1,721,262 research outputs found
LTI Stochastic Processes: a Behavioral Perspective
Full text linkThis paper revisits the definition of linear time-invariant (LTI) stochastic process within a behavioral systems framework. Building on Willems (2013), we derive a canonical representation of an LTI stochastic process and a physically grounded notion of interconnection between independent stochastic processes. We use this framework to analyze the invariance properties enjoyed by distances between spectral densities of LTI processes
AR Identification of Latent-Variable Graphical Models
Full text linkThe paper proposes an identification procedure for autoregressive Gaussian stationary stochastic processes under the assumption that the manifest (or observed) variables are nearly independent when conditioned on a limited number of latent (or hidden) variables. The method exploits the sparse plus low-rank decomposition of the inverse of the manifest spectral density and the efficient convex relaxations recently proposed for such decompositions
Inductive geometric matrix midranges
No full textCovariance data as represented by symmetric positive definite (SPD) matrices are ubiquitous throughout technical study as efficient descriptors of interdependent systems. Euclidean analysis of SPD matrices, while computationally fast, can lead to skewed and even unphysical interpretations of data. Riemannian methods preserve the geometric structure of SPD data at the cost of expensive eigenvalue computations. In this paper, we propose a geometric method for unsupervised clustering of SPD data based on the Thompson metric. This technique relies upon a novel "inductive midrange"centroid computation for SPD data, whose properties are examined and numerically confirmed. We demonstrate the incorporation of the Thompson metric and inductive midrange into X-means and K-means++ clustering algorithms. </p
Differential geometry with extreme eigenvalues in the positive semidefinite cone
No full textDifferential geometric approaches to the analysis and processing of data in the form of symmetric positive definite (SPD) matrices have had notable successful applications to numerous fields including computer vision, medical imaging, and machine learning. The dominant geometric paradigm for such applications has consisted of a few Riemannian geometries associated with spectral computations that are costly at high scale and in high dimensions. We present a route to a scalable geometric framework for the analysis and processing of SPD-valued data based on the efficient computation of extreme generalized eigenvalues through the Hilbert and Thompson geometries of the semidefinite cone. We explore a particular geodesic space structure based on Thompson geometry in detail and establish several properties associated with this structure. Furthermore, we define a novel iterative mean of SPD matrices based on this geometry and prove its existence and uniqueness for a given finite collection of points. Finally, we state and prove a number of desirable properties that are satisfied by this mean
Factor analysis of moving average processes
No full textAbstract — The paper considers an extension of factor analy-sis to moving average processes. The problem is formulated as a rank minimization of a suitable spectral density. It is shown that it can be efficiently solved via a trace norm convex relaxation. I
Time-optimal control of a 3-level quantum system and its generalization
Full text linkpeer reviewedWe solve the problem of steering a three-level quantum system from one eigen-
state to another in minimum time and study its possible extension to the time-optimal
control problem for a general n-level quantum system. For the three-level system we find all
optimal controls by finding two types of symmetry in the problems: Z2 ×S3 discrete sym-
metry and S1 continuous symmetry, and exploiting them to solve the problem through
discrete reduction and symplectic reduction. We then study the geometry, in the same
framework, which occurs in the time-optimal control of a general n-level quantum system
Sparse plus low-rank autoregressive identification in neuroimaging time series
Full text linkpeer reviewe
Target formation on the circle by monotone system design
Full text linkPositivity and Perron-Frobenius theory provide an elegant framework for the convergence analysis of linear consensus algorithms. Here we consider a generalization of these ideas to the analysis of nonlinear consensus algorithms on the circle and establish tools for the design of consensus protocols that monotonically converge to target formations on the circle
Towards coordination algorithms on compact Lie groups
Full text linkThe present work considers the design of control algorithms to coordinate a swarm of identical, autonomous, cooperating agents that evolve on compact Lie groups. The objective is that the agents reach a so-called consensus state without using any external reference. In the same line of thought, a leader-follower approach where ’follower’ agents would track one ’leader’ agent is excluded, in favor of a fully cooperative strategy. Moreover, the presence of communication links between agents is explicitly restricted, leading to undirected, directed and/or time-varying communication structures.
Two levels of complexity are considered for the models of the agents. First, they are modeled as simple integrators on Lie groups. This setting is meaningful in a trajectory-planning context for swarms of mechanical vehicles, or to solve algorithmic problems involving multiple agent coordination. In a second step, the model of Newtonian mechanics is used for Lie group solids, which correspond to the abstraction of the Euler laws for the rotation of a rigid body to general Lie groups. This setting is relevant for the actual control of mechanical vehicles through torques and forces.
As a common starting point, the consensus problem is formulated in terms of the extrema of a cost function. This cost function is linked to a specific centroid definition on manifolds, which is referred to in this work as the induced arithmetic mean, that is easily computable in closed form and hence may be of independent interest. Using the integrator model, this naturally leads to efficient gradient algorithms to synchronize (i.e. maximizing the consensus) or balance (i.e. minimizing the consensus) the agents; the latter however can only implement the corresponding control laws if the communication graph is fixed and undirected. For directed and/or varying communication graphs, a convenient adaptation of the gradient algorithms is obtained using auxiliary estimator variables that evolve in an embedding vector space. An extension of these results to homogeneous manifolds is briefly discussed. For the mechanical model, the coordination objective is specialized to coordinated motion (i.e. moving such that the relative positions of the agents are conserved) and synchronization (i.e. having all the agents at the same position on the Lie group). Control laws are derived using two classical approaches of nonlinear control - tracking and energy shaping. They are both based on the ideas developed in the first part.
For the sake of easier understanding and given its practical importance as representing orientations of rigid bodies in 3-dimensional space, the group SO(3) (or more generally SO(n)) is used as a running example throughout this report. Other examples are the circle SO(2) and, for the extension to homogeneous manifolds, the Grassmann manifolds Grass(p, n).
As this report is written in the middle of research activities, it closes with several future research directions that can be explored in the continuity of the present work
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