text: In this paper, we propose a simple and effective geometric model fitting method to fit and segment multi-structure data even in the presence of severe outliers. We cast the task of geometric model fitting as a representative mode-seeking problem on hypergraphs. Specifically, a hypergraph is first constructed, where the vertices represent model hypotheses and the hyperedges denote data points. The hypergraph involves higher-order similarities (instead of pairwise similarities used on a simple graph), and it can characterize complex relationships between model hypotheses and data points. In addition, we develop a hypergraph reduction technique to remove “insignificant” vertices while retaining as many “significant” vertices as possible in the hypergraph. Based on the simplified hypergraph, we then propose a novel mode-seeking algorithm to search for representative modes within reasonable time. Finally, the proposed mode-seeking algorithm detects modes according to two key elements, i.e., the weighting scores of vertices and the similarity analysis between vertices. Overall, the proposed fitting method is able to efficiently and effectively estimate the number and the parameters of model instances in the data simultaneously. Experimental results demonstrate that the proposed method achieves significant superiority over several state-of-the-art model fitting methods on both synthetic data and real images.
  @cite_29: The extension of conventional clustering to hypergraph clustering, which involves higher order similarities instead of pairwise similarities, is increasingly gaining attention in computer vision. This is due to the fact that many grouping problems require an affinity measure that must involve a subset of data of size more than two, i.e., a hyperedge. Almost all previous works, however, have considered the smallest possible hyperedge size, due to a lack of study into the potential benefits of large hyperedges and effective algorithms to generate them. In this paper, we show that large hyperedges are better from both theoretical and empirical standpoints. We then propose a novel guided sampling strategy for large hyperedges, based on the concept of random cluster models. Our method can generate pure large hyperedges that significantly improve grouping accuracy without exponential increases in sampling costs. In the important applications of face clustering and motion segmentation, our method demonstrates substantially better accuracy and efficiency.
  @cite_32: Motion segmentation based on point trajectories can integrate information of a whole video shot to detect and separate moving objects. Commonly, similarities are defined between pairs of trajectories. However, pairwise similarities restrict the motion model to translations. Non-translational motion, such as rotation or scaling, is penalized in such an approach. We propose to define similarities on higher order tuples rather than pairs, which leads to hypergraphs. To apply spectral clustering, the hypergraph is transferred to an ordinary graph, an operation that can be interpreted as a projection. We propose a specific nonlinear projection via a regularized maximum operator, and show that it yields significant improvements both compared to pairwise similarities and alternative hypergraph projections.
  @cite_6: In this paper, we propose an efficient method to detect the underlying structures in data. The same as RANSAC, we randomly sample MSSs (minimal size samples) and generate hypotheses. Instead of analyzing each hypothesis separately, the consensus information in all hypotheses is naturally fused into a hypergraph, called random consensus graph, with real structures corresponding to its dense subgraphs. The sampling process is essentially a progressive refinement procedure of the random consensus graph. Due to the huge number of hyperedges, it is generally inefficient to detect dense subgraphs on random consensus graphs. To overcome this issue, we construct a pairwise graph which approximately retains the dense subgraphs of the random consensus graph. The underlying structures are then revealed by detecting the dense subgraphs of the pair-wise graph. Since our method fuses information from all hypotheses, it can robustly detect structures even under a small number of MSSs. The graph framework enables our method to simultaneously discover multiple structures. Besides, our method is very efficient, and scales well for large scale problems. Extensive experiments illustrate the superiority of our proposed method over previous approaches, achieving several orders of magnitude speedup along with satisfactory accuracy and robustness.
  @cite_0: Hypergraph partitioning lies at the heart of a number of problems in machine learning and network sciences. Many algorithms for hypergraph partitioning have been proposed that extend standard approaches for graph partitioning to the case of hypergraphs. However, theoretical aspects of such methods have seldom received attention in the literature as compared to the extensive studies on the guarantees of graph partitioning. For instance, consistency results of spectral graph partitioning under the stochastic block model are well known. In this paper, we present a planted partition model for sparse random non-uniform hypergraphs that generalizes the stochastic block model. We derive an error bound for a spectral hypergraph partitioning algorithm under this model using matrix concentration inequalities. To the best of our knowledge, this is the first consistency result related to partitioning non-uniform hypergraphs.
  @cite_15: The higher-order clustering problem arises when data is drawn from multiple subspaces or when observations fit a higher-order parametric model. Most solutions to this problem either decompose higher-order similarity measures for use in spectral clustering or explicitly use low-rank matrix representations. In this paper we present our approach of Sparse Grassmann Clustering (SGC) that combines attributes of both categories. While we decompose the higher order similarity tensor, we cluster data by directly finding a low dimensional representation without explicitly building a similarity matrix. By exploiting recent advances in online estimation on the Grassmann manifold (GROUSE) we develop an efficient and accurate algorithm that works with individual columns of similarities or partial observations thereof. Since it avoids the storage and decomposition of large similarity matrices, our method is efficient, scalable and has low memory requirements even for large-scale data. We demonstrate the performance of our SGC method on a variety of segmentation problems including planar segmentation of Kinect depth maps and motion segmentation of the Hopkins 155 dataset for which we achieve performance comparable to the state-of-the-art.
summary: Compared with the hypergraph constructed in the previous methods (e.g., @cite_15 @cite_6 @cite_32 @cite_29 ), where a hyperedge is constrained to connect with a fixed number of vertices, the hyperedge of hypergraphs constructed in this paper can connect with a varying number of vertices (that is we construct non-uniform hypergraphs as those in @cite_0 ) . In addition, the vertices of the hypergraph constructed in the previous methods (e.g., @cite_15 @cite_6 @cite_32 @cite_29 ) represent data points, while the vertices of the hypergraph constructed in this paper denote model hypotheses. Therefore, we can directly deal with the model fitting problem in the parameter space.