Functional Analytic Perspectives on Nonparametric Density Estimation.

Nonparametric density estimation is a classic problem in statistics. In the standard estimation setting, when one has access to iid samples from an unknown distribution, there exist several established and well-studied nonparametric density estimators. Yet there remains interesting alternative settings which are less well-studied. This work considers two such settings. First we consider the case where the data contains some contamination, i.e. a portion of the data is not distributed according to the density we would like to estimate. In this setting one would like an estimator which is robust to the contaminating data. An approach to this was suggested in Kim and Scott (2012). The estimator in that paper was analytically and experimentally shown to be robust, but no consistency result was presented. In Chapter II it is demonstrated that this estimator is indeed consistent for a class of convex losses. Chapter III introduces a new robust kernel density estimator based on scaling and projection in Hilbert space. This estimator is proven to be consistent and will converge to the true density provided certain assumptions on the contaminating distribution. Its efficacy is demonstrated experimentally by applying it to several datasets. Chapter IV considers a different setting which can be thought of as nonparametric mixture modelling. Here one would like to estimate multiple densities with access to groups of samples where each sample in a group is known to be distributed according the same unknown density. Tight identifiability bounds and a highly general algorithm for recovery of the densities are presented for this setting. Functional analysis is a unifying theme of these problems. Hilbert spaces in particular are used extensively for the construction of estimators and mathematical analysis.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133205/1/rvdm_1.pd

Generalized Identifiability Bounds for Mixture Models with Grouped Samples

Recent work has shown that finite mixture models with $m$ components are identifiable, while making no assumptions on the mixture components, so long as one has access to groups of samples of size $2m-1$ which are known to come from the same mixture component. In this work we generalize that result and show that, if every subset of $k$ mixture components of a mixture model are linearly independent, then that mixture model is identifiable with only $(2m-1)/(k-1)$ samples per group. We further show that this value cannot be improved. We prove an analogous result for a stronger form of identifiability known as "determinedness" along with a corresponding lower bound. This independence assumption almost surely holds if mixture components are chosen randomly from a $k$-dimensional space. We describe some implications of our results for multinomial mixture models and topic modeling

Balanced task allocation by partitioning the multiple traveling salesperson problem

Task assignment and routing are tightly coupled problems for teams of mobile agents. To fairly balance the workload, each agent should be assigned a set of tasks which take a similar amount of time to complete. The completion time depends on the time needed to travel between tasks which depends on the order of tasks. We formulate the task assignment problem as the minimum Hamiltonian partition problem (MHPP) form agents, which is equivalent to the minmax multiple traveling salesperson problem (m-TSP). While the MHPP’s cost function depends on the order of tasks, its solutions are similar to solutions of the average Hamiltonian partition problem (AHPP) whose cost function is order-invariant. We prove that the AHPP is NP-hard and present an effective heuristic, AHP, for solving it. AHP improves a partitions of a graph using a series of transfer and swap operations which are guaranteed to improve the solution’s quality. The solution generated by AHP is used as an initial partition for an algorithm, AHP-mTSP, which solves the combined task assignment and routing problems to near optimality. For n tasks and m agents, each iteration of AHP is O(n2) and AHP-mTSP has an average run-time that scales with n2.11m0.33. Compared to state-of-the-art approaches, our approach found approximately 10% better solutions for large problems in a similar run-time

Thinking Outside of the Blue Marble: Novel Ocean Applications Using the VIIRS Sensor

While planning for future space-borne sensors will increase the quality, quantity, and duration of ocean observations in the years to come, efforts to extend the limits of sensors currently in orbit can help shed light on future scientific gains as well as associated uncertainties. Here, we present several applications that are unique to the polar orbiting Visual Infrared Imaging Radiometer Suite (VIIRS), each of which challenge the threshold capabilities of the sensor and provide lessons for future missions. For instance, while moderate resolution polar orbiters typically have a one day revisit time, we are able to obtain multiple looks of the same area by focusing on the extreme zenith angles where orbital views overlap, and pair these observations with those from other sensors to create pseudo-geostationary data sets. Or, by exploiting high spatial resolution (imaging) channels and analyzing patterns of synoptic covariance across the visible spectrum, we can obtain higher spatial resolution bio-optical products. Alternatively, non-traditional products can illuminate important biological interactions in the ocean, such as the use of the Day-Night-Band to provide some quantification of phototactic behavior of marine life along light polluted beaches, as well as track the location of marine fishing vessel fleets along ocean fronts. In this talk, we explore ways to take full advantage of the capabilities of existing sensors in order to maximize insights for future missions