## Abstract

We suggest a new optimization technique for minimizing the sum ∑ i=1n gi(x)∑i=1ngi(x) of n non-convex real functions that satisfy a property that we call piecewise log-Lipschitz. This is by forging links between techniques in computational geometry, combinatorics and convex optimization. As an example application, we provide the first constant-factor approximation algorithms whose running-times are polynomial in n for the fundamental problem of Points-to-Lines alignment: Given n points p 1,\p n p1,..,pn and n lines ℓ_1,\ℓ n ℓ1,..,ℓn on the plane and z>0 z>0, compute the matching π:[n]\to [n] π:[n]→[n] and alignment (rotation matrix R and translation vector t t) that minimize the sum of euclidean distances ∑ i=1n dist (Rp_i-t,ℓ π (i))z∑i=1n dist (Rpi-t,ℓπ(i))z between each point to its corresponding line. This problem is non-trivial even if z=1 and the matching ππ is given. If π is given, our algorithms run in O(n3)O(n3) time, and even near-linear in n using core-sets that support: streaming, dynamic, and distributed parallel computations in poly-logarithmic update time. Generalizations for handling e.g., outliers or pseudo-distances such as M-estimators for the problem are also provided. Experimental results and open source code show that our algorithms improve existing heuristics also in practice. A companion demonstration video in the context of Augmented Reality shows how such algorithms may be used in real-time systems.

Original language | English |
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Pages (from-to) | 138-149 |

Number of pages | 12 |

Journal | IEEE Transactions on Knowledge and Data Engineering |

Volume | 34 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2022 |

### Bibliographical note

Publisher Copyright:© 1989-2012 IEEE.

## Keywords

- Approximation algorithms
- Coresets
- Non-convex optimization
- Points-to-lines alignment
- Visual tracking

## ASJC Scopus subject areas

- Information Systems
- Computer Science Applications
- Computational Theory and Mathematics