Topology Design for Optimal Network Coherence

We consider a network topology design problem in which an initial undirected graph underlying the network is given and the objective is to select a set of edges to add to the graph to optimize the coherence of the resulting network. We show that network coherence is a submodular function of the network topology. As a consequence, a simple greedy algorithm is guaranteed to produce near optimal edge set selections. We also show that fast rank one updates of the Laplacian pseudoinverse using generalizations of the Sherman-Morrison formula and an accelerated variant of the greedy algorithm can speed up the algorithm by several orders of magnitude in practice. These allow our algorithms to scale to network sizes far beyond those that can be handled by convex relaxation heuristics

Inference of single-cell phylogenies from lineage tracing data using Cassiopeia.

The pairing of CRISPR/Cas9-based gene editing with massively parallel single-cell readouts now enables large-scale lineage tracing. However, the rapid growth in complexity of data from these assays has outpaced our ability to accurately infer phylogenetic relationships. First, we introduce Cassiopeia-a suite of scalable maximum parsimony approaches for tree reconstruction. Second, we provide a simulation framework for evaluating algorithms and exploring lineage tracer design principles. Finally, we generate the most complex experimental lineage tracing dataset to date, 34,557 human cells continuously traced over 15 generations, and use it for benchmarking phylogenetic inference approaches. We show that Cassiopeia outperforms traditional methods by several metrics and under a wide variety of parameter regimes, and provide insight into the principles for the design of improved Cas9-enabled recorders. Together, these should broadly enable large-scale mammalian lineage tracing efforts. Cassiopeia and its benchmarking resources are publicly available at www.github.com/YosefLab/Cassiopeia

Static and dynamic optimization problems in cooperative multi-agent systems

This dissertation focuses on challenging static and dynamic problems encountered in cooperative multi-agent systems. First, a unified optimization framework is proposed for a wide range of tasks including consensus, optimal coverage, and resource allocation problems. It allows gradient-based algorithms to be applied to solve these problems, all of which have been studied in a separate way in the past. Gradient-based algorithms are shown to be distributed for a subclass of problems where objective functions can be decoupled. Second, the issue of global optimality is studied for optimal coverage problems where agents are deployed to maximize the joint detection probability. Objective functions in these problems are non-convex and no global optimum can be guaranteed by gradient-based algorithms developed to date. In order to obtain a solution close to the global optimum, the selection of initial conditions is crucial. The initial state is determined by an additional optimization problem where the objective function is monotone submodular, a class of functions for which the greedy solution performance is guaranteed to be within a provable bound relative to the optimal performance. The bound is known to be within 1 − 1/e of the optimal solution and is improved by exploiting the curvature information of the objective function. The greedy solution is subsequently used as an initial point of a gradient-based algorithm for the original optimal coverage problem. In addition, a novel method is proposed to escape a local optimum in a systematic way instead of randomly perturbing controllable variables away from a local optimum. Finally, optimal dynamic formation control problems are addressed for mobile leader-follower networks. Optimal formations are determined by maximizing a given objective function while continuously preserving communication connectivity in a time-varying environment. It is shown that in a convex mission space, the connectivity constraints can be satisfied by any feasible solution to a Mixed Integer Nonlinear Programming (MINLP) problem. For the class of optimal formation problems where the objective is to maximize coverage, the optimal formation is proven to be a tree which can be efficiently constructed without solving a MINLP problem. In a mission space constrained by obstacles, a minimum-effort reconfiguration approach is designed for obtaining the formation which still optimizes the objective function while avoiding the obstacles and ensuring connectivity

Gossip Algorithms for Distributed Signal Processing

Gossip algorithms are attractive for in-network processing in sensor networks because they do not require any specialized routing, there is no bottleneck or single point of failure, and they are robust to unreliable wireless network conditions. Recently, there has been a surge of activity in the computer science, control, signal processing, and information theory communities, developing faster and more robust gossip algorithms and deriving theoretical performance guarantees. This article presents an overview of recent work in the area. We describe convergence rate results, which are related to the number of transmitted messages and thus the amount of energy consumed in the network for gossiping. We discuss issues related to gossiping over wireless links, including the effects of quantization and noise, and we illustrate the use of gossip algorithms for canonical signal processing tasks including distributed estimation, source localization, and compression.Comment: Submitted to Proceedings of the IEEE, 29 page

Solving Jigsaw Puzzles By the Graph Connection Laplacian

We propose a novel mathematical framework to address the problem of automatically solving large jigsaw puzzles. This problem assumes a large image, which is cut into equal square pieces that are arbitrarily rotated and shuffled, and asks to recover the original image given the transformed pieces. The main contribution of this work is a method for recovering the rotations of the pieces when both shuffles and rotations are unknown. A major challenge of this procedure is estimating the graph connection Laplacian without the knowledge of shuffles. We guarantee some robustness of the latter estimate to measurement errors. A careful combination of our proposed method for estimating rotations with any existing method for estimating shuffles results in a practical solution for the jigsaw puzzle problem. Numerical experiments demonstrate the competitive accuracy of this solution, its robustness to corruption and its computational advantage for large puzzles