Algorithms for Geometric Matching, Clustering, and Covering

With the popularization of GPS-enabled devices such as mobile phones, location data are becoming available at an unprecedented scale. The locations may be collected from many different sources such as vehicles moving around a city, user check-ins in social networks, and geo-tagged micro-blogging photos or messages. Besides the longitude and latitude, each location record may also have a timestamp and additional information such as the name of the location. Time-ordered sequences of these locations form trajectories, which together contain useful high-level information about people's movement patterns.

The first part of this thesis focuses on a few geometric problems motivated by the matching and clustering of trajectories. We first give a new algorithm for computing a matching between a pair of curves under existing models such as dynamic time warping (DTW). The algorithm is more efficient than standard dynamic programming algorithms both theoretically and practically. We then propose a new matching model for trajectories that avoids the drawbacks of existing models. For trajectory clustering, we present an algorithm that computes clusters of subtrajectories, which correspond to common movement patterns. We also consider trajectories of check-ins, and propose a statistical generative model, which identifies check-in clusters as well as the transition patterns between the clusters.

The second part of the thesis considers the problem of covering shortest paths in a road network, motivated by an EV charging station placement problem. More specifically, a subset of vertices in the road network are selected to place charging stations so that every shortest path contains enough charging stations and can be traveled by an EV without draining the battery. We first introduce a general technique for the geometric set cover problem. This technique leads to near-linear-time approximation algorithms, which are the state-of-the-art algorithms for this problem in either running time or approximation ratio. We then use this technique to develop a near-linear-time algorithm for this

shortest-path cover problem.

Stimulus Integration and Parsing in the Primate Auditory Midbrain

Integrating information from multiple sources is a crucial function of the brain. Examples of such integration include multiple stimuli of different modalties, such as visual and auditory, multiple stimuli of the same modality, such as auditory and auditory, and integrating stimuli from the sensory organs (i.e. ears) with stimuli delivered from brain-machine interfaces.

The overall aim of this body of work is to empirically examine stimulus integration in these three domains to inform our broader understanding of how and when the brain combines information from multiple sources.

First, I examine visually-guided auditory, a problem with implications for the general problem in learning of how the brain determines what lesson to learn (and what lessons not to learn). For example, sound localization is a behavior that is partially learned with the aid of vision. This process requires correctly matching a visual location to that of a sound. This is an intrinsically circular problem when sound location is itself uncertain and the visual scene is rife with possible visual matches. Here, we develop a simple paradigm using visual guidance of sound localization to gain insight into how the brain confronts this type of circularity. We tested two competing hypotheses. 1: The brain guides sound location learning based on the synchrony or simultaneity of auditory-visual stimuli, potentially involving a Hebbian associative mechanism. 2: The brain uses a ‘guess and check’ heuristic in which visual feedback that is obtained after an eye movement to a sound alters future performance, perhaps by recruiting the brain’s reward-related circuitry. We assessed the effects of exposure to visual stimuli spatially mismatched from sounds on performance of an interleaved auditory-only saccade task. We found that when humans and monkeys were provided the visual stimulus asynchronously with the sound but as feedback to an auditory-guided saccade, they shifted their subsequent auditory-only performance toward the direction of the visual cue by 1.3-1.7 degrees, or 22-28% of the original 6 degree visual-auditory mismatch. In contrast when the visual stimulus was presented synchronously with the sound but extinguished too quickly to provide this feedback, there was little change in subsequent auditory-only performance. Our results suggest that the outcome of our own actions is vital to localizing sounds correctly. Contrary to previous expectations, visual calibration of auditory space does not appear to require visual-auditory associations based on synchrony/simultaneity.

My next line of research examines how electrical stimulation of the inferior colliculus influences perception of sounds in a nonhuman primate. The central nucleus of the inferior colliculus is the major ascending relay of auditory information before it reaches the forebrain, and thus an ideal target for understanding low-level information processing prior to the forebrain, as almost all auditory signals pass through the central nucleus of the inferior colliculus before reaching the forebrain. Thus, the inferior colliculus is the ideal structure to examine to understand the format of the inputs into the forebrain and, by extension, the processing of auditory scenes that occurs in the brainstem. Therefore, the inferior colliculus was an attractive target for understanding stimulus integration in the ascending auditory pathway.

Moreover, understanding the relationship between the auditory selectivity of neurons and their contribution to perception is critical to the design of effective auditory brain prosthetics. These prosthetics seek to mimic natural activity patterns to achieve desired perceptual outcomes. We measured the contribution of inferior colliculus (IC) sites to perception using combined recording and electrical stimulation. Monkeys performed a frequency-based discrimination task, reporting whether a probe sound was higher or lower in frequency than a reference sound. Stimulation pulses were paired with the probe sound on 50% of trials (0.5-80 µA, 100-300 Hz, n=172 IC locations in 3 rhesus monkeys). Electrical stimulation tended to bias the animals’ judgments in a fashion that was coarsely but significantly correlated with the best frequency of the stimulation site in comparison to the reference frequency employed in the task. Although there was considerable variability in the effects of stimulation (including impairments in performance and shifts in performance away from the direction predicted based on the site’s response properties), the results indicate that stimulation of the IC can evoke percepts correlated with the frequency tuning properties of the IC. Consistent with the implications of recent human studies, the main avenue for improvement for the auditory midbrain implant suggested by our findings is to increase the number and spatial extent of electrodes, to increase the size of the region that can be electrically activated and provide a greater range of evoked percepts.

My next line of research employs a frequency-tagging approach to examine the extent to which multiple sound sources are combined (or segregated) in the nonhuman primate inferior colliculus. In the single-sound case, most inferior colliculus neurons respond and entrain to sounds in a very broad region of space, and many are entirely spatially insensitive, so it is unknown how the neurons will respond to a situation with more than one sound. I use multiple AM stimuli of different frequencies, which the inferior colliculus represents using a spike timing code. This allows me to measure spike timing in the inferior colliculus to determine which sound source is responsible for neural activity in an auditory scene containing multiple sounds. Using this approach, I find that the same neurons that are tuned to broad regions of space in the single sound condition become dramatically more selective in the dual sound condition, preferentially entraining spikes to stimuli from a smaller region of space. I will examine the possibility that there may be a conceptual linkage between this finding and the finding of receptive field shifts in the visual system.

In chapter 5, I will comment on these findings more generally, compare them to existing theoretical models, and discuss what these results tell us about processing in the central nervous system in a multi-stimulus situation. My results suggest that the brain is flexible in its processing and can adapt its integration schema to fit the available cues and the demands of the task.