Computing optimal k-regret minimizing sets with top-k depth contours

TL;DR

This paper introduces a novel approach to computing optimal k-regret minimizing sets using geometric top-k depth contours, providing algorithms for both 2D and higher-dimensional cases to improve data summarization.

Contribution

It generalizes regret minimizing sets to k-regret, adapts geometric methods for optimal set computation, and proposes algorithms for different dimensions.

Findings

O(cn^2) plane sweep algorithm for 2D cases

Greedy algorithm for higher dimensions

Effective comparison of sets using L2 distance and depth contours

Abstract

Regret minimizing sets are a very recent approach to representing a dataset D with a small subset S of representative tuples. The set S is chosen such that executing any top-1 query on S rather than D is minimally perceptible to any user. To discover an optimal regret minimizing set of a predetermined cardinality is conjectured to be a hard problem. In this paper, we generalize the problem to that of finding an optimal k$regret minimizing set, wherein the difference is computed over top-k queries, rather than top-1 queries. We adapt known geometric ideas of top-k depth contours and the reverse top-k problem. We show that the depth contours themselves offer a means of comparing the optimality of regret minimizing sets using L2 distance. We design an O(cn^2) plane sweep algorithm for two dimensions to compute an optimal regret minimizing set of cardinality c. For higher dimensions, we introduce a greedy algorithm that progresses towards increasingly optimal solutions by exploiting the transitivity of L2 distance.