Hybrid Linear Modeling via Local Best-fit Flats

TL;DR

This paper introduces a fast geometric approach for modeling data as a union of affine subspaces using local best-fit flats, with automatic neighborhood size selection and applications to motion segmentation and face clustering.

Contribution

The paper proposes a novel method combining local affine approximations with automatic neighborhood selection, improving accuracy and speed in hybrid linear modeling tasks.

Findings

Achieves state-of-the-art accuracy in motion segmentation and face clustering.

Demonstrates fast performance on synthetic and real datasets like MNIST.

Provides a method for rapid determination of the number of affine subspaces.

Abstract

We present a simple and fast geometric method for modeling data by a union of affine subspaces. The method begins by forming a collection of local best-fit affine subspaces, i.e., subspaces approximating the data in local neighborhoods. The correct sizes of the local neighborhoods are determined automatically by the Jones' $\beta_2$ numbers (we prove under certain geometric conditions that our method finds the optimal local neighborhoods). The collection of subspaces is further processed by a greedy selection procedure or a spectral method to generate the final model. We discuss applications to tracking-based motion segmentation and clustering of faces under different illuminating conditions. We give extensive experimental evidence demonstrating the state of the art accuracy and speed of the suggested algorithms on these problems and also on synthetic hybrid linear data as well as the MNIST handwritten digits data; and we demonstrate how to use our algorithms for fast determination of the number of affine subspaces.