Convex Hulls under Uncertainty

We study the convex-hull problem in a probabilistic setting, motivated by the need to handle data uncertainty inherent in many applications, including sensor databases, location-based services and computer vision. In our framework, the uncertainty of each input site is described by a probability distribution over a finite number of possible locations including a \emph{null} location to account for non-existence of the point. Our results include both exact and approximation algorithms for computing the probability of a query point lying inside the convex hull of the input, time-space tradeoffs for the membership queries, a connection between Tukey depth and membership queries, as well as a new notion of \some-hull that may be a useful representation of uncertain hulls

Probabilistic Bisimulations for PCTL Model Checking of Interval MDPs

Verification of PCTL properties of MDPs with convex uncertainties has been investigated recently by Puggelli et al. However, model checking algorithms typically suffer from state space explosion. In this paper, we address probabilistic bisimulation to reduce the size of such an MDPs while preserving PCTL properties it satisfies. We discuss different interpretations of uncertainty in the models which are studied in the literature and that result in two different definitions of bisimulations. We give algorithms to compute the quotients of these bisimulations in time polynomial in the size of the model and exponential in the uncertain branching. Finally, we show by a case study that large models in practice can have small branching and that a substantial state space reduction can be achieved by our approach.Comment: In Proceedings SynCoP 2014, arXiv:1403.784

On the expected diameter, width, and complexity of a stochastic convex-hull

We investigate several computational problems related to the stochastic convex hull (SCH). Given a stochastic dataset consisting of $n$ points in $\mathbb{R}^d$ each of which has an existence probability, a SCH refers to the convex hull of a realization of the dataset, i.e., a random sample including each point with its existence probability. We are interested in computing certain expected statistics of a SCH, including diameter, width, and combinatorial complexity. For diameter, we establish the first deterministic 1.633-approximation algorithm with a time complexity polynomial in both $n$ and $d$. For width, two approximation algorithms are provided: a deterministic $O(1)$-approximation running in $O(n^{d+1} \log n)$ time, and a fully polynomial-time randomized approximation scheme (FPRAS). For combinatorial complexity, we propose an exact $O(n^d)$-time algorithm. Our solutions exploit many geometric insights in Euclidean space, some of which might be of independent interest

HARPS: An Online POMDP Framework for Human-Assisted Robotic Planning and Sensing

Autonomous robots can benefit greatly from human-provided semantic characterizations of uncertain task environments and states. However, the development of integrated strategies which let robots model, communicate, and act on such 'soft data' remains challenging. Here, the Human Assisted Robotic Planning and Sensing (HARPS) framework is presented for active semantic sensing and planning in human-robot teams to address these gaps by formally combining the benefits of online sampling-based POMDP policies, multimodal semantic interaction, and Bayesian data fusion. This approach lets humans opportunistically impose model structure and extend the range of semantic soft data in uncertain environments by sketching and labeling arbitrary landmarks across the environment. Dynamic updating of the environment model while during search allows robotic agents to actively query humans for novel and relevant semantic data, thereby improving beliefs of unknown environments and states for improved online planning. Simulations of a UAV-enabled target search application in a large-scale partially structured environment show significant improvements in time and belief state estimates required for interception versus conventional planning based solely on robotic sensing. Human subject studies in the same environment (n = 36) demonstrate an average doubling in dynamic target capture rate compared to the lone robot case, and highlight the robustness of active probabilistic reasoning and semantic sensing over a range of user characteristics and interaction modalities