Randomized hybrid linear modeling by local best-fit flats

TL;DR

This paper introduces a simple geometric algorithm for hybrid linear modeling that automatically determines local neighborhood sizes and accurately identifies affine subspaces in high-dimensional data.

Contribution

It presents a novel local best-fit flats method using l1 error minimization and Jones' beta numbers for neighborhood selection, improving accuracy and speed.

Findings

Achieves state-of-the-art accuracy on synthetic and real data.

Demonstrates fast determination of the number of affine subspaces.

Proves geometric conditions for the existence of good local neighborhoods.

Abstract

The hybrid linear modeling problem is to identify a set of d-dimensional affine sets in a D-dimensional Euclidean space. It arises, for example, in object tracking and structure from motion. The hybrid linear model can be considered as the second simplest (behind linear) manifold model of data. In this paper we will present a very simple geometric method for hybrid linear modeling based on selecting a set of local best fit flats that minimize a global l1 error measure. The size of the local neighborhoods is determined automatically by the Jones' l2 beta numbers; it is proven under certain geometric conditions that good local neighborhoods exist and are found by our method. We also demonstrate how to use this algorithm for fast determination of the number of affine subspaces. We give extensive experimental evidence demonstrating the state of the art accuracy and speed of the algorithm on synthetic and real hybrid linear data.