Observability of Lattice Graphs

We consider a graph observability problem: how many edge colors are needed for an unlabeled graph so that an agent, walking from node to node, can uniquely determine its location from just the observed color sequence of the walk? Specifically, let G(n,d) be an edge-colored subgraph of d-dimensional (directed or undirected) lattice of size n^d = n * n * ... * n. We say that G(n,d) is t-observable if an agent can uniquely determine its current position in the graph from the color sequence of any t-dimensional walk, where the dimension is the number of different directions spanned by the edges of the walk. A walk in an undirected lattice G(n,d) has dimension between 1 and d, but a directed walk can have dimension between 1 and 2d because of two different orientations for each axis. We derive bounds on the number of colors needed for t-observability. Our main result is that Theta(n^(d/t)) colors are both necessary and sufficient for t-observability of G(n,d), where d is considered a constant. This shows an interesting dependence of graph observability on the ratio between the dimension of the lattice and that of the walk. In particular, the number of colors for full-dimensional walks is Theta(n^(1/2)) in the directed case, and Theta(n) in the undirected case, independent of the lattice dimension. All of our results extend easily to non-square lattices: given a lattice graph of size N = n_1 * n_2 * ... * n_d, the number of colors for t-observability is Theta (N^(1/t))

The Optical Frequency Comb as a One-Way Quantum Computer

In the one-way model of quantum computing, quantum algorithms are implemented using only measurements on an entangled initial state. Much of the hard work is done up-front when creating this universal resource, known as a cluster state, on which the measurements are made. Here we detail a new proposal for a scalable method of creating cluster states using only a single multimode optical parametric oscillator (OPO). The method generates a continuous-variable cluster state that is universal for quantum computation and encoded in the quadratures of the optical frequency comb of the OPO. This work expands on the presentation in Phys. Rev. Lett. 101, 130501 (2008).Comment: 20 pages, 8 figures. v2 corrects minor error in published versio

Slingshot: cell lineage and pseudotime inference for single-cell transcriptomics.

BackgroundSingle-cell transcriptomics allows researchers to investigate complex communities of heterogeneous cells. It can be applied to stem cells and their descendants in order to chart the progression from multipotent progenitors to fully differentiated cells. While a variety of statistical and computational methods have been proposed for inferring cell lineages, the problem of accurately characterizing multiple branching lineages remains difficult to solve.ResultsWe introduce Slingshot, a novel method for inferring cell lineages and pseudotimes from single-cell gene expression data. In previously published datasets, Slingshot correctly identifies the biological signal for one to three branching trajectories. Additionally, our simulation study shows that Slingshot infers more accurate pseudotimes than other leading methods.ConclusionsSlingshot is a uniquely robust and flexible tool which combines the highly stable techniques necessary for noisy single-cell data with the ability to identify multiple trajectories. Accurate lineage inference is a critical step in the identification of dynamic temporal gene expression

Contours in Visualization

This thesis studies the visualization of set collections either via or defines as the relations among contours. In the first part, dynamic Euler diagrams are used to communicate and improve semimanually the result of clustering methods which allow clusters to overlap arbitrarily. The contours of the Euler diagram are rendered as implicit surfaces called blobs in computer graphics. The interaction metaphor is the moving of items into or out of these blobs. The utility of the method is demonstrated on data arising from the analysis of gene expressions. The method works well for small datasets of up to one hundred items and few clusters. In the second part, these limitations are mitigated employing a GPU-based rendering of Euler diagrams and mixing textures and colors to resolve overlapping regions better. The GPU-based approach subdivides the screen into triangles on which it performs a contour interpolation, i.e. a fragment shader determines for each pixel which zones of an Euler diagram it belongs to. The rendering speed is thus increased to allow multiple hundred items. The method is applied to an example comparing different document clustering results. The contour tree compactly describes scalar field topology. From the viewpoint of graph drawing, it is a tree with attributes at vertices and optionally on edges. Standard tree drawing algorithms emphasize structural properties of the tree and neglect the attributes. Adapting popular graph drawing approaches to the problem of contour tree drawing it is found that they are unable to convey this information. Five aesthetic criteria for drawing contour trees are proposed and a novel algorithm for drawing contour trees in the plane that satisfies four of these criteria is presented. The implementation is fast and effective for contour tree sizes usually used in interactive systems and also produces readable pictures for larger trees. Dynamical models that explain the formation of spatial structures of RNA molecules have reached a complexity that requires novel visualization methods to analyze these model\''s validity. The fourth part of the thesis focuses on the visualization of so-called folding landscapes of a growing RNA molecule. Folding landscapes describe the energy of a molecule as a function of its spatial configuration; they are huge and high dimensional. Their most salient features are described by their so-called barrier tree -- a contour tree for discrete observation spaces. The changing folding landscapes of a growing RNA chain are visualized as an animation of the corresponding barrier tree sequence. The animation is created as an adaption of the foresight layout with tolerance algorithm for dynamic graph layout. The adaptation requires changes to the concept of supergraph and it layout. The thesis finishes with some thoughts on how these approaches can be combined and how the task the application should support can help inform the choice of visualization modality

Learning to Transform Time Series with a Few Examples

We describe a semi-supervised regression algorithm that learns to transform one time series into another time series given examples of the transformation. This algorithm is applied to tracking, where a time series of observations from sensors is transformed to a time series describing the pose of a target. Instead of defining and implementing such transformations for each tracking task separately, our algorithm learns a memoryless transformation of time series from a few example input-output mappings. The algorithm searches for a smooth function that fits the training examples and, when applied to the input time series, produces a time series that evolves according to assumed dynamics. The learning procedure is fast and lends itself to a closed-form solution. It is closely related to nonlinear system identification and manifold learning techniques. We demonstrate our algorithm on the tasks of tracking RFID tags from signal strength measurements, recovering the pose of rigid objects, deformable bodies, and articulated bodies from video sequences. For these tasks, this algorithm requires significantly fewer examples compared to fully-supervised regression algorithms or semi-supervised learning algorithms that do not take the dynamics of the output time series into account

Stability Margin Scaling Laws for Distributed Formation Control as a Function of Network Structure

We consider the problem of distributed formation control of a large number of vehicles. An individual vehicle in the formation is assumed to be a fully actuated point mass. A distributed control law is examined: the control action on an individual vehicle depends on (i) its own velocity and (ii) the relative position measurements with a small subset of vehicles (neighbors) in the formation. The neighbors are defined according to an information graph. In this paper we describe a methodology for modeling, analysis, and distributed control design of such vehicular formations whose information graph is a D-dimensional lattice. The modeling relies on an approximation based on a partial differential equation (PDE) that describes the spatio-temporal evolution of position errors in the formation. The analysis and control design is based on the PDE model. We deduce asymptotic formulae for the closed-loop stability margin (absolute value of the real part of the least stable eigenvalue) of the controlled formation. The stability margin is shown to approach 0 as the number of vehicles N goes to infinity. The exponent on the scaling law for the stability margin is influenced by the dimension and the structure of the information graph. We show that the scaling law can be improved by employing a higher dimensional information graph. Apart from analysis, the PDE model is used for a mistuning-based design of control gains to maximize the stability margin. Mistuning here refers to small perturbation of control gains from their nominal symmetric values. We show that the mistuned design can have a significantly better stability margin even with a small amount of perturbation. The results of the analysis with the PDE model are corroborated with numerical computation of eigenvalues with the state-space model of the formation.Comment: This paper is the expanded version of the paper with the same name which is accepted by the IEEE Transactions on Automatic Control. The final version is updated on Oct. 12, 201