Sam Safavi

Sam Safavi, Usman A. Khan, Soummya Kar et al. · cmu

Fifth generation~(5G) networks providing much higher bandwidth and faster data rates will allow connecting vast number of static and mobile devices, sensors, agents, users, machines, and vehicles, supporting Internet-of-Things (IoT), real-time dynamic networks of mobile things. Positioning and location awareness will become increasingly important, enabling deployment of new services and contributing to significantly improving the overall performance of the 5G~system. Many of the currently talked about solutions to positioning in~5G are centralized, mostly requiring direct line-of-sight (LoS) to deployed access nodes or anchors at the same time, which in turn requires high-density deployments of anchors. But these LoS and centralized positioning solutions may become unwieldy as the number of users and devices continues to grow without limit in sight. As an alternative to the centralized solutions, this paper discusses distributed localization in a 5G enabled IoT environment where many low power devices, users, or agents are to locate themselves without global or LoS access to anchors. Even though positioning is essentially a non-linear problem (solving circle equations by trilateration or triangulation), we discuss a cooperative \textit{linear} distributed iterative solution with only local measurements, communication and computation needed at each agent. Linearity is obtained by reparametrization of the agent location through barycentric coordinate representations based on local neighborhood geometry that may be computed in terms of certain Cayley-Menger determinants involving relative local inter-agent distance measurements.

Sam Safavi, Usman A. Khan

In this paper, we develop a \textit{distributed} algorithm to localize an arbitrary number of agents moving in a bounded region of interest. We assume that the network contains \textit{at least one} agent with known location (hereinafter referred to as an anchor), and each agent measures a noisy version of its motion and the distances to the nearby agents. We provide a~\emph{geometric approach}, which allows each agent to: (i) continually update the distances to the locations where it has exchanged information with the other nodes in the past; and (ii) measure the distance between a neighbor and any such locations. Based on this approach, we provide a \emph{linear update} to find the locations of an arbitrary number of mobile agents when they follow some convexity in their deployment and motion. Since the agents are mobile, they may not be able to find nearby nodes (agents and/or anchors) to implement a distributed algorithm. To address this issue, we introduce the notion of a \emph{virtual convex hull} with the help of the aforementioned geometric approach. In particular, each agent keeps track of a virtual convex hull of other nodes, which may not physically exist, and updates its location with respect to its neighbors in the virtual hull. We show that the corresponding localization algorithm, in the absence of noise, can be abstracted as a Linear Time-Varying (LTV) system, with non-deterministic system matrices, which asymptotically tracks the true locations of the agents. We provide simulations to verify the analytical results and evaluate the performance of the algorithm in the presence of noise on the motion as well as on the distance measurements.

Surjyendu Ray, Bei Jia, Sam Safavi et al.

Motivation: Many inference tools use the Perfect Phylogeny Model (PPM) to learn trees from noisy variant allele frequency (VAF) data. Learning in this setting is hard, and existing tools use approximate or heuristic algorithms. An algorithmic improvement is important to help disentangle the limitations of the PPM's assumptions from the limitations in our capacity to learn under it. Results: We make such improvement in the scenario, where the mutations that are relevant for evolution can be clustered into a small number of groups, and the trees to be reconstructed have a small number of nodes. We use a careful combination of algorithms, software, and hardware, to develop EXACT: a tool that can explore the space of all possible phylogenetic trees, and performs exact inference under the PPM with noisy data. EXACT allows users to obtain not just the most-likely tree for some input data, but exact statistics about the distribution of trees that might explain the data. We show that EXACT outperforms several existing tools for this same task. Availability: https://github.com/surjray-repos/EXACT

Sam Safavi, José Bento

Graphs are used in almost every scientific discipline to express relations among a set of objects. Algorithms that compare graphs, and output a closeness score, or a correspondence among their nodes, are thus extremely important. Despite the large amount of work done, many of the scalable algorithms to compare graphs do not produce closeness scores that satisfy the intuitive properties of metrics. This is problematic since non-metrics are known to degrade the performance of algorithms such as distance-based clustering of graphs (Stratis and Bento 2018). On the other hand, the use of metrics increases the performance of several machine learning tasks (Indyk et al. 1999, Clarkson et al. 1999, Angiulli et al. 2002, Ackermann et al. 2010). In this paper, we introduce a new family of multi-distances (a distance between more than two elements) that satisfies a generalization of the properties of metrics to multiple elements. In the context of comparing graphs, we are the first to show the existence of multi-distances that simultaneously incorporate the useful property of alignment consistency (Nguyen et al. 2011), and a generalized metric property. Furthermore, we show that these multi-distances can be relaxed to convex optimization problems, without losing the generalized metric property.

Sam Safavi, Bikash Joshi, Guilherme França et al.

The framework of Integral Quadratic Constraints (IQC) introduced by Lessard et al. (2014) reduces the computation of upper bounds on the convergence rate of several optimization algorithms to semi-definite programming (SDP). In particular, this technique was applied to Nesterov's accelerated method (NAM). For quadratic functions, this SDP was explicitly solved leading to a new bound on the convergence rate of NAM, and for arbitrary strongly convex functions it was shown numerically that IQC can improve bounds from Nesterov (2004). Unfortunately, an explicit analytic solution to the SDP was not provided. In this paper, we provide such an analytical solution, obtaining a new general and explicit upper bound on the convergence rate of NAM, which we further optimize over its parameters. To the best of our knowledge, this is the best, and explicit, upper bound on the convergence rate of NAM for strongly convex functions.

Sam Safavi, Usman Khan

In this paper, we develop a \textcolor{black}{\emph{distributed}} algorithm to localize a network of robots moving arbitrarily in a bounded region. In the case of such mobile networks, the main challenge is that the robots may not be able to find nearby robots to implement a distributed algorithm. We address this issue by providing an opportunistic algorithm that only implements a location update when there are nearby robots and does not update otherwise. We assume that each robot measures a noisy version of its motion and the distances to the nearby robots. To localize a network of mobile robots in~$\mathbb{R}^m$, we provide a simple \emph{linear} update, which is based on barycentric coordinates and is linear-convex. We abstract the corresponding localization algorithm as a Linear Time-Varying (LTV) system and show that it asymptotically converges to the true locations~of~the robots. We first focus on the noiseless case, where the distance and motion vectors are known (measured) perfectly, and provide sufficient conditions on the convergence of the algorithm. We then evaluate the performance of the algorithm in the presence of noise and provide modifications to counter the undesirable effects of noise. \textcolor{black}{We further show that our algorithm precisely tracks a mobile network as long as there is at least one known beacon (a node whose location is perfectly known).