Abstract
A coreset of a dataset is a small weighted set, such that querying the coreset provably yields a ()-factor approximation to the original (full) dataset, for a given family of queries. This paper suggests accurate coresets () that are subsets of the input for fundamental optimization problems. These coresets enabled us to implement a “Guardian Angel” system that computes pose-estimation in a rate frames per second. It tracks a toy quadcopter which guides guests in a supermarket, hospital, mall, airport, and so on. We prove that any set of n matrices in whose sum is a matrix S of rank r, has a coreset whose sum has the same left and right singular vectors as S, and consists of matrices, independent of n. This implies the first (exact, weighted subset) coreset of points to problems such as linear regression, PCA/SVD, and Wahba’s problem, with corresponding streaming, dynamic, and distributed versions. Our main tool is a novel usage of the Caratheodory Theorem for coresets, an algorithm that computes its set in time that is linear in its cardinality. Extensive experimental results on both synthetic and real data, companion video of our system, and open code are provided.
Highlights
Introduction and MotivationCoresets is a powerful technique for data reduction that was originally used to improve the running time of algorithms in computational geometry (e.g., [1,2,3,4,5,6,7])
Sensors 2020, 20, 3042 coreset for pose-estimation based on the alignment between two paired sets. We show how this coreset enables us to compute the orientation of a rigid body, in particular a moving robot, which is a fundamental question in Simultaneous Localization And Mapping (SLAM) and computer vision; see references in [28]
We prove that the result of running the classic Kabsch algorithm on the entire input, would yield the same result when applied on the coreset only
Summary
Introduction and MotivationCoresets is a powerful technique for data reduction that was originally used to improve the running time of algorithms in computational geometry (e.g., [1,2,3,4,5,6,7]). (i) Introduce coresets to the robotics community and show how their theory can be applied in real-time systems and in the context of machine learning or theoretical computer science. To obtain goal (i), we suggest a simple but powerful and generic coreset that approximates the center of mass of a set of points, using a sparse distribution on a small subset of the input points. While this “mean coreset” has many other applications, to obtain goal (ii) we use it to design a novel Coresets appear in machine learning conferences [12,13,14,15,16,17,18,19,20,21] with robotics [12,13,15,16,18,20,21,22,23,24] and image [25,26,27] applications.
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