By an undirected rigid formation of mobile autonomous agents is meant a formation based on graph rigidity in which each pair of "neighboring" agents is responsible for maintaining a prescribed target distance between them. In a recent paper a systematic method was proposed for devising gradient control laws for asymptotically stabilizing a large class of rigid, undirected formations in two-dimensional space assuming all agents are described by kinematic point models. The aim of this paper is to explain what happens to such formations if neighboring agents have slightly different understandings of what the desired distance between them is supposed to be or equivalently if neighboring agents have differing estimates of what the actual distance between them is. In either case, what one would expect would be a gradual distortion of the formation from its target shape as discrepancies in desired or sensed distances increase. While this is observed for the gradient laws in question, something else quite unexpected happens at the same time. It is shown that for any rigidity-based, undirected formation of this type which is comprised of three or more agents, that if some neighboring agents have slightly different understandings of what the desired distances between them are suppose to be, then almost for certain, the trajectory of the resulting distorted but rigid formation will converge exponentially fast to a closed circular orbit in two-dimensional space which is traversed periodically at a constant angular speed.

Spectral methods are of fundamental importance in statistics and machine learning, because they underlie algorithms from classical principal components analysis to more recent approaches that exploit manifold structure. In most cases, the core technical problem can be reduced to computing a low-rank approximation to a positive-definite kernel. For the growing number of applications dealing with very large or high-dimensional datasets, however, the optimal approximation afforded by an exact spectral decomposition is too costly, because its complexity scales as the cube of either the number of training examples or their dimensionality. Motivated by such applications, we present here 2 new algorithms for the approximation of positive-semidefinite kernels, together with error bounds that improve on results in the literature. We approach this problem by seeking to determine, in an efficient manner, the most informative subset of our data relative to the kernel approximation task at hand. This leads to two new strategies based on the Nyström method that are directly applicable to massive datasets. The first of these-based on sampling-leads to a randomized algorithm whereupon the kernel induces a probability distribution on its set of partitions, whereas the latter approach-based on sorting-provides for the selection of a partition in a deterministic way. We detail their numerical implementation and provide simulation results for a variety of representative problems in statistical data analysis, each of which demonstrates the improved performance of our approach relative to existing methods.statistical data analysis | kernel methods | low-rank approximation S pectral methods hold a central place in statistical data analysis. Indeed, the spectral decomposition of a positive-definite kernel underlies a variety of classical approaches such as principal components analysis (PCA), in which a low-dimensional subspace that explains most of the variance in the data is sought; Fisher discriminant analysis, which aims to determine a separating hyperplane for data classification; and multidimensional scaling (MDS), used to realize metric embeddings of the data. Moreover, the importance of spectral methods in modern statistical learning has been reinforced by the recent development of several algorithms designed to treat nonlinear structure in data-a case where classical methods fail. Popular examples include isomap (1), spectral clustering (2), Laplacian (3) and Hessian (4) eigenmaps, and diffusion maps (5). Though these algorithms have different origins, each requires the computation of the principal eigenvectors and eigenvalues of a positive-definite kernel.Although the computational cost (in both space and time) of spectral methods is but an inconvenience for moderately sized datasets, it becomes a genuine barrier as data sizes increase and new application areas appear. A variety of techniques, spanning fields from classical linear algebra to theoretical computer science (6), have been proposed to trade off analysis precision ...

This paper considers the discrete-time version of Altafini's model for opinion dynamics in which the interaction among a group of agents is described by a time-varying signed digraph. Prompted by an idea from [1], exponential convergence of the system is studied using a graphical approach. Necessary and sufficient conditions for exponential convergence with respect to each possible type of limit states are provided. Specifically, under the assumption of repeatedly jointly strong connectivity, it is shown that (1) a certain type of two-clustering will be reached exponentially fast for almost all initial conditions if, and only if, the sequence of signed digraphs is repeatedly jointly structurally balanced corresponding to that type of two-clustering; (2) the system will converge to zero exponentially fast for all initial conditions if, and only if, the sequence of signed digraphs is repeatedly jointly structurally unbalanced. An upper bound on the convergence rate is also provided. neighbors. Specifically, for those neighbors with friendship, the agent will trust their opinions; for those neighbors with antagonism, the agent will not trust their opinions and, instead, the agent will take the opposite of their opinions in updating, which is the key difference between the Altafini model and other opinion dynamics models.The continuous-time Altafini model has been studied in [31][32][33]38], and papers [1, 34-37, 39] have studied the discrete-time counterpart. This paper will focus on the latter and present a more comprehensive treatment of the work in [39]. Specifically, the paper provides proofs for the main theorems, addresses the convergence rate issue, and discusses a time-invariant case with a less restrictive connectivity condition, which were not included in [39].The most general result in the literature regarding the discrete-time version of Altafini's model is that for any "repeated jointly strongly connected" sequence of signed digraphs, the absolute values of all the agents' opinions will asymptotically reach a consensus, which is called "modulus consensus", having standard consensus and "bipartite consensus" as special cases [37]. The result is independent of the structure of signs in the digraphs which can be described by the term structural balance (or structural unbalance) from social sciences [40] in that different types of structural balance correspond to different clusterings of opinions in the network (Section 3.1). Notwithstanding this, the following questions remain. What are necessary and sufficient conditions on the sequence of signed digraphs that will lead to a specific clustering? When will the convergence be exponentially fast and how can the rate of convergence be characterized? What will happen if the assumption of strong connectivity is relaxed? This paper aims to answer these questions and will appeal to a graphical approach prompted by an idea introduced in [1], as further discussed below.In the recent work by Hendrickx [1], a very nice lifting approach was proposed to establish the e...

Formation control deals with the design of decentralized control laws that stabilize mobile, autonomous agents at prescribed distances from each other. We call any configuration of the agents a target configuration if it satisfies the inter-agent distance conditions. It is well-known that when the distance conditions are defined by a rigid graph, there is a finite number of target configurations modulo rotations and translations of the entire formation. We can thus recast the objective of formation control as stabilizing one or many of the target configurations. A major issue is that such control laws will also have equilibria corresponding to configurations which do not meet the desired inter-agent distance conditions; we refer to these as undesirable configurations. The undesirable configurations become problematic if they are also stable. Designing decentralized control laws whose stable equilibria are all target configurations in the case of a general rigid graph is still an open problem. We provide here a new point of view on this problem, and propose a partial solution by exhibiting a class of rigid graphs and control laws for which all stable equilibria are target configurations.