public 01:34:46

Paul Bendich : Topology and Geometry for Tracking and Sensor Fusion

  -   Graduate/Faculty Seminar ( 205 Views )

Many systems employ sensors to interpret the environment. The target-tracking task is to gather sensor data from the environment and then to partition these data into tracks that are produced by the same target. The goal of sensor fusion is to gather data from a heterogeneous collection of sensors (e.g, audio and video) and fuse them together in a way that enriches the performance of the sensor network at some task of interest. This talk summarizes two recent efforts that incorporate mildly sophisticated mathematics into the general sensor arena, and also comments on the joys and pitfalls of trying to apply math for customers who care much more about the results than the math. First, a key problem in tracking is to 'connect the dots:' more precisely, to take a piece of sensor data at a given time and associate it with a previously-existing track (or to declare that this is a new object). We use topological data analysis (TDA) to form data-association likelihood scores, and integrate these scores into a well-respected algorithm called Multiple Hypothesis Tracking. Tests on simulated data show that the TDA adds significant value over baseline, especially in the context of noisy sensor data. Second, we propose a very general and entirely unsupervised sensor fusion pipeline that uses recent techniques from diffusion geometry and wavelet theory to compress and then fuse time series of arbitrary dimension arising from disparate sensor modalities. The goal of the pipeline is to differentiate classes of time-ordered behavior sequences, and we demonstrate its performance on a well-studied digit sequence database. This talk represents joint work with many people. including Chris Tralie, Nathan Borggren, Sang Chin, Jesse Clarke, Jonathan deSena, John Harer, Jay Hineman, Elizabeth Munch, Andrew Newman, Alex Pieloch, David Porter, David Rouse, Nate Strawn, Adam Watkins, Michael Williams, Lihan Yao, and Peter Zulch.

public 01:14:48

Pam Miao Gu : Factorization tests and algorithms arising from counting modular forms and automorphic representations

  -   Graduate/Faculty Seminar ( 203 Views )

A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight $k$ on~$\Gamma_0(N)$ to a simpler function of $k$ and~$N$, showing that the two are equal whenever $N$ is squarefree. We prove the converse of this theorem (with one small exception), thus providing a characterization of squarefree integers. We also establish a similar characterization of prime numbers in terms of the number of Hecke newforms of weight $k$ on~$\Gamma_0(N)$. It follows that a hypothetical fast algorithm for computing the number of such automorphic representations for even a single weight $k$ would yield a fast test for whether $N$ is squarefree. We also show how to obtain bounds on the possible square divisors of a number $N$ that has been found to not be squarefree via this test, and we show how to probabilistically obtain the complete factorization of the squarefull part of $N$ from the number of such automorphic representations for two different weights. If in addition we have the number of such Hecke newforms for even a single weight $k$, then we show how to probabilistically factor $N$ entirely. All of these computations could be performed quickly in practice, given the number(s) of automorphic representations and modular forms as input. (Joint work with Greg Martin.)