Hybrid Linear Modeling via Local Best-fit Flats (1010.3460v2)

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.

• The paper presents a novel LBF algorithm that uses Jones' β2 numbers to select optimal local neighborhoods for affine subspace approximation.
• The paper introduces SLBF, a spectral clustering method that infers global subspace structures from local best-fit flats.
• The paper validates its approach on datasets like Hopkins 155 and extended Yale, demonstrating superior runtime efficiency and clustering accuracy.

Hybrid Linear Modeling via Local Best-fit Flats

The paper presents a sophisticated geometric approach to Hybrid Linear Modeling (HLM), aimed at modeling data by a union of affine subspaces. This technique is relevant in applications such as motion segmentation and face clustering. The proposed method identifies local affine subspaces that best fit data within local neighborhoods, determined using Jones' $\beta_2$ numbers, ensuring the correct sizes of these neighborhoods.

The proposed method involves two key algorithms: Local Best-fit Flats (LBF) and Spectral LBF (SLBF). The LBF algorithm uses a greedy energy minimization approach to process local subspaces, while SLBF opts for a spectral clustering method. In addition, the authors implement a process for estimating the number of affine subspaces, promising efficient determination of the model parameters.

Methodology and Algorithms:

• Local Best-fit Flats (LBF): The LBF algorithm identifies candidate flats by selecting neighborhoods based on the $\beta_2$ numbers, ensuring optimal representation of local structures. The selected flats are further refined through a greedy approach to minimize energy, i.e., the sum of distances from points to their closest flats.
• Spectral LBF (SLBF): This algorithm incorporates spectral clustering, constructing an affinity matrix from distances between points and their local affine approximations. This matrix is then subjected to spectral decomposition to infer global subspace structures.

The authors offer a theoretical framework guaranteeing the neighborhood selection's accuracy under specific geometric conditions, alongside comprehensive experimental validation. The experiments show the algorithm's robustness and speed across different datasets like motion segmentation in video sequences, face clustering, and synthetic data. Notably, LBF shows impressive speed, often outperforming other methods in terms of runtime, while SLBF excels in accuracy.

Results and Implications:

Through extensive testing on motion segmentation datasets like Hopkins 155, and real-world data from the MNIST and extended Yale face datasets, the proposed models demonstrate remarkable accuracy. They efficiently manage scenarios with intersecting subspaces and substantial outliers.

The implications of this research are multifaceted. Primarily, it underscores a methodological advancement in data modeling, where the balance between computational efficiency and clustering accuracy is well-managed. The method's adaptability, making it suitable for large and complex datasets, highlights its practical utility in real-world applications, particularly within the realms of computer vision and machine learning.

Future Prospects:

The paper hints at possible future directions, such as adapting this approach for multi-manifold clustering—a task demanding the handling of non-linear subspaces. The claim to establish a comprehensive theoretical backbone to justify the observed empirical robustness against noise, and potentially augment the algorithm to support manifold clustering, would be pivotal advancements.

In summary, this paper provides a substantial contribution to the field of subspace clustering, with a well-founded theoretical basis and empirical success. The tools and techniques developed are poised to influence further research and applications in geometric data analysis and beyond.