We develop a new data-dependent distance for regression problems to compare two regressors (hyperplanes that fits or divides a data set). Most distances between objects attempt to capture the intrinsic geometry of these objects and measure how similar that geometry is. However, we argue that existing measures are inappropriate for regressors to a data set . We introduce a family of new distances that measure how similarly two regressors interact with the data they attempt to fit. For the variant we advocate we show it is a metric (under mild assumptions), induces metric balls with bounded VC-dimension, it is robust to changes in the corresponding data , and can be approximated quickly. We show a simple extension to trajectories that inherits these properties, as well as several other algorithmic applications. Moreover, in order to develop efficient approximation algorithms for this distance we formalize the relationship between sensitivity and leverage scores. This may be of independent interest .
We develop a new class of distances for objects including lines, hyperplanes, and 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 a geometry distorting dual (common in the line or halfspace case) or complicated alignment (common in trajectory case). We show reasonable and often simple conditions under which these distances are metrics .
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data-dependent distance for regression problems to compare two regressors (hyperplanes that fits or divides a data set). Most distances between objects attempt to capture the intrinsic geometry of these objects and measure how similar that geometry is. However, we argue that existing measures are inappropriate for regressors to a data set . We introduce a family of new distances that measure how similarly two regressors interact with the data they attempt to fit. For the variant we advocate we show it is a metric (under mild assumptions), induces metric balls with bounded VC-dimension, it is robust to changes in the corresponding data , and can be approximated quickly. We show a simple extension to trajectories that inherits these properties, as well as several other algorithmic applications. Moreover, in order to develop efficient approximation algorithms for this distance we formalize the relationship between sensitivity and leverage scores. This may be of independent interest
class of distances for objects including lines, hyperplanes, and 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 a geometry distorting dual (common in the line or halfspace case) or complicated alignment (common in trajectory case). We show reasonable and often simple conditions under which these distances are metrics