Lastly, we explain how the proposed GEM can be designed, and present a pedagogical example to understand the advantages of the proposed approach.

Orlando Romero, Sarthak Chatterjee, Sérgio Pequito

In this paper, we propose a dynamical systems perspective of the Expectation-Maximization (EM) algorithm. More precisely, we can analyze the EM algorithm as a nonlinear state-space dynamical system. The EM algorithm is widely adopted for data clustering and density estimation in statistics, control systems, and machine learning. This algorithm belongs to a large class of iterative algorithms known as proximal point methods. In particular, we re-interpret limit points of the EM algorithm and other local maximizers of the likelihood function it seeks to optimize as equilibria in its dynamical system representation. Furthermore, we propose to assess its convergence as asymptotic stability in the sense of Lyapunov. As a consequence, we proceed by leveraging recent results regarding discrete-time Lyapunov stability theory in order to establish asymptotic stability (and thus, convergence) in the dynamical system representation of the EM algorithm.

Sarthak Chatterjee, Sérgio Pequito

Fractional-order dynamical systems are used to describe processes that exhibit long-term memory with power-law dependence. Notable examples include complex neurophysiological signals such as electroencephalogram (EEG) and blood-oxygen-level dependent (BOLD) signals. When analyzing different neurophysiological signals and other signals with different origin (for example, biological systems), we often find the presence of artifacts, that is, recorded activity that is due to external causes and does not have its origins in the system of interest. In this paper, we consider the problem of estimating the states of a discrete-time fractional-order dynamical system when there are artifacts present in some of the sensor measurements. Specifically, we provide necessary and sufficient conditions that ensure we can retrieve the system states even in the presence of artifacts. We provide a state estimation algorithm that can estimate the states of the system in the presence of artifacts. Finally, we present illustrative examples of our main results using real EEG data.

Jingqi Li, Ximing Chen, Sérgio Pequito et al.

In this paper, we study the target controllability problem of networked dynamical systems, in which we are tasked to steer a subset of network states towards a desired objective. More specifically, we derive necessary and sufficient conditions for the structural target controllability problem of linear time-invariant (LTI) systems with symmetric state matrices, such as undirected dynamical networks with unknown link weights. To achieve our goal, we first characterize the generic rank of symmetrically structured matrices, as well as the modes of any numerical realization. Subsequently, we provide a graph-theoretic necessary and sufficient condition for the structural controllability of undirected networks with multiple control nodes. Finally, we derive a graph-theoretic necessary and sufficient condition for structural target controllability of undirected networks. Remarkably, apart from the standard reachability condition, only local topological information is needed for the verification of structural target controllability.