Georgia Institute of Technology College of Sciences

Dynamical processes, such as information diffusion in social networks, gene regulation in biological systems and functional collaborations between brain regions, generate a large volume of high dimensional “asynchronous” and “interdependent” time-stamped event data. This type of timing information is rather different from traditional iid. data and discrete-time temporal data, which calls for new models and scalable algorithms for learning, analyzing and utilizing them. In this talk, I will present methods based on multivariate point processes, high dimensional sparse recovery, and randomized algorithms for addressing a sequence of problems arising from this context. As a concrete example, I will also present experimental results on learning and optimizing information cascades in web logs, including estimating hidden diffusion networks and influence maximization with the learned networks. With both careful model and algorithm design, the framework is able to handle millions of events and millions of networked entities.

We consider a variational model for image segmentation which takes into account the occlusions between different objects. The model consists in minimizing a functional which depends on: (i) a partition (segmentation) of the image domain constituted by partially overlapping regions; (ii) a piecewise constant function which gives information about the visible portions of objects; (iii) a piecewise constant function which constitutes an approximation of a given image. The geometric part of the energy functional depends on the curvature of the boundaries of the overlapping regions. Some variational properties of the model are discussed with the aim of investigating the reconstruction capabilities of occluded boundaries of shapes. Joint work with Giovanni Bellettini.

From bird flocks to ungulate herds to fish schools, nature abounds with examples of biological aggregations that arise from social interactions. These interactions take place over finite (rather than infinitesimal) distances, giving rise to nonlocal models. In this modeling-based talk, I will discuss two projects on insect swarms in which nonlocal social interactions play a key role. The first project examines desert locusts. The model is a system of nonlinear partial integrodifferential equations of advection-reaction type. I find conditions for the formation of an aggregation, demonstrate transiently traveling pulses of insects, and find hysteresis in the aggregation's existence. The second project examines the pea aphid. Based on experiments that motion track aphids walking in a circular arena, I extract a discrete, stochastic model for the group. Each aphid transitions randomly between a moving and a stationary state. Moving aphids follow a correlated random walk. The probabilities of motion state transitions, as well as the random walk parameters, depend strongly on distance to an aphid’s nearest neighbor. For large nearest neighbor distances, when an aphid is isolated, its motion is ballistic and it is less likely to stop. In contrast, for short nearest neighbor distances, aphids move diffusively and are more likely to become stationary; this behavior constitutes an aggregation mechanism.

The Alpert multiwavelets are an extension of the Haar wavelet to higher degree piecewise polynomials thereby giving higher approximation order. This system has uses in numerical analysis in problems where shocks develop. An orthogonal basis of scaling functions for this system are the Legendre polynomials and we will examine the consequence of this. In particular we will show that the coefficients in the refinement equation can be written in terms of Jacobi polynomials with varying parameters. Difference equationssatisfied by these coefficients will be exhibited that give rise to generalized eigenvalue problems. Furthermore an orthogonal basis of wavelet functions will be discussed that have explicit formulas as hypergeometric polynomials.