Shape Interaction Matrix Revisited and Robustified: Efficient Subspace Clustering with Corrupted and Incomplete Data

The Shape Interaction Matrix (SIM) is one of the earliest approaches to performing subspace clustering (i.e., separating points drawn from a union of subspaces). In this paper, we revisit the SIM and reveal its connections to several recent subspace clustering methods. Our analysis lets us derive a simple, yet effective algorithm to robustify the SIM and make it applicable to realistic scenarios where the data is corrupted by noise. We justify our method by intuitive examples and the matrix perturbation theory. We then show how this approach can be extended to handle missing data, thus yielding an efficient and general subspace clustering algorithm. We demonstrate the benefits of our approach over state-of-the-art subspace clustering methods on several challenging motion segmentation and face clustering problems, where the data includes corrupted and missing measurements.Comment: This is an extended version of our iccv15 pape

A Compact Fourth-order Gas-kinetic Scheme for the Euler and Navier-Stokes Solutions

In this paper, a fourth-order compact gas-kinetic scheme (GKS) is developed for the compressible Euler and Navier-Stokes equations under the framework of two-stage fourth-order temporal discretization and Hermite WENO (HWENO) reconstruction. Due to the high-order gas evolution model, the GKS provides a time dependent gas distribution function at a cell interface. This time evolution solution can be used not only for the flux evaluation across a cell interface and its time derivative, but also time accurate evolution solution at a cell interface. As a result, besides updating the conservative flow variables inside each control volume, the GKS can get the cell averaged slopes inside each control volume as well through the differences of flow variables at the cell interfaces. So, with the updated flow variables and their slopes inside each cell, the HWENO reconstruction can be naturally implemented for the compact high-order reconstruction at the beginning of next step. Therefore, a compact higher-order GKS, such as the two-stages fourth-order compact scheme can be constructed. This scheme is as robust as second-order one, but more accurate solution can be obtained. In comparison with compact fourth-order DG method, the current scheme has only two stages instead of four within each time step for the fourth-order temporal accuracy, and the CFL number used here can be on the order of $0.5$ instead of $0.11$ for the DG method. Through this research, it concludes that the use of high-order time evolution model rather than the first order Riemann solution is extremely important for the design of robust, accurate, and efficient higher-order schemes for the compressible flows

Neural Collaborative Subspace Clustering

We introduce the Neural Collaborative Subspace Clustering, a neural model that discovers clusters of data points drawn from a union of low-dimensional subspaces. In contrast to previous attempts, our model runs without the aid of spectral clustering. This makes our algorithm one of the kinds that can gracefully scale to large datasets. At its heart, our neural model benefits from a classifier which determines whether a pair of points lies on the same subspace or not. Essential to our model is the construction of two affinity matrices, one from the classifier and the other from a notion of subspace self-expressiveness, to supervise training in a collaborative scheme. We thoroughly assess and contrast the performance of our model against various state-of-the-art clustering algorithms including deep subspace-based ones.Comment: Accepted to ICML 201

Unsupervised Deep Epipolar Flow for Stationary or Dynamic Scenes

Unsupervised deep learning for optical flow computation has achieved promising results. Most existing deep-net based methods rely on image brightness consistency and local smoothness constraint to train the networks. Their performance degrades at regions where repetitive textures or occlusions occur. In this paper, we propose Deep Epipolar Flow, an unsupervised optical flow method which incorporates global geometric constraints into network learning. In particular, we investigate multiple ways of enforcing the epipolar constraint in flow estimation. To alleviate a "chicken-and-egg" type of problem encountered in dynamic scenes where multiple motions may be present, we propose a low-rank constraint as well as a union-of-subspaces constraint for training. Experimental results on various benchmarking datasets show that our method achieves competitive performance compared with supervised methods and outperforms state-of-the-art unsupervised deep-learning methods.Comment: CVPR 201