We consider sketch vectors of geometric objects $J$ through the \mindist function \[ v_i(J) = \inf_{p \in J} \|p-q_i\| \] for $q_i \in Q$ from a point set $Q$. Collecting the vector of these sketch values induces a simple, effective, and powerful distance: the Euclidean distance between these sketched vectors. This paper shows how large this set $Q$ needs to be under a variety of shapes and scenarios. For hyperplanes we provide direct connection to the sensitivity sample framework, so relative error can be preserved in $d$ dimensions using $Q = O(d/\varepsilon^2)$. However, for other shapes, we show we need to enforce a minimum distance parameter $ρ$, and a domain size $L$. For $d=2$ the sample size $Q$ then can be $\tilde{O}((L/ρ) \cdot 1/\varepsilon^2)$. For objects (e.g., trajectories) with at most $k$ pieces this can provide stronger \emph{for all} approximations with $\tilde{O}((L/ρ)\cdot k^3 / \varepsilon^2)$ points. Moreover, with similar size bounds and restrictions, such trajectories can be reconstructed exactly using only these sketch vectors.

We develop a new class of distances for trajectories, based on the distance to a set of landmarks. These distances easily and interpretably map objects to a Euclidean space, are simple to compute, and perform well in data analysis tasks. For trajectories, they match and in some cases significantly out-perform all state-of-the-art other metrics, can effortlessly be used in k-means clustering, and directly plugged into approximate nearest neighbor approaches which immediately out-perform the best recent advances in trajectory similarity search by several orders of magnitude. These distances do not require complicated alignment (common in trajectory case). We show reasonable and often simple conditions under which these distances are metrics.

Robust estimators, like the median of a point set, are important for data analysis in the presence of outliers. We study robust estimators for locationally uncertain points with discrete distributions. That is, each point in a data set has a discrete probability distribution describing its location. The probabilistic nature of uncertain data makes it challenging to compute such estimators, since the true value of the estimator is now described by a distribution rather than a single point. We show how to construct and estimate the distribution of the median of a point set. Building the approximate support of the distribution takes near-linear time, and assigning probability to that support takes quadratic time. We also develop a general approximation technique for distributions of robust estimators with respect to ranges with bounded VC dimension. This includes the geometric median for high dimensions and the Siegel estimator for linear regression.

We formalize notions of robustness for composite estimators via the notion of a breakdown point. A composite estimator successively applies two (or more) estimators: on data decomposed into disjoint parts, it applies the first estimator on each part, then the second estimator on the outputs of the first estimator. And so on, if the composition is of more than two estimators. Informally, the breakdown point is the minimum fraction of data points which if significantly modified will also significantly modify the output of the estimator, so it is typically desirable to have a large breakdown point. Our main result shows that, under mild conditions on the individual estimators, the breakdown point of the composite estimator is the product of the breakdown points of the individual estimators. We also demonstrate several scenarios, ranging from regression to statistical testing, where this analysis is easy to apply, useful in understanding worst case robustness, and sheds powerful insights onto the associated data analysis.