## Abstract

A coreset of a dataset is a small weighted set, such that querying the coreset provably yields a (1 + ε)-factor approximation to the original (full) dataset, for a given family of queries. This paper suggests accurate coresets (ε = 0) 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 > 20 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 R^{dxd} 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 O(dr) = O(d^{2}) matrices, independent of n. This implies the first (exact, weighted subset) coreset of O(d^{2}) 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.

Original language | English |
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Article number | 3042 |

Number of pages | 20 |

Journal | Sensors |

Volume | 20 |

Issue number | 11 |

DOIs | |

State | Published - 1 Jun 2020 |

### Bibliographical note

Publisher Copyright:© 2020 by the authors. Licensee MDPI, Basel, Switzerland.

## Keywords

- Autonomous sensors for micro drones
- Caratheodory
- Coresets
- Indoor navigation and mapping
- Localization
- Pose estimation

## ASJC Scopus subject areas

- Analytical Chemistry
- Information Systems
- Atomic and Molecular Physics, and Optics
- Biochemistry
- Instrumentation
- Electrical and Electronic Engineering